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Fixed Point Theoryand ApplicationsEditor-in-Chief: Ravi P. Agarwal

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VOLUME 2006

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Special IssueNielsen Theory and Related Topics

Guest Editors Philip R. Heath, Robert F. Brown, and Edward C. Keppelmann

Fixed Point Theory and Applications

Editor-in-Chief

Ravi P. AgarwalFlorida Institute of Technology, [email protected]

Associate Editors

Thomas BartschUniversitaet Giessen, [email protected]

Tomas Dominguez BenavidesFacultad de Matematicas, Seville (Spain)[email protected]

J. M. BorweinDalhousie University, [email protected]

R. F. BrownUniversity of California, [email protected]

D. G. de FigueiredoIMECC-UNICAMP, [email protected]

P. M. FitzpatrickUniversity of Maryland, [email protected]

Helene FrankowskaCREA, Ecole Polytechnique, [email protected]

M. FuriUniversita di Firenze, [email protected]

L. GorniewiczNicholas Copernicus University, [email protected]

Evelyn HartColgate University, [email protected]

Nan-jing HuangSichuan University, [email protected]

Jerzy JezierskiUniversity of Agriculture, [email protected]

W. A. KirkUniversity of Iowa, [email protected]

V. LakshmikanthamFlorida Institute of Technology, [email protected]

J. MawhinUniversite Catholique de Louvain, [email protected]

R. D. NussbaumRutgers University, [email protected]

D. O’ReganNational University of Ireland, Galway, [email protected]

S. ReichTechnion–Israel Institute of Technology, [email protected]

B. E. RhoadesIndiana University, [email protected]

Klaus SchmittUniversity of Utah, [email protected]

B. SimsThe University of Newcastle, [email protected]

Andrzej SzulkinStockholm University, [email protected]

W. TakahashiTokyo Institute of Technology, [email protected]

J. R. L. WebbUniversity of Glasgow, [email protected]

Fabio ZanolinUniversity of Udine, [email protected]

Fixed Point Theory and ApplicationsVolume 2006

Special IssueNielsen Theory and Related TopicsGuest Editors: Philip R. Heath, Robert F. Brown, and Edward C. Keppelmann

Contents

Nielsen theory and related topics (Editorial),Philip R. Heath, Robert F. Brown, and Edward C. KeppelmannVolume 2006, Article ID 73530, 2 pages

Duan’s fixed point theorem: proof and generalization, Martin ArkowitzVolume 2006, Article ID 17563, 10 pages

Fixed point indices and manifolds with collars,Chen-Farng Benjamin and Daniel Henry GottliebVolume 2006, Article ID 87657, 8 pages

Epsilon Nielsen fixed point theory, Robert F. BrownVolume 2006, Article ID 29470, 10 pages

A base-point-free definition of the Lefschetz invariant, Vesta CoufalVolume 2006, Article ID 34143, 20 pages

The Anosov theorem for infranilmanifolds with an odd-orderAbelian holonomy group,K. Dekimpe, B. De Rock, and H. PouseeleVolume 2006, Article ID 63939, 12 pages

Wecken type problems for self-maps of the Klein bottle,D. L. Goncalves and M. R. KellyVolume 2006, Article ID 75848, 15 pages

Algebraic periods of self-maps of a rational exteriorspace of rank 2, Grzegorz Graff

Volume 2006, Article ID 80521, 9 pages

Nielsen number of a covering map, Jerzy JezierskiVolume 2006, Article ID 37807, 11 pages

Geometric and homotopy theoretic methods in Nielsencoincidence theory, Ulrich KoschorkeVolume 2006, Article ID 84093, 15 pages

Fixed point sets of maps homotopic to a given map,Christina L. SoderlundVolume 2006, Article ID 46052, 20 pages

Coincidence classes in nonorientable manifolds,Daniel VendruscoloVolume 2006, Article ID 68513, 9 pages

Reducing the number of fixed points of some homeomorphismson nonprime 3-manifolds, Xuezhi ZhaoVolume 2006, Article ID 25897, 19 pages

EditorialNIELSEN THEORY AND RELATED TOPICS

PHILIP R. HEATH, ROBERT F. BROWN, AND EDWARD C. KEPPELMANN

Received 14 July 2005; Accepted 14 July 2005

Copyright © 2006 Philip R. Heath et al. This is an open access article distributed underthe Creative Commons Attribution License, which permits unrestricted use, distribution,and reproduction in any medium, provided the original work is properly cited.

The International Conference on Nielsen Theory and Related Topics took place fromJune 28 through July 2, 2004 at Memorial University, St. John’s, Newfoundland, Canada.This was the 13th such conference in a series that began in 1977 with a conference inOberwolfach, Germany.

Nielsen theory is named after Jakob Nielsen who, in the 1920s, turned the focus offixed point theory from the existence of fixed points (as, e.g., in the famous Lefschetz fixedpoint theorem) to the problem of estimating the actual number of such points within thehomotopy class of a given map. He did this by introducing what is now called the Nielsennumber of a self map, a homotopy invariant lower bound for the number of fixed pointsof the map. After important initial contributions by Reidemeister and Wecken, there waslittle activity in Nielsen theory until the 1960s when a breakthrough by Boju Jiang allowedfor easy calculations of the Nielsen number for maps on Lie groups and some other in-teresting kinds of spaces. It was these and other important examples that would guidethe direction of research. As the present collection of papers illustrates, the frontiers ofthe subject now involve an impressive variety and interplay of algebraic and geometrictechniques, on a wide class of spaces. Consequently, there continue to be an increasingnumber of interesting areas for future investigations, both in the areas of computationand the development of new invariants.

As research in Nielsen theory progressed, its concern with fixed points expanded, onthe one hand, into related issues such as coincidences, periodic points, and roots and, onthe other, into various refinements for restricted classes of maps and homotopies such asthose that occur in the fiber space, relative and equivariant contexts. Just about all of theseaspects of Nielsen theory were represented by the talks in Newfoundland. The conference

Hindawi Publishing CorporationFixed Point Theory and ApplicationsVolume 2006, Article ID 73530, Pages 1–2DOI 10.1155/FPTA/2006/73530

2 Editorial

was attended by 36 people from 14 different countries and five of the six inhabitable con-tinents; this comprises almost all mathematicians who are currently active in research inNielsen theory. The participants displayed a range of experience ranging from graduatestudents to people who had attended the previous Canadian Nielsen theory conference,in Sherbrooke, Quebec, in 1980. The group also included a number of experienced math-ematical researchers who are new to Nielsen theory. Their varied expertise is providingimportant new directions for research in our subject.

The major financial support for the conference was furnished by Canada’s Atlantic As-sociation for Research in the Mathematical Sciences (AARMS) which, in turn, is partiallyfunded by Memorial University of Newfoundland (MUN). We are profoundly gratefulfor the generous support from AARMS. Substantial funding for the attendance of par-ticipants from developing countries came from the Commission on Development andExchanges (CDE) of the International Mathematical Union (IMU). We thank Prof. Her-bert Clemens, Secretary of the CDE, and also the Executive Committee of the IMU whichmodified its rules in order to support these mathematicians at a conference not held ina developing country. Special thanks are due to the secretaries of the MUN MathematicsDepartment, in particular to Wanda Heath and Ros English, for their outstanding helpin organizing the conference.

We are also grateful to Prof. Ravi Agarwal and the publishers of the journal “FixedPoint Theory and Applications” for the opportunity to celebrate the occasion of the New-foundland conference by publishing the collection of research papers in Nielsen theorythat occupy this issue.

Philip R. HeathRobert F. Brown

Edward C. Keppelmann

DUAN’S FIXED POINT THEOREM: PROOF ANDGENERALIZATION

MARTIN ARKOWITZ

Received 25 July 2004; Revised 6 January 2005; Accepted 21 July 2005

Let X be an H-space of the homotopy type of a connected, finite CW-complex, f : X → Xany map and pk : X → X the kth power map. Duan proved that pk f : X → X has a fixedpoint if k ≥ 2. We give a new, short and elementary proof of this. We then use rationalhomotopy to generalize to spaces X whose rational cohomology is the tensor product ofan exterior algebra on odd dimensional generators with the tensor product of truncatedpolynomial algebras on even dimensional generators. The role of the power map is playedby a θ-structure μθ : X → X as defined by Hemmi-Morisugi-Ooshima. The conclusion isthat μθ f and f μθ each has a fixed point.

Copyright © 2006 Martin Arkowitz. This is an open access article distributed under theCreative Commons Attribution License, which permits unrestricted use, distribution,and reproduction in any medium, provided the original work is properly cited.

1. Introduction

Let G be a topological group and f : G→ G a map (i.e., a continuous function). Letpk : G→ G be the kth power map defined by pk(x) = xk. Recall that a fixed point of fis an element x0 ∈ G such that f (x0) = x0. In 1993 Duan Haibo proved the followinginteresting fixed point theorem.

Theorem 1.1 [1]. If G is a compact, connected topological group and f : G→ G is a map,then for any k ≥ 2, the map pk f :G→G has a fixed point.

This theorem was proved more generally for homotopy-associative H-spaces havingthe homotopy type of a finite, connected CW-complex (Theorem 2.2). In 1996, Luptonand Oprea [2] gave a new proof of Duan’s theorem using rational homotopy theory. In1997, Hemmi-Morisugi-Ooshima [3] extended Duan’s theorem to spaces more generalthan homotopy-associative H-spaces. In all of the above results, the existence of a fixedpoint of a map was obtained by showing the Lefschetz number of the map is non-zero.

The purpose of this paper is two-fold. First, we give a new, short proof of Duan’stheorem. The proof is elementary in that the only non-trivial result required is the


2 Duan’s fixed point theorem: proof and generalization

Hopf-Leray-Samelson theorem on the rational cohomology of a homotopy-associativeH-space. Secondly, we use rational methods, in particular, a result of Halperin [4], andideas from [3] to generalize Duan’s theorem.

2. Duan’s theorem

We begin by briefly discussing the Lefschetz number and H-spaces. All spaces will beassumed to have the homotopy type of a finite, connected CW-complex (though this as-sumption can be weakened). The cohomology of a space with coefficients in the additivegroup of rationals will be written H∗(X) = {Hn(X)}, so that cohomology will alwaysbe taken with rational coefficients. A map f : X → X induces a linear transformationf ∗n :Hn(X)→Hn(X). The Lefschetz number is defined by

L( f )=N∑

n=0

(−1)nTr(f ∗n

), (2.1)

where Hi(X)= 0, for i > N , and Tr denotes the trace. Lefschetz’s famous fixed point the-orem asserts that if L( f )�= 0, then f has a fixed point [5].

Next we state some basic facts about H-spaces. An H-space consists of a space X anda map m : X ×X → X (called the multiplication) such that m restricted to each factor ishomotopic to the identity map id. For an H-space X , the power map pk : X → X , k ≥ 1, isinductively defined as follows: p1 = id, and pk is the composition

XΔ

X ×X pk−1×idX ×X m

X , (2.2)

where Δ is the diagonal map. The multiplication m induces a homomorphism m∗ :H∗(X) → H∗(X ×X) ≈ H∗(X)⊗H∗(X). An element x ∈ Hn(X) is called primitive ifm∗(x) = x⊗ 1 + 1⊗ x. If x ∈ Hn(X) is primitive, then it follows immediately from thedefinitions that

p∗k (x)= kx. (2.3)

The H-space X is said to be homotopy-associative if the maps m(m× id), m(id×m) :X ×X ×X → X are homotopic. The Hopf-Leray-Samelson theorem ([6, page 268] and[7, Theorem 7.20]) asserts that if X is a homotopy-associative H-space, then

H∗(X)=Λ(x1,x2, . . . ,xr

), (2.4)

an exterior algebra on odd degree generators x1,x2, . . . ,xr which are primitive.With these generalities out of the way, we state an obvious lemma and proceed with

Duan’s theorem and its proof.

Martin Arkowitz 3

Lemma 2.1. IfA is an n×nmatrix of rationals and B is a diagonal n×nmatrix of rationals,

A=

⎛⎜⎜⎜⎜⎝

a11 a12 . . . a1n

a21 a22 . . . a2n...

.... . .

...an1 an2 . . . ann

⎞⎟⎟⎟⎟⎠

, B =

⎛⎜⎜⎜⎜⎝

b1 0 . . . 00 b2 . . . 0...

.... . .

...0 0 . . . bn

⎞⎟⎟⎟⎟⎠

, (2.5)

then Tr(AB)= a11b1 + a22b2 + ···+ annbn = Tr(BA).

We write the diagonal matrix B as diag(b1,b2, . . . ,bn).

Theorem 2.2 [1]. If X is a homotopy-associative H-space, f : X → X any map and pk :X → X the kth power map, k ≥ 2, then pk f : X → X has a fixed point.

Proof. We show that L(pk f )�= 0. For this we consider the trace of (pk f )∗n = f ∗np∗nk

Hn(X)p∗nk

Hn(X)f ∗n

Hn(X). (2.6)

By the theorem of Hopf-Leray-Samelson,

H∗(X)=Λ(x1,x2, . . . ,xr

), (2.7)

where the xi are primitive elements of odd degree |xi| =mi. If n≥ 1 then a basis ofHn(X)consists of elements

yi1i2···il = xi1xi2 ···xil , (2.8)

where l ≥ 1, 1≤ i1 < i2 < ··· < il ≤ r andmi1 +mi2 + ···+mil = n. We examine the matrixof p∗nk and f ∗n with respect to this basis. Since p∗k (xi)= kxi,

p∗nk(yi1i2···il

)= kl yi1i2···il . (2.9)

Now suppose that there are b(n)1 basis elements in Hn(X) of length one (i.e., those of the

form yi), b(n)2 basis elements in Hn(X) of length two (i.e., those of the form yi1i2 , i1 < i2),

etc., where b(n)i ≥ 0. Then the matrix B of p∗nk is diagonal,

B = diag

(k, . . . ,k︸︷︷︸b(n)

1

,k2, . . . ,k2︸︷︷︸

b(n)2

, . . .

). (2.10)

Next we consider the matrix A of f ∗n with respect to this basis. Now f ∗n is obtainedby taking the homomorphism on integral n-dimensional cohomology induced by f and

tensoring it with the rationals. Thus A is a matrix of integers. Let e(n)1 be the sum of the

first b(n)1 diagonal entries of A, e(n)

2 the sum of the next b(n)2 diagonal entries, etc. Then by

Lemma 2.1,

Tr((pk f

)∗n)= Tr(AB)= ke(n)1 + k2e(n)

2 + ···+ kre(n)r . (2.11)


Thus Tr((pk f )∗n)≡ 0(mod k) for n≥ 1, and so L(pk f )≡ 1(mod k). Since k ≥ 2, we haveL(pk f )�= 0. This completes the proof. �

Remark 2.3. There are some simple extensions of Theorem 2.2:(1) If the H-space X has a homotopy inverse, one can define pk for all integers k.

Theorem 2.2 then holds for all |k| ≥ 2.(2) Theorem 2.2 holds for the map f pk : X → X since it is well known that L( f pk)=

L(pk f ).(3) Let k be an integer such that |k| ≥ 2. Suppose that X is a homotopy-associative H-

space and there is a map μ : X → X such that μ∗(xi) = aixi, where ai ≡ 0(mod k),for i= 1,2, . . . ,r. Then the previous proof shows that if f : X → X is any map, thenμ f and f μ each has a fixed point. We will return to this in Section 4.

We note the following immediate consequence of Duan’s theorem which appears in[5, Theorem 1, page 49].

Corollary 2.4. Let G be a compact, connected topological group, a ∈ G and k ≥ 2. Thenthere exists x0 ∈G such that xk0 = a.

Proof. Let La :G→G be left multiplication by a. By Duan’s theorem, La−1 pk+1 has a fixedpoint x0. �

3. Fixed points and eigenvalues

In this section we consider spaces with restricted cohomology and state a result on theLefschetz number of self maps of such spaces. This result, Theorem 3.1, which may be ofsome interest in itself, will be used to generalize Duan’s theorem in Section 4.

Let Y be a space and consider the vector space I∗(H∗(Y))= {In(H∗(Y))} of indecom-posables of H∗(Y) defined by

I∗(H∗(Y)

)= H+(Y)H+(Y) ·H+(Y)

, (3.1)

where H+ denotes positive-dimensional cohomology. A map f : Y → Y ′ induces f ∗ :H∗(Y ′) → H∗(Y) and this induces a linear transformation I∗( f ∗) : I∗(H∗(Y ′)) →I∗(H∗(Y)).

For the rest of this section we consider spaces X whose cohomology has the followingform

H∗(X)=Λ(x1,x2, . . . ,xr

)⊗P(y1, y2, . . . , ys)/⟨yn1

1 , yn22 , . . . , ynss

⟩, (3.2)

where Λ(x1,x2, . . . ,xr) is an exterior algebra on odd dimensional generators x1,x2, . . . ,xr ,P(y1, y2, . . . , ys) is a polynomial algebra on even dimensional generators y1, y2, . . . , ys and〈yn1

1 , yn22 , . . . , ynss 〉 is the ideal generated by the powers yn1

1 , yn22 , . . . , ynss . In short, H∗(X) is a

tensor product of monogenic algebras.We will always assume for a space X which satisfies (3.2) that 1 < n1 < n2 < ··· < ns.We give examples of such spaces in Examples 3.3(1).

Martin Arkowitz 5

Now let X be a space satisfying (3.2) and f : X → X a map. The vector space of inde-composables I∗(H∗(X)) can be split into its odd and even degree parts

I∗(H∗(X)

)=V ⊕W , (3.3)

where V =⊕iodd Ii(H∗(X)) and W =⊕ieven I

i(H∗(X)). Then I∗( f ∗) : I∗(H∗(X)) →I∗(H∗(X)) induces linear transformations

fV :V −→V , fW :W −→W. (3.4)

The following theorem will be proved in Section 5.

Theorem 3.1. Let X be a space satisfying (3.2) and f : X → X a map. Suppose that −1 isnot an eigenvalue of fW . Then

L( f )�= 0⇐⇒ fV has no eigenvalue equal to 1. (3.5)

We make some remarks on the theorem.

Remarks 3.2. (1) The matrices of the linear transformations fV and fW can be determinedfrom the induced linear transformation f ∗ applied to the algebra generators x1,x2, . . . ,xr , y1, y2, . . . , ys of H∗(X). In general, it is difficult to calculate the eigenvalues of a lineartransformation since this requires finding the roots of the characteristic polynomial. Inapplying Theorem 3.1 to show L( f ) �= 0, however, it is only necessary to show that −1and 1 are not roots of the appropriate characteristic polynomials. This is much easier todo.

(2) The theorem holds if r or s= 0. In addition, the conclusion L( f )�= 0 holds withoutthe hypothesis that fV has no eigenvalue equal to 1, provided all ni are odd. This can beseen from the proof.

(3) A result similar to Theorem 3.1 has been proved by Lupton and Oprea [2]. InRemark 5.3 we discuss the relation of their result to our work.

We next give some examples related to Theorem 3.1.

Examples 3.3. (1) We indicate one way (though not the only way) to construct spaces Xsatisfying (3.2). LetA be a space such thatH∗(A)=Λ(x1, . . . ,xr). For example,A could bethe product of any number of the following spaces: homotopy-associative H-spaces andodd dimensional spheres. Let B be a space such that H∗(B)= P(y1, . . . , ys)/〈yn1

1 , . . . , ynss 〉.For example, B could be the product of any number of the following spaces: projectivespaces and even dimensional spheres. Then X = A×B is a space which satisfies (3.2).

(2) We next show that the hypothesis that −1 is not an eigenvalue of fW is necessaryin general. Let X be the complex projective space CP2n+1 and let f : X → X be a map ofdegree −1, that is, f ∗2(u)=−u for every u∈H2(X). Then 1 is not an eigenvalue of fV ,−1 is an eigenvalue of fW and L( f )= 0.

(3) Here we show that the strict inequality 1 < n1 < n2 < ··· < nr is necessary inTheorem 3.1. Define f : S2 × S2 → S2 × S2 by f (x, y) = (y,−x). Let {u,v} ⊆H2(S2 × S2)be the basis corresponding to the two 2-spheres. Then f ∗2(u) = v, f ∗2(v) = −u andf ∗4(uv)=−uv, and so L( f )= 0.


4. Theta spaces

In this section we will use Theorem 3.1 to extend Duan’s theorem to spaces X which sat-isfy (3.2). In order to do this it is necessary to describe a map X → X which plays the roleof the power map of H-spaces. This has been done by Hemmi-Morisugi-Ooshima [3]. Webegin this section by summarizing their work (with some small changes in terminology).

For the remainder of the paper we will use X to denote a space which satisfies (3.2) ofSection 3 and will use Y to denote an arbitrary space (of the homotopy type of a finite,connected CW-complex).

Definition 4.1. Let Y be a space and {m1,m2, . . . ,mt} a set of positive integers defined asfollows:

Im(H∗(Y)

)�= 0⇐⇒m=mi, for some i= 1,2, . . . , t. (4.1)

Let θ : {m1,m2, . . . ,mt} → Z be an integer-valued function. Then a θ-structure on Y is amap μθ : Y → Y such that

Imi(μ∗θ)(y)= θ(mi

)(y), (4.2)

for every y ∈ Imi(H∗(Y)). The pair (Y ,μθ) (or just Y) is called a θ-space. A constant θ-structure is one such that θ(mi)= k, for all i, where k ∈ Z is a fixed integer.

There is a long list of θ-spaces in [3] and we mention some of them below. All θ func-tions in the following list have the form θ(mi)= ke(mi), for some integer k and function e.

(i) H-spaces and co-H-spaces have constant θ-structure given by the power map.(ii) Semi-simple Lie groups G and their classifying spaces BG have θ-structure given

by the unstable Adams operations ψk on BG and Ωψk on ΩBG=G, for certain k.(iii) Complex and Quaternionic Grassman manifolds Gp,q with some restrictions on

p and q have θ-structure.(iv) The Stiefel manifolds U(2n+ 2)/U(2n) have constant θ-structure k if and only if

k ≡ 0,1,5(mod 8).In addition, the existence of θ-structure on a large class of spaces is obtained from the

following corollary of Theorem 1 in [3]:If X is a simply-connected space which satisfies (3.2) of Section 3, then there exists

infinitely many θ-structures on X .In Theorem 2 of [3] the authors consider self maps f : Y → Y of a θ-space, for certain

θ, and show the existence of fixed points of f μθ and μθ f . The restrictions on θ are thatθ(mi) = ke(mi), where e(mi) = (b− a)mi + 2a− b, for b ≥ a ≥ 1 and |k| ≥ 2. We prove asimilar theorem below (by different methods) which restricts the spaces to those satisfy-ing (3.2) but allows a much larger class of functions θ.

Theorem 4.2. Let X be any space satisfying (3.2) and f : X → X any map. Let{m1,m2, . . . ,mt} be the set of degrees of the non-zero indecomposables of H∗(X) and letθ : {m1,m2, . . . ,mt} → Z−{0,±1} be any function. If μθ : X → X is a θ-structure on X , thenf μθ and μθ f each has a fixed point.

Martin Arkowitz 7

Proof. We apply Theorem 3.1 to μθ f : X → X . We decompose I∗(H∗(X))= V ⊕W intoodd and even parts and first consider I∗( f ∗μ∗θ ) |W= (μθ f )W : W → W . Supposew ∈Wmi is an eigenvector of (μθ f )W with eigenvalue −1. Then

fW(θ(mi)(w)

)= Imi(f ∗μ∗θ

)(w)= (μθ f

)W (w)=−w, (4.3)

and so fW (w) = (−1/θ(mi))w. Thus −1/θ(mi) is an eigenvalue of fW which is a ratio-nal number. But fW is induced by a map f : X → X and so, as noted in the proof ofTheorem 2.2, with respect to some basis of W , fW is represented by an integral matrix(see also [2, §3]). But the only rational eigenvalues of an integral matrix are integers [8,Theorem 4.16]. Therefore −1/θ(mi) cannot be an eigenvalue of fW since θ(mi) �= ±1.Thus −1 is not an eigenvalue of fW . A similar argument shows that 1 is not an eigenvalueof fV . Therefore by Theorem 3.1, L(μθ f )�= 0. An analogous argument holds for f μθ . �

Remark 4.3. We illustrate how Theorem 4.2 can be used in some concrete examples. LetX = A×B be a space of the type discussed in Examples 3.3(1). If μθ is a θ-structure onA and μθ′ is a θ-structure on B, then μθ ×μθ′ is a θ-structure on A×B. More specifically,suppose A is a homotopy-associative H-space with μθ the kth power map and B is aproduct of even dimensional spheres and projective spaces with μθ′ a θ-structure whichis constant at l (for example, the product of maps of degree l). If k and l are both�= 0, ±1,then Theorem 4.2 applies to the θ-structure μθ ×μθ′ on X .

5. Proof of Theorem 3.1

We state a special case of a theorem of Halperin which will be needed to prove Theorem3.1. This requires the use of rational homotopy theory, in particular, Sullivan minimalmodels (see [9] and [10]). For a space X which satisfies (3.2), one can construct the min-imal model � of X . This has the following properties: � is a free-commutative, graded,differential algebra with generators x1, . . . ,xr , y1, . . . , ys (which are in one-one correspon-dence with the generators of H∗(X) and have the same degree) and generators z1, . . . ,zswith |zi| = |yi|ni− 1. Then

� =Λ(x1, . . . ,xr ,z1, . . . ,zs

)⊗P(y1, . . . , ys), (5.1)

with |xi| and |zi| odd and |yi| even. Note that a vector space basis for � consists of all

xη1

1 ···xηrr yλ11 ··· yλss zτ1

1 ···zτss , where 0≤ ηi, τi ≤ 1 and 0≤ λi. The differential d on � isdefined by: dxi = 0, dyi = 0 and dzi = ynii . ClearlyH∗(�,d)=H∗(X). We split the vectorspace I∗(�) of indecomposables of � into the direct sum of an odd degree partO and aneven degree part E. We identify O = 〈x1, . . . ,xr ,z1, . . . ,zs〉 and E = 〈y1, . . . , ys〉. A map f :X → X induces a homomorphism φ : � →�. This determines I∗(φ) : I∗(�)→ I∗(�)and thence φO :O→O and φE : E→ E. We now state a special case of Halperin’s theoremfor spaces which satisfy (3.2).

Theorem 5.1 [4, Theorem 3]. The number of eigenvalues of φO which are 1 equals thenumber of eigenvalues of φE which are 1 if and only if L( f )�= 0.


Using this theorem, we now prove Theorem 3.1.

Proof. We fix l, 1≤ l ≤ s, and write φ(zl) as a linear combination of basis elements in thevector space �,

φ(zl)=∑

i

alizi +∑

j

bl jx j + εl, (5.2)

where εl is decomposable and |zl| = |zi| = |xj| for all i and j in the above sums. We letI = {i | |zi| = |zl|} and apply d to both sides of (5.2) to obtain

φ(dzl)=

∑

i∈Iali y

nii +dεl . (5.3)

But φ(dzl)= φ(ynll )= (φ(yl))nl , and so (5.3) yields

(φ(yl))nl =

∑

i∈Iali y

nii +dεl . (5.4)

Next we write φ(yl) as a linear combination of basis elements

φ(yl)=

∑

k∈Kclk yk + δl, (5.5)

where K = {k | |yk| = |yl|} and δl is decomposable. Since n1 < ··· < ns, it follows thatI ∩K = {l}. Then we obtain from (5.4) and (5.5),

(∑

k∈Kclk yk + δl

)nl=∑

i∈Iali y

nii +dεl . (5.6)

Consider any t with 1≤ t ≤ s. We will equate the terms which are linear combinations ofyat for all a > 0 on the left side of (5.6) with those on the right side of (5.6). For this itis necessary to analyze the elements dεl and δl in terms of the vector space basis above,noting that εl and δl are decomposable and that |dεl| = |zl|+ 1 and |δl| = |yl|. Now δlmay contain a term of the form uay

at , where ua is a rational and a ≥ 2. Thus the only

possible terms on the left side of (5.6) which are powers of yt are cnllt ynlt and unla y

anlt . For

the right side of (5.6) note that εl may contain a term of the form vb ybt zt, where b > 0

and vb is a rational. Then dεl will contain vb ynt+bt . Thus the only possible terms which are

powers of yt on the right side of (5.6) are alt yntt and vb y

nt+bt . Now suppose t ∈ K and t �= l.

Then t /∈ I and (5.6) yields

cnllt ynlt = vb ynt+bt . (5.7)

Thus if t > l, then nl < nt, and so clt = 0. Next suppose t ∈ I and t �= l. Then t /∈ K and(5.6) yields

unla yanlt = alt yntt . (5.8)

Martin Arkowitz 9

If l > t, then anl > nt, and so alt = 0. Finally, if l = t, then cnlll = all. Putting this informationinto (5.2) and (5.5), we have

φO(zl)=

∑

i≥lalizi +

∑

j

bl jx j , (5.9)

φE(yl)=

∑

k≤lclk yk, (5.10)

where all = cnlll . Now E = 〈y1, . . . , ys〉 and φE : E→ E. From (5.10), the eigenvalues of φE(in degree |yl|) are rational numbers of the form cii. These are the same as the eigenvaluesof fW . By hypothesis, none of these eigenvalues equals −1. Clearly I∗(φ) : 〈x1, . . . ,xr〉 →〈x1, . . . ,xr〉. By examining the matrix of φO, we see that the eigenvalues of φO consist ofthose of the form aii = cniii together with the eigenvalues of φO|〈x1, . . . ,xr〉. Now aii = 1 ifand only if cii = 1. Thus the number of eigenvalues of φO which are 1 equals the numberof eigenvalues of φE which are 1 plus the number of eigenvalues of φO|〈x1, . . . ,xr〉 whichare 1. But the eigenvalues of φO|〈x1, . . . ,xr〉 are just the eigenvalues of fV . By Halperin’stheorem, L( f )�= 0 if and only if no eigenvalue of fV equals 1. �

Remark 5.2. Halperin’s theorem as stated and proved in [4] is more general in two distinctways than what we have stated above. First of all, the theorem applies to elliptic spaces.These are spaces whose (rational) cohomology and rational homotopy groups (i.e., ho-motopy groups tensored with the rationals) vanish in all sufficiently high dimensions.The spaces which satisfy (3.2) are elliptic spaces. Secondly, the theorem gives a formulafor the Lefschetz number of a map f in terms of the eigenvalues of fV and fW . Since weare interested in fixed points of maps, we have only considered the case where L( f )�= 0.This has led to a simplified statement of the theorem.

Remark 5.3. Lupton and Oprea consider an elliptic space X whose minimal model � isoddly graded, that is, � = Λ(x1, . . . ,xr), an exterior algebra on odd dimensional gener-ators. It is not assumed that the differential d = 0. If f : X → X is a map such that theinduced map φO = I∗(φ) : I∗(�) → I∗(�) does not have 1 as an eigenvalue, then themain result of [2, § 5] asserts that L( f ) �= 0. It is possible to modify the statement andproof of Theorem 3.1 slightly so as to include this result. One assumes that the minimalmodel � of X has the form

� =Λ(x1, . . . ,xr ,z1, . . . ,zs

)⊗P(y1, . . . , ys), (5.11)

with |xi| and |zi| odd and |yi| even. Furthermore, dxi ∈Λ(x1, . . . ,xr), dzi = yni and dyi =0 with n1 < ··· < ns. We assume that φE : E→ E does not have −1 as an eigenvalue. Thenthe modified version of Theorem 3.1 asserts that L( f )�= 0 if and only if φO does not have 1as an eigenvalue. For the proof it is only necessary to show that φO(xi)∈ 〈x1, . . . ,xr〉whichrequires straightforward arguments similar to those given in the proof of Theorem 3.1.

Acknowledgment

We would like to thank Robert Brown, Gregory Lupton, Donald Stanley, and Peter Wongfor several helpful comments.


References

[1] D. Haibao, A characteristic polynomial for self-maps of H-spaces, The Quarterly Journal of Math-ematics. Oxford. Second Series (2) 44 (1993), no. 175, 315–325.

[2] G. Lupton and J. Oprea, Fixed points and powers of self-maps of H-spaces, Proceedings of theAmerican Mathematical Society 124 (1996), no. 10, 3235–3239.

[3] Y. Hemmi, K. Morisugi, and H. Ooshima, Self maps of spaces, Journal of the Mathematical Soci-ety of Japan 49 (1997), no. 3, 438–453.

[4] S. Halperin, Spaces whose rational homology and de Rham homotopy are both finite-dimensional,Algebraic Homotopy and Local Algebra (Luminy, 1982), Asterisque, vol. 113-114, Soc. Math.France, Paris, 1984, pp. 198–205.

[5] R. F. Brown, The Lefschetz Fixed Point Theorem, Scott, Foresman, Illinois, 1971.[6] E. H. Spanier, Algebraic Topology, McGraw-Hill, New York, 1966.[7] J. W. Milnor and J. C. Moore, On the structure of Hopf algebras, Annals of Mathematics. Second

Series (2) 81 (1965), 211–264.[8] T. Hungerford, Abstract Algebra: An Introduction, Saunders college, Pennsylvania, 1990.[9] P. A. Griffiths and J. W. Morgan, Rational Homotopy Theory and Differential Forms, Progress in

Mathematics, vol. 16, Birkhauser, Massachusetts, 1981.[10] Y. Felix, S. Halperin, and J.-C. Thomas, Rational Homotopy Theory, Graduate Texts in Mathe-

matics, vol. 205, Springer, New York, 2001.

Martin Arkowitz: Department of Mathematics, Dartmouth College, Hanover, NH 03755, USAE-mail address: [email protected]

FIXED POINT INDICES AND MANIFOLDS WITH COLLARS

CHEN-FARNG BENJAMIN AND DANIEL HENRY GOTTLIEB

Received 7 December 2004; Revised 25 April 2005; Accepted 24 July 2005

This paper concerns a formula which relates the Lefschetz number L( f ) for a map f :M →M′ to the fixed point index I( f ) summed with the fixed point index of a derivedmap on part of the boundary of ∂M. Here M is a compact manifold and M′ is M with acollar attached.

Copyright © 2006 C.-F. Benjamin and D. H. Gottlieb. This is an open access article dis-tributed under the Creative Commons Attribution License, which permits unrestricteduse, distribution, and reproduction in any medium, provided the original work is prop-erly cited.

1. Introduction

This paper represents the first third of a Ph.D. thesis [2] written by the first author underthe direction of the second author at Purdue University in 1990. The thesis is entitled“Fixed Point Indices and Transfers, and Path Fields” and it contains, in addition to thecontents of this manuscript, a formula analogous to (1.1), which relates to Dold’s fixedpoint transfers and a study of path fields of differential manifolds in order to relate theformula in this manuscript with an analogous formula involving indices of vector fields.These results are related to the papers [1, 3, 4, 7, 8, 14, 16]

Let M be a compact differentiable manifold with or without boundary ∂M. Assume Vis a vector field on M with only isolated zeros. If M is with boundary ∂M and V pointsoutward at all boundary points, then the index of the vector field V equals Euler char-acteristic of the manifold M. This is the classical Poincare-Hopf index theorem. (A 2-dimensional version of this theorem was proven by Poincare in 1885; in full generalitythe theorem was proven by Hopf [13] in 1927). In particular, the index is a topologicalinvariant of M; it does not depend on the particular choice of a vector field on M.

Morse [15] extended this result to vector fields under more general boundary condi-tions, namely, to any vector field without zeros on the boundary ∂M; he discovered thefollowing formula:

Ind(V) + Ind(∂ V

)= χ(M), (1.1)


2 Fixed point indices and manifolds with collars

where χ(M) denotes the Euler characteristic of M and ∂ V is defined as follows. Let ∂ Mbe the open subset of the boundary ∂M containing all the points m for which the vectorsV(m) point inward, and let ∂V be the vector field on the boundary ∂M obtained by firstrestricting V to the boundary and then projecting V |∂M to its component field tangent tothe boundary. Then ∂ V = ∂V |∂ M . Furthermore, in the same paper, Morse generalizedhis result to indices of vector fields with nonisolated zeros. This is the formula (1.1). Now(1.1) was rediscovered by Gottlieb [10] and Pugh [17]. Gottlieb further found furtherinteresting applications in [9, 11, 12]. Throughout this paper, we will call formula (1.1)the Morse formula for indices of vector fields.

We consider maps f :M →M′ from a compact topological manifold M to M′, whereM′ is obtained by attaching a collar ∂M × [0,1] to M. If f has no fixed points on theboundary ∂M, we prove Theorem 3.1 which is the fixed point version of the Morse for-mula:

I( f ) + I(r ◦ f |∂ M

)= L(r ◦ f ), (1.2)

where I denotes the fixed point index, r is a retraction of M′ onto M which maps thecollar ∂M × I onto the boundary ∂M, ∂ M is an open subset of ∂M containing all thepoints x ∈ ∂M mapped outside of M under f , and L(r ◦ f ) is the Lefschetz number ofthe composite map r ◦ f .

In particular, if the map r ◦ f is homotopic to the identity map, we have

I( f ) + I(r ◦ f |∂ M

)= χ(M), (1.3)

which is similar to the Morse formula; and the map r ◦ f | ∂ M is analogous to the vectorfield ∂ V .

Formula (1.3) was independently obtained by A. Dold (private letter to D. Gottlieb).This paper is organized as follows: in Section 2, we list some properties of fixed point

indices; our first main result, Theorem 3.1, is proven in Section 3.

2. Fixed point index and its properties

In this section, we use the definition of fixed point index and some well-known resultson fixed point index given by Dold in [5] or [6, Chapter 7] to obtain an equation forfixed point indices (Theorem 3.1) analogous to the Morse equation for vector field indicesdescribed in the introduction.

Let X be an Euclidean neighborhood retract (ENR). Consider maps f from an opensubset V of X into X whose fixed point set F( f )= {x ∈ V | f (x)= x} is compact. Dold[5] defined the fixed point index I( f ) and proved the following properties.

Localization 2.1. Let f : V → X be a map such that F( f ) is compact, then I( f )= I( f |W )for any open neighborhood W of F( f ) in V .

C.-F. Benjamin and D. H. Gottlieb 3

Additivity 2.2. Given a map f :V → X and V is a union of open subsets Vj , j = 1,2, . . . ,n,such that the fixed point sets F j = F( f )∩Vj , are mutually disjoint. Then for each j,I( f |Vj ) is defined and

I( f )=n∑

j=1

I(f |Vj

). (2.1)

Units 2.3. Let f :V → X be a constant map. Then

I( f )= 1 if f (V)= p ∈V ,

I( f )= 0 if f (V)= p /∈V. (2.2)

Normalization 2.4. If f is a map from a compact ENR X to itself, then I( f )= L( f ), whereL( f ) is the Lefschetz number of the map f .

Multiplicativity 2.5. Let f : V → X and f ′ : V ′ → X ′ be maps such that the fixed pointsets F( f ) and F( f ′) are compact, then fixed point index of the product f × f ′ :V ×V ′ →X ×X ′ is defined and

I( f × f ′)= I( f ) · I( f ′). (2.3)

Commutativity Axiom 2.6. If f : U → X ′ and g : U ′ → X are maps where U ⊆ X andU ′ ⊆ X ′ are open subsets, then the two composites g f : V = f −1(U ′)→ X and f g : V ′ =g−1(U)→ X ′ have homeomorphic fixed point sets. In particular, I( f g) is defined if andonly if I(g f ) is defined, in that case,

I( f g)= I(g f ). (2.4)

Homotopy Invariance 2.7. Let H :V × I → X be a homotopy between the maps f0 and f1.Assume the set F = {x ∈V |H(x, t)= x for some t} is compact, then

I(f0)= I( f1

). (2.5)

For our purposes, it is useful to reformulate the properties of Additivity 2.2 andHomotopy Invariance 2.7 in the form of the following propositions. These reformula-tions are found in Brown’s book [4], and they form part of an axiom system for the fixedpoint index. The five axioms are a subset of Dold’s properties. They consist of localiza-tion, homotopy invariance , addititvity, normalization, and commutivity. We will showthat the main formula will follow from these axioms. We will give an alternate proof inSection 3.

Proposition 2.8. Assume X is compact and V is an open subset of X . Let f : V → X be amap without fixed points on Bd(V). If {Vj}, j = 1,2, . . . ,n are mutually disjoint open subsetsof V and whose union

⋃nj=1Vj contains all the fixed points of f , then

I(f |V)=

n∑

j=1

I(f |Vj

). (2.6)


Proposition 2.9. Assume X is compact and V is an open subset of X . Let H : V × I → Xbe a homotopy from f 0 and f 1, where f 0 and f 1 are maps from V , the closure of V to X . IfH(x, t)�= x for all x ∈ Bd(V) and for all t, then

I(f0)= I( f1

)where f0 = f 0 |V , f1 = f 1 |V. (2.7)

Proof. Since H =H|V×I is a homotopy from f0 to f1, it suffices to verify that the set F ={x ∈ V | H(x, t) = x for some t} is compact. Let {xj} be a sequence in F converging tox ∈V =V ∪Bd(V). There exists a subsequence {t j} of those t’s in I such that H(xj , t j)=xj . Since I is compact, a subsequence of {t j} converges to a point t ∈ I . By the continuityofH , we haveH(x, t)= x. On the other hand, we know that H(x, t)�= x for all x ∈ Bd(V);thus, x ∈ V and H(x, t) = x. Consequently, x ∈ F. Therefore, F is a closed subset of acompact space, hence F is compact. This proves the proposition. �

3. The main formula

Consider a compact topological manifold M with boundary ∂M. We attach a collar to Mand call the resulting manifold M′ : M′ =M∪∂M∼∂M×{0} ∂M× [0,1]. Let f : M →M′ bea map such that f (x)�= x for all x ∈ ∂M. Since M is compact, the fixed point set F( f ) is

a compact set contained in◦M =M\∂M. For such f : M →M′, we define the index of f ,

denoted by I( f ), to be the fixed point index of the map f | ◦M

given in Section 1.For specificity, we define the retraction r: let r :M′ →M be the retraction from M′ to

M given by the formula,

r(m)=m for m∈M,

r(b, t)= (b,0)∼ b for (b, t)∈ ∂M× [0,1].(3.1)

Now we can formulate the main result of the section.Now, assume r′ is any retraction fromM′ toM such that r′ maps the collar ∂M× [0,1]

into the boundary ∂M. Then the following theorem is true.

Theorem 3.1. One has that

I( f ) + I(r′ f |∂ M

)= L(r′ f ). (3.2)

Furthermore,

L(r f )= L(r′ f ),

I(r f |∂ M

)= I(r′ f |∂ M),

(3.3)

where r is the standard retraction defined above and where L(r f ) denotes the Lefschetz num-ber of r f :M→M and ∂ M = {x ∈ ∂M | f (x) /∈M}.Proof. First, we prove the formula I( f ) + I(r′ f |∂ M)= L(r′ f ). Let V1 = {x ∈M | f (x)∈◦M} and V2 = {x ∈M | f (x) ∈M′\M}, then V1 and V2 are disjoint open subsets of the


manifold M and V1 ∪V2 contains all the fixed points of the map r′ f . Indeed, if x /∈(V1 ∪V2), then f (x) ∈ ∂M, and hence r′ f (x) = f (x) �= x. Proposition 2.8 implies theequation

I(r′ f )= I(r′ f |V1

)+ I(r′ f |V2

). (3.4)

Since r′ f is a self-map from M to M, so

I(r′ f )= L(r′ f ). (3.5)

We have

L(r′ f )= I(r′ f |V1

)+ I(r′ f |V2

). (3.6)

Now, since r′ f |V1 = f |V1 and F( f )⊆V1, then

I(r′ f |V1

)= I( f |V1

)= I( f ). (3.7)

Let us decompose the map r′ f |V2 :

r′ f |V2 :V2f |V2−→ ∂M× [0,1]

r′−→ ∂Mi−→M. (3.8)

The Commutativity 2.6 implies that

I(r′ f |V2

)= I(ir′ f |V2

)= I(r′ f i|i−1(V2))= I(r′ f |∂ M

). (3.9)

Combining (3.6), (3.7), and (3.9), we obtain

I( f ) + I(r′ f |∂ M

)= L(r′ f ). (3.10)

This completes the proof of the formula holding for any retraction r′. The following twolemmas will show that the terms in (3.10) are the same no matter which retraction r′ ischosen. �

Lemma 3.2. The retraction r is homotopic to r′.

Proof. Consider the homotopy Ht :M′ →M, 0≤ t ≤ 1, defined as follows:

Ht(m)=m for m∈M,

Ht(b,s)= r′(b,st) for (b,s)∈ ∂M× [0,1].(3.11)

Clearly, H0 = r and H1 = r′. So, r f and r′ f are homotopic. �

Lemma 3.3. L(r′ f )= L(r f ) and I(r f |∂ M)= I(r′ f |∂ M).

Proof. By Lemma 3.2, r f and r′ f are homotopic and, consequently,

L(r f )= L(r′ f ) (3.12)

since the Lefschetz number L is a homotopy invariant.


Equations (3.10) and (3.12) with r replacing r′ imply that

I(r f |∂ M

)= I(r′ f |∂ M). (3.13)

This concludes the proof of Theorem 3.1. �

Corollary 3.4. If f :M →M′ is a map such that f (x) /∈M for any x ∈ ∂M, then I( f )=L(r f )−L(r f |∂M).

Corollary 3.5. If f : M →M′ is without fixed points on the boundary ∂M and f (∂M)⊂M, then I( f )= L(r f ).

Example 3.6. Consider a map f : Dn → Rn. Here Dn is the unit ball and Sn−1 is the unitboundary sphere, so we can think of Rn as Dn with an open collar attached.

(i) If f (Sn−1)⊂Dn, then f has a fixed point.(ii) If f (Sn−1) ⊂ Rn\Dn, then Corollary 3.4 implies that I( f ) = L(r f )− L(r f |Sn−1 ) =

1− (1 + (−1)n−1 deg(r f |Sn−1 ))= (−1)ndeg(r f |Sn−1 ).

Corollary 3.7. If f : M →M′ is homotopic to the inclusion map M↩M′, then I( f ) +I(r f |∂ M)= χ(M), where χ(M) denotes the Euler characteristic of M.

Proof. If f : M →M′ is homotopic to the inclusion map M↩M′, then the compositemap r f :M→M is homotopic to the identity map. Therefore L(r f )= L(Id)= χ(M). �

Remark 3.8. Here is a more geometric proof of the main theorem (Theorem 3.1).

Proof. Let DM be the double of M, that is, the union of two copies of M intersecting ontheir boundaries. Let R : DM →M be the retraction which takes the second copy ontothe first. Now f ◦R :DM →M. Then the Lefschetz numbers L( f )= L( f ◦R) since R is aretraction, which splits the homology of DM, so that the traces of the induced map mustbe calculated only on the first copy M of DM.

Also we consider M ⊂M′ ⊂DM. Then R restricted to M′ is equal to r. Now the fixedpoint set of f ◦R consists of the fixed point set of f , in the interior of M, and the fixedpoint set F( f ◦R)= F( f ◦ r) contained in ∂ M. Now the index of r ◦ f calculated on theopen set ∂M is equal to the index calculated on a small open set V of M′ containing ∂ Mwhich follows from the next lemma. �

Lemma 3.9. One has that

I(r ◦ f |∂ M

)= I(r ◦ f |V). (3.14)

Proof. Commutativity 2.6 implies that

I(r ◦ f |V

)= I( f ◦ r|r−1(V)). (3.15)

It is easy to see that the fixed point set of the map f ◦ r|r−1(V) is {(b, t) ∈ ∂M × (0,1] | f (b) = (b, t)} and the fixed point set of the map r f |V is {b ∈ ∂M | f (b) = (b, t)for some t}.


We now define a homotopy Gs, 0≤ s≤ 1, as the composite of the following maps

∂ M× I r−→ ∂ Mf−→ ∂M× I Hs−→ ∂M× I , (3.16)

where the map Hs is defined as follows:

Hs(b, t)= (b,st+ (1− s)t), where t is a constant, 0 < t ≤ 1. (3.17)

Since the map H0 = Identity, we have

G0(x, t)=H0(f r(x, t)

)= f r(x, t),

G1(x, t)=H1(f r(x, t)

)= (r f × g)(x, t),(3.18)

where r ◦ f is a map from ∂ M to ∂M and g : I → I , g(t) = t, is the constant map. Fur-thermore, the restriction Gs|Bd(∂ M×I) has no fixed points for any 0 ≤ s ≤ 1. To see this,we look at a point x ∈ Bd(∂ M). We know then that f (x) ∈ ∂M and r f (x) = f (x) �= x,therefore, Gs(x, t)=Hs( f r(x, t))=Hs( f (x))=Hs( f (x),0)= ( f (x),st)�= (x, t).

Now the Axioms 2.9, 2.5, and 2.3 imply that

I(f r|∂ M×(0,1]

)= I(r f |∂ M) · I(g)= I(r f |∂ M

) · 1,

I(r f |V

)= I( f r|r−1(V))= I( f r|∂ M×(0,1]

).

(3.19)

The last equality holds because ∂ M× (0,1] contains the fixed point set of ( f r|r−1(V)).Thus, I(r f |V )= I(r f |∂ M). �

Proof of Theorem 3.1. Consider the composite Mf→M′ r→M. Let V be the open set as

in Lemma 3.3, then V and◦M are two open subsets of M such that V ∪

◦M =M. Clearly,

F(r f )∩◦M and F(r f )∩V are disjoint. Using Additivity 2.2 and Normalization 2.4 of the

fixed point indices, we have

I(r f | ◦

M

)+ I(r f |V

)= I(r f )= L(r f ). (3.20)

Lemmas 3.2 and 3.3 then imply the equation

I( f ) + I(r f |∂ M

)= L(r f ). (3.21)�

References

[1] J. C. Becker and D. H. Gottlieb, Vector fields and transfers, Manuscripta Mathematica 72 (1991),no. 2, 111–130.

[2] C.-F. Benjamin, Fixed point indices, transfers and path fields, Ph.D. thesis, Purdue University,Indiana, 1990.

[3] R. F. Brown, Path fields on manifolds, Transactions of the American Mathematical Society 118(1965), 180–191.

[4] , The Lefschetz Fixed Point Theorem, Scott, Foresman, Illinois, 1971.


[5] A. Dold, Fixed point index and fixed point theorem for Euclidean neighborhood retracts, Topology.An International Journal of Mathematics 4 (1965), 1–8.

[6] , Lectures on Algebraic Topology, Die Grundlehren der mathematischen Wissenschaften,vol. 200, Springer, New York, 1972.

[7] , The fixed point transfer of fibre-preserving maps, Mathematische Zeitschrift 148 (1976),no. 3, 215–244.

[8] E. Fadell, Generalized normal bundles for locally-flat imbeddings, Transactions of the AmericanMathematical Society 114 (1965), 488–513.

[9] D. H. Gottlieb, A de Moivre like formula for fixed point theory, Fixed Point Theory and Its Applica-tions (Berkeley, CA, 1986) (R. F. Brown, ed.), Contemp. Math., vol. 72, American MathematicalSociety, Rhode Island, 1988, pp. 99–105.

[10] , A de Moivre formula for fixed point theory, ATAS de 5◦ Encontro Brasiliero de Topologia53 (1988), 59–67, Universidade de Sao Paulo, Sao Carlos S. P., Brasil.

[11] , On the index of pullback vector fields, Differential Topology (Siegen, 1987) (U.Koschorke, ed.), Lecture Notes in Math., vol. 1350, Springer, Berlin, 1988, pp. 167–170.

[12] , Zeroes of pullback vector fields and fixed point theory for bodies, Algebraic Topology(Evanston, IL, 1988), Contemp. Math., vol. 96, American Mathematical Society, Rhode Island,1989, pp. 163–180.

[13] H. Hopf, Abbildungsklassen n-dimensionaler Mannigfaltigkeiten, Mathematische Annalen 96(1927), no. 1, 209–224 (German).

[14] S. T. Hu, Fibrings of enveloping spaces, Proceedings of the London Mathematical Society. ThirdSeries 11 (1961), 691–707.

[15] M. Morse, Singular points of vector fields under general boundary conditions, American Journal ofMathematics 51 (1929), no. 2, 165–178.

[16] J. Nash, A path space and the Stiefel-Whitney classes, Proceedings of the National Academy ofSciences of the United States of America 41 (1955), 320–321.

[17] C. C. Pugh, A generalized Poincare index formula, Topology. An International Journal of Mathe-matics 7 (1968), 217–226.

Chen-Farng Benjamin: 705 Sugar Hill Drive, West Lafayette, IN 47906, USAE-mail address: [email protected]

Daniel Henry Gottlieb: Mathematics Department, Purdue University, West Lafayette, IN 47907, USAE-mail address: [email protected]

EPSILON NIELSEN FIXED POINT THEORY

ROBERT F. BROWN

Received 11 October 2004; Revised 17 May 2005; Accepted 21 July 2005

Let f : X → X be a map of a compact, connected Riemannian manifold, with or withoutboundary. For ε > 0 sufficiently small, we introduce an ε-Nielsen number Nε( f ) that isa lower bound for the number of fixed points of all self-maps of X that are ε-homotopicto f . We prove that there is always a map g : X → X that is ε-homotopic to f such that ghas exactly Nε( f ) fixed points. We describe procedures for calculating Nε( f ) for maps of1-manifolds.

Copyright © 2006 Robert F. Brown. This is an open access article distributed under theCreative Commons Attribution License, which permits unrestricted use, distribution,and reproduction in any medium, provided the original work is properly cited.

1. Introduction

Forster has applied Nielsen fixed point theory to the study of the calculation by computerof multiple solutions of systems of polynomial equations, using a Nielsen number toobtain a lower bound for the number of distinct solutions [4]. Because machine accuracyis finite, the solution procedure requires approximations, but Forster’s information is stillapplicable to the original problem. The reason is that sufficiently close functions on well-behaved spaces are homotopic and the Nielsen number is a homotopy invariant.

The point of view of numerical analysis concerning accuracy is described by Hilde-brand in his classic text [5] in the following way. “Generally the numerical analyst doesnot strive for exactness. Instead, he attempts to devise a method which will yield an ap-proximation differing from exactness by less than a specified tolerance.” The work ofForster does not involve an initially specified tolerance. In particular, although the homo-topy between two sufficiently close maps is through maps that are close to both, Forsterputs no limitation on the homotopies he employs. The purpose of this paper is to intro-duce a type of Nielsen fixed point theory that does assume that a specified tolerance forerror must be respected.

If distortion is limited to a pre-assigned amount, then it may not be possible, withoutexceeding the limit, to deform a map f so that it has exactly N( f ) fixed points. For avery simple example, consider a map f : I → I = [0,1] such that f (0) = f (2/3) = 1 and


2 Epsilon Nielsen fixed point theory

f (1/3) = f (1) = 0. If a map g has N( f ) = 1 fixed point, then there must be some t ∈ Isuch that | f (t)− g(t)| > 1/3.

This example suggests a concept of the geometric minimum (fixed point) number ofa map f : X → X different from the one, MF[ f ], that is the focus of Nielsen fixed pointtheory, namely,

MF[ f ]=min{

#Fix(g) : g is homotopic to f}

, (1.1)

where #Fix(g) denotes the cardinality of the fixed point set. The distance d( f ,g) betweenmaps f ,g : Z → X , where Z is compact and X is a metric space with distance function d,is defined by

d( f ,g)=max{d(f (z),g(z)

): z ∈ Z}. (1.2)

Given ε > 0, a homotopy {ht} : Z → X is an ε-homotopy if d(ht,ht′) < ε for all t, t′ ∈ I . Fora given ε > 0, we define the ε-minimum (fixed point) number MFε( f ) of a map f : X → Xof a compact metric space by

MFε( f )=min{

#(

Fix(g))

: g is ε-homotopic to f}. (1.3)

Note that the concept of ε-homotopic maps does not give an equivalence relation.The notationMF[ f ] for the minimum number incorporates the symbol [ f ], generally

used to denote the homotopy class of f , because MF[ f ] is a homotopy invariant. Wedo not use the corresponding notation for the ε-minimum number because it is notinvariant on the homotopy class of f . For instance, although a constant map k of I ishomotopic to the map f of the example, for whichMFε( f )= 3 for any ε ≤ 1/3, obviouslyMFε(k)= 1 for any choice of ε.

Let f : X → X be a map of a compact manifold. Just as the Nielsen number N( f ) hasthe property N( f ) ≤MF[ f ], in the next section we will introduce the ε-Nielsen num-ber Nε( f ), for ε sufficiently small, that has the property Nε( f ) ≤MFε( f ). Our mainresult, proved in Section 3, is a “minimum theorem”: given f : X → X , there exists g withd(g, f ) < ε such that g has exactly Nε( f ) fixed points. Wecken’s minimum theorem, thatif f : X → X is a map of an n-manifold, then there is a map g homotopic to f with exactlyN( f ) fixed points, requires that n�= 2. It is well known that on all but a few surfaces thereare maps f for which no map homotopic to f has onlyN( f ) fixed points, and indeed thegap between MF[ f ] and N( f ) can be made arbitrarily large [2]. In contrast to Wecken’stheorem, our result holds for manifolds of all dimensions. Finally, in Section 4, we discussthe problem of calculating Nε( f ).

2. The ε-Nielsen number

Throughout this paper, X is a compact, connected differentiable manifold, possibly withboundary. We introduce a Riemannian metric on X and denote the associated distancefunction by d. If the boundary of X is nonempty, we choose a product metric on a tubu-lar neighborhood of the boundary and then use a partition of unity to extend to a metricfor X . There is an ε > 0 small enough so that, if p,q ∈ X with d(p,q) < ε, then there

Robert F. Brown 3

is a unique geodesic cpq connecting them. This choice of ε is possible even though themanifold may have a nonempty boundary because the metric is a product on a neighbor-hood of the boundary. For the rest of this paper, ε > 0 will always be small enough so thatpoints within a distance of ε are connected by a unique geodesic. We view the geodesicbetween p and q as a path cpq(t) in X such that cpq(0)= p and cpq(1)= q. The functionthat takes the pair (p,q) to cpq is continuous. If x ∈ cpq then d(p,x)≤ d(p,q) because cpqis the shortest path from p to q (see [7, Corollary 10.8 on page 62]).

If f ,g : Z → X are maps with d( f ,g) < ε, then setting ht(z)= c f (z)g(z)(t) defines an ε-homotopy between f and g. Thus an equivalent definition of the ε-minimum number off : X → X is

MFε( f )=min{

#(

Fix(g))

: d( f ,g) < ε}. (2.1)

For a map f : X → X , let

Δε( f )= {x ∈ X : d(x, f (x)

)< ε

}. (2.2)

Theorem 2.1. The set Δε( f ) is open in X .

Proof. Let R+ denote the subspace of R of non-negative real numbers. Define Df : X →R+ by Df (x) = d(x, f (x)). Since [0,ε) is open in R+, it follows that Δε( f )= D−1

f ([0,ε))is an open subset of X . �

For a map f : X → X , define an equivalence relation on Fix( f ) as follows: x, y ∈ Fix( f )are ε-equivalent, if there is a path w : I → X from x to y such that d(w, f ◦w) < ε. Theequivalence classes will be called the ε-fixed point classes or, more briefly, the ε-fpc of f .

Theorem 2.2. Fixed points x, y of f : X → X are ε-equivalent if and only if there is acomponent of Δε( f ) that contains both of them.

Proof. Suppose x, y ∈ Fix( f ) are ε-equivalent and let w be a path in X from x to y suchthat d(w, f ◦w) < ε. Thus, for each s ∈ I we have d(w(s), f (w(s))) < ε and we see thatw(I)⊂ Δε( f ). Since w(I) is connected it is contained in some component of Δε( f ). Con-versely, suppose x, y ∈ Fix( f ) are in the same component of Δε( f ). The components ofΔε( f ) are pathwise connected so there is a path w in it from x to y. Since w is in Δε( f ),that means d(w, f ◦w) < ε and thus x and y are ε-equivalent. �

Theorems 2.1 and 2.2 imply that the ε-fpc are open in Fix( f ), so there are finitelymany of them Fε1, . . . ,Fεr . We denote the component of Δε( f ) that contains Fεj by Δεj ( f ).An ε-fpc Fεj = Fix( f )∩Δεj ( f ) is essential if the fixed point index i( f ,Δεj ( f )) �= 0. Theε-Nielsen number of f , denoted by Nε( f ), is the number of essential ε-fpc.

Theorem 2.3. If fixed points x and y of f : X → X are ε-equivalent, then x and y are inthe same (Nielsen) fixed point class. Therefore each fixed point class is a union of ε-fpc andNε( f )≥N( f ).

Proof. If x and y are ε-equivalent by means of a path w between them such that d(w, f ◦w) < ε then ht(s) = cw(s) f (w(s))(t) defines a homotopy, relative to the endpoints, betweenw and f ◦w so x and y are in the same fixed point class. Therefore a fixed point class


F of f is the union of ε-fpcs. If F is essential, the additivity property of the fixed pointindex implies that at least one of the ε-fpc it contains must be an essential ε-fpc. ThusNε( f )≥N( f ). �

The ε-Nielsen number is a local Nielsen number in the sense of [3], specificallyNε( f )= n( f ,Δε( f )). However, in local Nielsen theory, the domainU of the local Nielsennumber n( f ,U) is the same for all the maps considered whereas Δε( f ) depends on f .

Theorem 2.4. Let f : X → X be a map, then Nε( f )≤MFε( f ).

Proof. Given a map g : X → X with d( f ,g) < ε, let {ht} : X → X be the ε-homotopy withh0 = f and h1 = g defined by ht(x)= c f (x)g(x)(t). Theorem 2.1 implies that d(x, f (x))≥ εfor all x in the boundary of Δεj ( f ). Thus for x in the boundary of Δεj ( f ) and t ∈ I we have

d(x,ht(x)

)+d(ht(x), f (x)

)≥ d(x, f (x))≥ ε. (2.3)

Since {ht} is an ε-homotopy, d(ht(x), f (x)) = d(ht(x),h0(x)) < ε so d(x,ht(x)) > 0, thatis, ht has no fixed points on the boundary of Δεj ( f ). Therefore the homotopy property ofthe fixed point index implies that

i(f ,Δεj ( f )

)= i(g,Δεj ( f )). (2.4)

Consequently, if Fεj = Fix( f )∩Δεj ( f ) is an essential ε-fpc, then i(g ,Δεj ( f ))�= 0 so g has afixed point in Δεj ( f ). We conclude that g has at least Nε( f ) fixed points. �

Although Theorem 2.4 tells us that Nε( f ) is a lower bound for the number of fixedpoints of all maps g that are ε-homotopic to f , the number Nε( f ) is not itself invariantunder ε-homotopies. In fact it fails to be invariant under ζ-homotopies for ζ > 0 arbi-trarily small, as the following example demonstrates.

Example 2.5. Let f : I → I be the map whose graph is the solid line in Figure 2.1. Letg : I → I equal f except on the interval [p,q], where the graph of g is the line segmentconnecting (p, f (p)) and (q, f (q)). Given ζ > 0, we can adjust f so that setting ht(x) =tg(x) + (1− t) f (x) for x ∈ [p,q] and ht(x)= f (x)= g(x) elsewhere defines a ζ-homotopybetween f and g. However, Nε( f )= 3 whereas Nε(g)= 1.

Since N( f ) = 1 for any map of the interval I , this example also demonstrates that3=Nε( f ) > N( f ). For an example where Nε( f ) > N( f ) in which N( f ) > 1, we considerthe map of the circle described by Figure 2.2. (Note: the line in Figure 2.2 labelled ΓC isnot relevant to the description of the example. However, we will need it in Section 4 forthe algorithm that computes Nε( f ) for maps of the circle.)

Example 2.6. The circle S1 is represented in Figure 2.2 as I/{0,1}. The map f is of degree−3 soN( f )= 4. There are a total of seven ε-fpc; these consist of single fixed points exceptfor the ε-fpc on the left which is essential and the one on the right which is not. Thus wehave Nε( f )= 6.

Robert F. Brown 5

ε {

Δε1( f ) Δε2( f ) Δε3( f ) = Δε2(g)

Δε1(g)

I 0

︸︷︷︸ε p q

1� � � �

f

Figure 2.1. Map of the interval.

ε {

ε {

ε {0

1

S1 0 1

︸︷︷︸ε

︸︷︷︸ε

︸︷︷︸ε

︸︷︷︸︸︷︷︸� � � � � � � � � �

f

ΓC

Fix ( f )

v

Figure 2.2. Map of the circle.

3. The minimum theorem

Lemma 3.1. Let F be a closed subset of a compact manifoldX and letU be an open, connectedsubset of X that contains F, then there is an open, connected subset V of X containing F suchthat the closure of V is contained in U .


Proof. Since F and X −U are disjoint compact sets, there is an open set W containingF such that the closure of W is contained in U . There are finitely many componentsW1, . . . ,Wr of W that contain points of the compact set F. Let a1 be a path in U fromx1 ∈W1 ∩ F to x2 ∈W2 ∩ F and let A1 be an open subset of U containing a1 such thatthe closure of A1 is in U . Since a1 is connected, we may assume A1 is also connected.Continuing in this manner, we let

V =W1∪A1∪W2∪A2∪···∪Wr , (3.1)

which is connected. The closures of each of the Wi and Ai are in U so the closure of V isalso in U . �

Let Fεj = Fix( f )∩Δεj ( f ) be an ε-fpc. By Lemma 3.1, there is an open, connected subsetVj of Δεj ( f ) containing Fεj whose closure cl(Vj) is in Δεj ( f ). For the map Df : X → R+

defined by Df (x)= d(x, f (x)), we see that Df (cl(Vj))= [0,δj] where δj < ε. Choose αj >0 small enough so that δj + 2αj < ε.

Theorem 3.2 (Minimum Theorem). Given f : X → X , there exists g : X → X with d(g , f )< ε such that g has exactly Nε( f ) fixed points.

Proof. We will define g outside Δε( f ) to be a simplicial approximation to f such thatd(g, f ) < α, where α denotes the minimum of the αj . The proof then consists of describingg on each Δεj ( f ) so, to simplify notation, we will assume for now that Δε( f ) is connectedand thus we are able to suppress the subscript j. Triangulate X and take a subdivisionof such small mesh that if u is a simplicial approximation to f with respect to that tri-angulation, then d(u, f ) < α/2 and, for σ a simplex that intersects X − int(V), we haveu(σ)∩ σ =∅. By the Hopf construction, we may modify u, moving no point more thanα/2, so that it has finitely many fixed points, each of which lies in a maximal simplex in Vand therefore in the interior of X (see [1, Theorem 2 on page 118]). We will still call themodified map u, so we now have a map u with finitely many fixed points and it has theproperty that d(u, f ) < α.

Refine the triangulation of X so that the fixed points of u are vertices. Since V is aconnected n-manifold, we may connect the fixed points of u by paths in V , let P be theunion of all these paths. With respect to a sufficiently fine subdivision of the triangulationof X , the star neighborhood S(P) of P, which is a finite, connected polyhedron, has theproperty that the derived neighborhood of S(P) lies in V . Let T be a spanning tree for thefinite connected graph that is the 1-skeleton of S(P), then T contains Fix(u). Let R(T) bea regular neighborhood of T in V ∩ int(X) then, since T is collapsible, R(T) is an n-ballby [8, Corollary 3.27 on page 41]. Thus we have a subset W = int(R(T)) of V containingFix(u) and a homeomorphism φ : W → Rn. We may assume that φ(Fix(u)) lies in theinterior of the unit ball in Rn, which we denote by B1. Set φ−1(B1)= B∗1 . If x ∈ B∗1 , then

d(x,u(x)

)≤ d(x, f (x))

+d(f (x),u(x)

)< δ +α < ε (3.2)

so there is a unique geodesic cxu(x) connecting x to u(x). Consider the map H : B∗1 × I →X defined by H(x, t) = cxu(x)(t), then H−1(W) is an open subset of B∗1 × I containingB∗1 ×{0}. Therefore, there exists t0 > 0 such that H(B∗1 × [0, t0])⊂W .

Robert F. Brown 7

Denote the origin in Rn by 0 and let 0∗ = φ−1(0). Define a retraction ρ : B∗1 − 0∗ →∂B∗1 , the boundary of B∗1 , by

ρ(x)= φ−1

(1∣∣φ(x)∣∣φ(x)

). (3.3)

Define K : B∗1 × [0, t0]→W by setting K(0∗, t)= 0∗ for all t and, otherwise, let

K(x, t)= φ−1(∣∣φ(x)∣∣φ(H(ρ(x), t

))). (3.4)

The function K is continuous because φ(H(∂B∗1 × I)) is a bounded subset of Rn. NowdefineDK : B∗1 × [0, t0]→R+ byDK (x, t)= d(x,K(x, t)). SinceD−1

K ([0,η)) is an open sub-set of B∗1 × [0, t0] containing B∗1 ×{0}, there exists 0 < t1 < t0 such that d(x,K(x, t1)) < α.Define v : B∗1 → X by v(x)= K(x, t1).

Next we extend v to the set B∗2 consisting of x ∈W such that 0≤ |φ(x)| ≤ 2 by letting

v(x)= cxu(x)((

1− t1)∣∣φ(x)

∣∣+ 2t1− 1)

(3.5)

when 1≤ |φ(x)| ≤ 2. Noting that v(x)= u(x) if φ(x)= 2, we extend v to all ofX by settingv = u outside B∗2 .

The map v has a single fixed point at 0∗. If i( f ,Δε( f )) �= 0, we let g = v : X → X . Ifi( f ,Δε( f )) = 0, by [1, Theorem 4 on page 123], there is a map g : X → X , identical to voutside of B∗1 , such that g has no fixed point in B∗1 and d(g ,v) < α.

We claim that d(g, f ) < ε. For x �∈ B∗2 , we defined g(x)= u(x) where d(u, f ) < α < ε. Ifx ∈ B∗2 −B∗1 , then g(x)= v(x)∈ cxu(x) so d(v(x),u(x))≤ d(x,u(x)). Therefore,

d(g(x), f (x)

)= d(v(x), f (x))

≤ d(v(x),u(x))

+d(u(x), f (x)

)

≤ d(x,u(x))

+d(u(x), f (x)

)

≤ (d(x, f (x))

+d(f (x),u(x)

))+d(u(x), f (x)

)

< δ + 2α < ε.

(3.6)

Now suppose x ∈ B∗1 . If i( f ,Δε( f ))�= 0, then g(x)= v(x)= K(x, t1) so

d(g(x), f (x)

)= d(K(x, t1), f (x)

)

≤ d(K(x, t1),x)

+d(x, f (x)

)

< α+ δ < ε.(3.7)

If i( f ,Δε( f ))= 0 then

d(g(x), f (x)

)≤ d(g(x),v(x))

+d(v(x), f (x)

)

= d(g(x),v(x))

+d(K(x, t1

), f (x)

)

≤ α+ (α+ δ) < ε

(3.8)

which completes the proof that d(g, f ) < ε.


We return now to the general case, in which Δε( f ) may not be connected. Applyingthe construction above to each Δεj ( f ) gives us a map g : X → X with exactly Nε( f ) fixedpoints. For x �∈ Δε( f ) we defined g to be a simplicial approximation with d(g(x), f (x)) <α < ε. For x ∈ Δεj ( f ), the argument just concluded proves that

d(g(x), f (x)

)≤ 2αj + δj < ε (3.9)

because α is the minimum of the αj , so we know that d(g , f ) < ε. �

Theorem 3.2 throws some light on the failure of the Wecken property for surfaces [2].For instance, consider the celebrated example of Jiang [6], of a map f of the pants surfacewith N( f ) = 0 but MF[ f ] = 2. The fixed point set of f consists of three points, oneof them of index zero. The other two fixed points, y1 and y2, are of index +1 and −1respectively and Jiang described a path, call it σ , from y1 to y2 such that σ is homotopicto f ◦ σ relative to the endpoints. Suppose ε > 0 is small enough so that points in thepants surface that are within ε of each other are connected by a unique geodesic. If therewere a path τ from y1 to y2 such that τ and f ◦ τ were ε-homotopic, then Nε( f )= 0 andtherefore, by Theorem 3.2, there would be a fixed point free map homotopic to f . SinceJiang proved that no map homotopic to f can be fixed point free, we conclude that nosuch path τ exists. In other words, for any path τ from y1 to y2 that is homotopic to f ◦ τrelative to the endpoints, it must be that d(τ, f ◦ τ) > ε.

4. Calculation of the ε-Nielsen number

In some cases, the ε-Nielsen theory does not differ from the usual theory. If a map f :X → X has only one fixed point, as a constant map does for example, then there is onlyone ε-fpc soNε( f )=N( f ). For another instance, let 1X : X → X denote the identity map.Again we have only one ε-fpc for any ε > 0 because Δε(1X)= X .

However, in general we would expect Nε( f ) > N( f ) and Example 2.5 can easily bemodified to produce a map of the interval for which Nε( f )−N( f ) is arbitrarily large.The problem of calculating the ε-Nielsen number appears to be even more difficult thanthat for the usual Nielsen number because Nε( f ) is not homotopy invariant so it doesnot seem that the tools of algebraic topology can be applied. The goal then is to obtainenough information from the given map f itself to determine Nε( f ). As in the usualNielsen theory, even a complete description of the fixed point set Fix( f ) is not sufficient,except in extreme cases such as those we noted, without information about the fixed pointclass structure on Fix( f ), which generally has to be obtained in some indirect manner.

We will next present a procedure that determines Nε( f ) for a map f : I → I just bysolving equations involving the map f itself. Let Aε denote the set of solutions to theequation f (x) = x + ε and let Bε be the set of solutions to f (x) = x− ε. Thus, lookingback at Figure 2.1, Aε corresponds to the intersection of the graph of f and the boundaryof the ε-neighborhood of the diagonal that lies above the diagonal and Bε correspondsto the intersection of the graph of f and the boundary of that neighborhood that liesbelow the diagonal. The set Qε = I − (Aε ∪ Bε) is a union of intervals that are open inI . We define the essential intervals in Qε to be the intervals with one endpoint in Aε and

Robert F. Brown 9

the other in Bε together with the interval [0,x) if x ∈ Bε and (x,1] if x ∈ Aε. AlthoughQε may consist of infinitely many intervals, only finitely many of them can be essential.Otherwise, let Aε0 ⊆ Aε be the endpoints of the essential intervals, then Aε0 contains asequence converging to some point a0 ∈ Aε. Thus every neighborhood of a0 containspoints of Aε0, but it also contains points of Bε, which contradicts the continuity of f . Thereason is that, by the definition of essential interval, for any set of three successive pointsa1 < a2 < a3 in Aε0 there must be at least one b ∈ Bε such that a1 < b < a3.

We claim that Nε( f ) equals the number of essential intervals. Note that Δε( f ) is aunion of intervals of Qε. For J an interval of Qε, we write its closure as cl(J)= [ j0, j1]. IfJ is an essential interval, then one of the points ( j0, f ( j0)) and ( j1, f ( j1)) must lie abovethe diagonal and the other below it, and therefore f has a fixed point in J . The graph off restricted to cl(J) can be deformed vertically, keeping the endpoints fixed, to the linesegment connecting ( j0, f ( j0)) and ( j1, f ( j1)) so, by the homotopy property of the fixedpoint index, i( f , J)=±1. Now let K be an interval of Qε that is not essential and write itsclosure as cl(K)= [k0,k1]. Either both of (k0, f (k0)) and (k1, f (k1)) lie above the diagonalor both lie below it and thus the restriction of the graph of f to cl(K) can be deformedvertically, keeping the endpoints fixed, to the line segment connecting (k0, f (k0)) and(k1, f (k1)). Since the components of the complement of the diagonal in I × I are convex,the line segment does not intersect the diagonal and therefore i( f ,K)= 0. We have provedthat the essential intervals in Qε ⊂ I are the Δεj that contain the essential ε-fpc of f andthat establishes our claim.

For an example of the use of this procedure, we return to Example 2.5, pictured inFigure 2.1. Denoting points of I that lie in Aε by a and those in Bε by b then, in theordering of I we have

0 < (a < b) < (b < a) < (a < b) < 1. (4.1)

We note that there are three essential intervals, as indicated by the parentheses, so againwe have Nε( f )= 3.

A modification of the previous procedure can be used for maps f : S1 → S1. In this case,the set of points (x, y)∈ S1× S1 such that d(x, y)= ε is the union of two disjoint simpleclosed curves, which we will call ΓA and ΓB, on the torus. We denote by Aε ⊂ S1 the pointsx such that (x, f (x))∈ ΓA and by Bε the points x ∈ S1 such that (x, f (x))∈ ΓB. Since thecomplement of the diagonal in S1 × S1 is connected, if an interval in S1 − (Aε ∪Bε) hasone endpoint in each of those sets, it does not necessarily contain a fixed point of f . Thus,in order to identify intervals of that type that do contain fixed points, we consider the setof points (x, y) ∈ S1 × S1 such that d(x, y) = 2ε. This set is the union of two disjointsimple closed curves and we choose one of them arbitrarily, calling it ΓC (see Figure 2.2).Denote by Cε ⊂ S1 the points x such that (x, f (x))∈ ΓC. The setQε = S1− (Aε ∪Bε ∪Cε)is a union of connected open subsets of S1 which we will refer to as open intervals in S1.Now we may call an open interval essential if one of its endpoints is in Aε and the other inBε. Again, there are only finitely many essential intervals and the number of them equalsNε( f ). The reasoning is similar to that for maps of the interval. An essential interval Jdoes contain fixed points and a homotopy shows that i( f , J)=±1. If an interval K in S1

is not essential, that means either that at least one of the endpoints of K lies in Cε or both


its endpoints lie in one of the sets Aε or Bε. Then there is a homotopy of f to a map thatis identical to f outside of K but has no fixed points inK , so we conclude that i( f ,K)= 0.

Referring to Figure 2.2 for Example 2.6, we can write

(1=)0 < (a < b) < c < (a < b) < (b < a) < (a < b)

< c < (a < b) < c < (a < b) < b < b < c < 1(= 0)(4.2)

and conclude that Nε( f )= 6.

5. Acknowledgments

I thank Robert Greene for geometric advice. I am also grateful to the referee whose con-scientious review lead to significant improvements in this paper.

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nois Journal of Mathematics 25 (1981), no. 4, 673–699.[4] W. Forster, Computing “all” solutions of systems of polynomial equations by simplicial fixed point

algorithms, The Computation and Modelling of Economic Equilibria (Tilburg, 1985) (D. Talmanand G. van der Laan, eds.), Contrib. Econom. Anal., vol. 167, North-Holland, Amsterdam, 1987,pp. 39–57.

[5] F. B. Hildebrand, Introduction to Numerical Analysis, 2nd ed., International Series in Pure andApplied Mathematics, McGraw-Hill, New York, 1974.

[6] B. Jiang, Fixed points and braids, Inventiones Mathematicae 75 (1984), no. 1, 69–74.[7] J. Milnor, Morse Theory, Annals of Mathematics Studies, no. 51, Princeton University Press, New

Jersey, 1963.[8] C. P. Rourke and B. J. Sanderson, Introduction to Piecewise-Linear Topology, Ergebnisse der

Mathematik und ihrer Grenzgebiete, vol. 69, Springer, New York, 1972.

Robert F. Brown: Department of Mathematics, University of California, Los Angeles,CA 90095-1555, USAE-mail address: [email protected]

A BASE-POINT-FREE DEFINITION OFTHE LEFSCHETZ INVARIANT

VESTA COUFAL

Received 30 November 2004; Accepted 21 July 2005

In classical Lefschetz-Nielsen theory, one defines the Lefschetz invariant L( f ) of an endo-morphism f of a manifold M. The definition depends on the fundamental group of M,and hence on choosing a base point∗∈M and a base path from∗ to f (∗). At times, it isinconvenient or impossible to make these choices. In this paper, we use the fundamentalgroupoid to define a base-point-free version of the Lefschetz invariant.

Copyright © 2006 Vesta Coufal. This is an open access article distributed under the Cre-ative Commons Attribution License, which permits unrestricted use, distribution, andreproduction in any medium, provided the original work is properly cited.

1. Introduction

In classical Lefschetz fixed point theory [3], one considers an endomorphism f :M→Mof a compact, connected polyhedron M. Lefschetz used an elementary trace construc-tion to define the Lefschetz invariant L( f )∈ Z. The Hopf-Lefschetz theorem states that ifL( f )�= 0, then every map homotopic to f has a fixed point. The converse is false. How-ever, a converse can be achieved by strengthening the invariant. To begin, one choosesa base point ∗ of M and a base path τ from ∗ to f (∗). Then, using the fundamen-tal group and an advanced trace construction one defines a Lefschetz-Nielsen invariantL( f ,∗,τ), which is an element of a zero-dimensional Hochschild homology group [4].Wecken proved that when M is a compact manifold of dimension n > 2, L( f ,∗,τ)= 0 ifand only if f is homotopic to a map with no fixed points.

We wish to extend Lefschetz-Nielsen theory to a family of manifolds and endomor-phisms, that is, a smooth fiber bundle p : E→ B together with a map f : E→ E such thatp = p ◦ f . One problem with extending the definitions comes from choosing base pointsin the fibers, that is, a section s of p, and the fact that f is not necessarily fiber homotopicto a map which fixes the base points (as is the case for a single path connected space and asingle endomorphism.) To avoid this difficulty, we reformulate the classical definitions ofthe Lefschetz-Nielsen invariant by employing a trace construction over the fundamentalgroupoid, rather than the fundamental group.


2 A base-point-free definition of the Lefschetz invariant

In Section 2, we describe the classical (strengthened) Lefschetz-Nielsen invariant fol-lowing the treatment given by Geoghegan [4] (see also Jiang [6], Brown [3] and Luck[8]). We also introduce the Hattori-Stallings trace, which will replace the usual trace inthe construction of the algebraic invariant.

In Section 3, we develop the background necessary to explain our base-point-free def-initions. This includes the general theory of groupoids and modules over ringoids, as wellas our version of the Hattori-Stallings trace.

In Section 4, we present our base-point-free definitions of the Lefschetz-Nielsen in-variant, and show that they are equivalent to the classical definitions.

2. The classical theory

2.1. The geometric invariant. In this section, Mn is a compact, connected manifold ofdimension n, and f :M→M is a continuous endomorphism.

The concatenation of two paths α : I → X and β : I → X such that α(1)= β(0) is definedby

α ·β(t)=

⎧⎪⎪⎨⎪⎪⎩

α(2t) if 0≤ t ≤ 12

,

β(2t− 1) if12≤ t ≤ 1.

(2.1)

The fixed point set of f is

Fix( f )= {x ∈M | f (x)= x}. (2.2)

Note that Fix( f ) is compact. Define an equivalence relation ∼ on Fix( f ) by letting x ∼ yif there is a path ν in M from x to y such that ν · ( f ◦ ν)−1 is homotopic to a constantpath.

Choose a base point∗∈M and a base path τ from∗ to f (∗). Let π = π1(M,∗). Giventhese choices, f induces a homomorphism

φ : π −→ π (2.3)

defined by

φ([w]

)= [τ · ( f ◦w) · τ−1], (2.4)

where [w] is the homotopy class of a path w rel endpoints. Define an equivalence relationon π by saying g,h ∈ π are equivalent if there is some w ∈ π such that h = wgφ(w)−1.The equivalence classes are called semiconjugacy classes; denote the set of semiconjugacyclasses by πφ.

Define a map

Φ : Fix( f )−→ πφ (2.5)

by

x −→ [μ · ( f ◦μ)−1 · τ−1], (2.6)

Vesta Coufal 3

where x ∈ Fix( f ) and μ is a path in M from ∗ to x. This map is well-defined and inducesan injection

Φ : Fix( f )/ ∼−→ πφ. (2.7)

It follows that Fix( f )/ ∼ is compact and discrete, and hence finite. Denote the fixed pointclasses by F1, . . . ,Fs.

Next, assume that the fixed point set of f is finite. Let x be a fixed point. Let U be anopen neighborhood of x in M and h :U →Rn a chart. Let V be an open n-ball neighbor-hood of x in U such that f (V) ⊂ U . Then the fixed point index of f at x, i( f ,x), is thedegree of the map of pairs

(id−h f h−1) :

(h(V),h(V)− {h(x)

})−→ (Rn,Rn−{0}). (2.8)

For a fixed point class Fk, define

i( f ,Fk)=∑

x∈Fki( f ,x)∈ Z. (2.9)

Definition 2.1. The classical geometric Lefschetz invariant of f with respect to the basepoint ∗ and the base path τ is

Lgeo( f ,∗,τ)=s∑

k=1

i( f ,Fk)Φ(Fk)∈ Zπφ, (2.10)

where Zπφ is the free abelian group generated by the set πφ.

2.2. The algebraic invariant. To construct the classical algebraic Lefschetz invariant, letM be a finite connected CW complex and f : M →M a cellular map. Again, choose abase point ∗ ∈M (a vertex of M) and a base path τ from ∗ to f (∗). Also, choose anorientation on each cell in M.

Let p : M →M be the universal cover of M. The CW structure on M lifts to a CWstructure on M. Choose a lift of the base point ∗ to a base point ∗ ∈ M, and lift the basepath τ to a path τ such that τ(0)= ∗. Then f lifts to a cellular map f : M→ M such that

f (∗)= τ(1).The group π = π1(M,∗) acts on M on the left by covering transformations. For each

cell σ in M, choose a lift σ in M and orient it compatibly with σ . Take the cellular chaincomplex C(M) of M. The action of π on M makes Ck(M) into a finitely generated freeleft Zπ-module with basis given by the chosen lifts of the oriented k-cells of M.

As in the geometric construction, f and τ induce a homomorphism φ : π → π. Since

f is cellular, it induces a chain map fk : Ck(M)→ Ck(M) which is φ-linear, namely if σ

is a k-cell of M and g ∈ π then fk(gσ) = φ(g) fk(σ). Classically, one represents fk by a

matrix over Zπ whose (i, j) entry is the coefficient of σ j in the chain fk(σi), where σi and

σ j are k-cells. For each k, one can now take the trace of fk, that is, the sum of the diagonal

entries of the matrix which represents fk.


Definition 2.2. The classical algebraic Lefschetz invariant of f with respect to the basepoint ∗ and the base path τ is

Lalg( f ,∗,τ)=∑

k≥0

(−1)kq(

trace(fk))∈ Zπφ, (2.11)

where q : Zπ → Zπφ is the map sending g ∈ π to its semiconjugacy class.

2.3. Hattori-Stallings trace. In the classical algebraic construction of the Lefschetz in-

variant above, Reidemeister viewed fk as a matrix and took its trace, the sum of thediagonal entries, to define Lalg( f ). In our generalizations, we will need to use a moresophisticated trace map, namely the Hattori-Stallings trace. Since on finitely generatedfree modules, the Hattori-Stallings trace agrees with the usual trace of a matrix, we coulduse it in the classical case as well. We introduce the classical Hattori-Stallings trace here.(For the special case when M = R, see [1, 2, 9].)

Let R be a ring, M an R-bimodule, and P a finitely generated projective left R-module.Let P∗ = HomR(P,R) be the dual of P. Let [R,M] denote the abelian subgroup of Mgenerated by elements of the form rm−mr, for r ∈ R and m∈M. The Hattori-Stallingstrace map, tr is given by the following composition:

HomR(P,M⊗R P

)

tr

P∗ ⊗R M⊗R P∼=

M/[R,M]

HH0(R;M)

(2.12)

The map P∗ ⊗R M ⊗R P →HomR(P,M ⊗R P) is given by α⊗m⊗ p → (p1 → α(p1)(m⊗p)). The map P∗ ⊗R M⊗R P→M/[R,M] is given by α⊗m⊗ p → α(p)m.

The fact that the first map is an isomorphism is an application of the following lemma.

Lemma 2.3. Let R be a ring, P a finitely generated projective right R-module, and N aleft R-module. Define fP : P∗ ⊗R N →HomR(P,N) by fP(α,n)(p) = α(p)n. Then fP is anisomorphism of groups.

Proof. Note that fR : R∗ ⊗R N →HomR(R,N) is an isomorphism with inverse given by (g :R→N)→ idR⊗Rg(1R). The result follows from the fact that f(−) : (−)∗ ⊗R N →HomR(−,N) preserves finite direct sums. �

3. Background on groups and ringoids

In this section, we generalize to the “oid” setting the basic algebraic definitions and re-sults which we will need for our constructions. This treatment is based on [7, Section 9],though we have developed additional material as needed. In particular, in Section 3.2, wegeneralize the Hattori-Stallings trace.

We use the following notation. If C is a category, denote the collection of objects in Cby Ob(C). If x and y are objects in C, denote the collection of maps from x to y in C byC(x, y). The category of sets will be denoted Sets, the category of abelian groups will bedenoted Ab, and the category of left R-modules will be denoted R-mod.

Throughout, “ring” will mean an associative ring with unit.

Vesta Coufal 5

3.1. General definitions and results

3.1.1. Groupoids and ringoids. Let G be a group. We may view G as a category, denotedby G, in which there is one object ∗, and for which all of the maps are isomorphisms.Each map corresponds to an element of G with composition of maps corresponding tothe multiplication in the group. This idea generalizes to define a groupoid.

Definition 3.1. A groupoid G is a small category (the objects form a set) such that allmaps are isomorphisms.

The analogous game can be played with rings in order to define a ringoid, also knownas a linear category or as a small category enriched in the category of abelian groups.

Definition 3.2. A ringoid � is a small category such that for each pair of objects x and y,�(x, y) is an abelian group and the composition function �(y,z)×�(x, y)→�(x,z) isbilinear.

Example 3.3. Recall that if H is a group, then the group ring ZH is the free abelian groupgenerated by H . This group ring construction can be generalized to a “groupoid ringoid”(though we will call it the group ring): let G be a groupoid and R a ring. The group ringof G with respect to R, denoted RG, is the category with the same objects as G, but withmaps given by RG(x, y)= R(G(x, y)), the free R-module generated by the set G(x, y).

3.1.2. Modules. For the remainder of this paper, unless otherwise noted, letG be a group-oid and let R be a commutative ring. While much of the following can be done in termsof a ringoid �, we will restrict our attention to group rings RG.

Definition 3.4. A left RG-module is a (covariant) functor M : G→ R-mod. A right RG-modules is a (covariant) functors Gop → R-mod.

Definition 3.5. LetM andN be RG-modules. An RG-module homomorphism fromM toN is a natural transformation from M to N . The set of all RG-module homomorphismsfrom M to N is denoted by HomRG(M,N).

Let RG-mod denote the category of left RG-modules, and let mod-RG denote the cat-egory of right RG-modules.

Definition 3.6. LetM andN be RG-modules. The direct sumM⊕N ofM andN is the leftRG-module defined on an object x by (M⊕N)(x)=M(x)⊕N(x) and on a map g : x→ yby (M⊕N)(g)=M(g)⊕N(g).

Definition 3.7. Let N be a left RG-module and M a right RG-module. Define the tensorproduct over RG of M and N to be the abelian group

M⊗RG N = P/Q, (3.1)

where P is the abelian group

P =⊕

x∈Ob(G)

M(x)⊗R N(x), (3.2)


and Q is the subgroup of P generated by

{M( f )(m)⊗n−m⊗N( f )(n) |m∈M(y), n∈N(x), f ∈ RG(x, y)

}. (3.3)

Proposition 3.8. Let M, N , and P be RG-modules. Then

HomRG(M⊕N ,P)∼=HomRG(M,P)⊕HomRG(N ,P). (3.4)

Proposition 3.9. Let M, N , and P be RG-modules. Then

(M⊕N)⊗RG P ∼=(M⊗RG P

)⊕ (N ⊗RG P). (3.5)

Definition 3.10. Given an RG-bimodule M, define M/[RG,M] to be the R-module

(⊕

x∈Ob(G)

M(x,x)

)/{m−M(g,g−1)(m) | g : x −→ y, m∈M(x,x)

}. (3.6)

Call this the zero dimensional Hochschild homology of RG with coefficients in M, de-noted by

HH0(RG;M). (3.7)

Next, we define free RG-modules. First, we need the following notions.Given a category C, we can view Ob(C) as the subcategory of C whose objects are the

same as the objects of C, but whose maps are only the identity maps. A covariant (con-travariant) functor Ob(C)→ Sets will be called a left (right) Ob(C)-set. A map of Ob(C)-sets is a natural transformation. Let Ob(C)-Sets denote the category of left Ob(C)-sets,and let Sets-Ob(C) denote the category of right Ob(C)-sets.

Given either a left or right Ob(C)-set B, let

�=⊔

x∈Ob(C)

B(x), (3.8)

where⊔

denotes disjoint union, and let

β : �−→Ob(C) (3.9)

send b to x if b ∈ B(x). Given Ob(C)-sets B and B′, we say B is an Ob(C)-subset of B′ iffor every x ∈Ob(C), B(x)⊂ B′(x).

Suppose C is a small category and D is a category equipped with a “forgetful functor”D→ Sets. For a functor F : C → D, let |F| : Ob(C)→ Sets be the composition Ob(C)↩C → D → Sets, where the functor D → Sets is the forgetful functor. In particular, |−| :RG-mod→Ob(C)-Sets and |−| : mod-RG→ Sets-Ob(G).

Definition 3.11. For each x ∈ Ob(G), define a left RG-module RGx = RG(x,−) byRGx(y) = RG(x, y). For a map g : y → z in G, let RGx(g) = g ◦ (−). Define a right RG-module RG

x = RG(−,x) similarly.

Vesta Coufal 7

Definition 3.12. Define a functor RG(−) : Ob(G)-Sets→ RG-mod by

RGB =⊕

b∈�

RGβ(b) =⊕

b∈�

RG(β(b),−). (3.10)

Similarly, define RG(−)

: Sets-Ob(G)→mod-RG by

RGB =

⊕

b∈�

RGβ(b) =

⊕

b∈�

RG(−,β(b)

). (3.11)

Proposition 3.13. The functor RG(−) is a left adjoint to the functor |−| : RG-mod →Ob(G)-Sets. The functor RG

(−)is a left adjoint to |−| : mod-RG→ Sets-Ob(G).

Proof. For an Ob(G)-set B and a left RG-module M, define a set map ψ = ψB,M :RG-mod(RGB,M) → Ob(G)-Sets(B,|M|) by ψ(η)y(b) = ηy(idy) ∈ |M(y)|, where η :RGB →M is a natural transformation and b ∈ B(y). Then ψ is a bijection whose inverseis defined in the most obvious way. �

Notice that for each Ob(G)-set B, we get a natural transformation ηB = ψ(idRGB) : B→

|RGB| which is universal. This leads to the following definition of a free RG-module withbase B.

Definition 3.14. An RG-module M is free with base an Ob(G)-set B ⊂ |M| if for eachRG-module N and natural transformation f : B→ |N| there is a unique natural transfor-mation F :M→N with |F| ◦ i= f , where i is the inclusion B→ |M|.Example 3.15. The RG-module RGx is a free left RG-module with base Bx : Ob(G)→ Setsgiven by

Bx(y)=⎧⎨⎩{x} if y = x,

∅ if y �= x. (3.12)

If B is any Ob(G)-set, RGB =⊕

b∈�RGβ(b) =⊕

b∈�RG(β(b),−) is a free RG-module withbase B.

Let M be an RG-module. Let S be an Ob(G)-subset of |M| and let Span(S) be thesmallest RG-submodule of M containing S,

Span(S)=∩{N |N is an RG-submodule of M, S⊂N}. (3.13)

Definition 3.16. Say that M is generated by S if M = Span(S), and M is finitely generatedif S is finite.

Proposition 3.17. If M is a left RG-module, and B is an Ob(G)-subset of |M|, thenSpan(B) is the image of the unique natural transformation τ : RGB →M extending id : B→B ⊂ |M|. Furthermore, M is generated by B if τ is surjective.

Proposition 3.18. Let B be an Ob(G)-set. If M is a free left RG-module with base B, thenM is generated by B. In particular, there is a natural equivalence τ : RGB →M.


Proof. Define τ : RGB →M. For x ∈Ob(G), let

τx : RGB(x)=⊕

b∈�

RG(β(b),x

)−→M(x) (3.14)

be given by (g : β(b) → x) →M(g)(b). To construct an inverse natural transformation,define η : B → |RGB| by setting ηx(b) = idx. Since M is free with base B, η extends to aunique natural transformation M→ RGB. �

Definition 3.19. An RG-module P is projective if it is the direct summand of a free RG-module.

3.1.3. Bimodules.

Definition 3.20. An RG-bimodule is a (covariant) functor

M :G×Gop −→ R-mod. (3.15)

Denote the category of RG-bimodules by RG-bimod.

Example 3.21. Let RG be RG with the following RG-bimodule structure. For (x, y) ∈G×Gop, set RG(x, y) = RG(y,x). Notice the change in the order of x and y. For mapsg : x→ x′ in G and h : y→ y′ in Gop, set RG(g,h)= g ◦ (−)◦h : RG(y,x)→ RG(y′,x′).

We would like to be able to view an RG-bimodule N as either a right or a left RG-module. However, there is no canonical way to do so as each choice of object in G pro-duces a different left and a right RG-module structure on N . Instead, we define two func-tors: (−)ad and ad(−). In essence, N ad encapsulates all of the right RG-module struc-tures on N induced by objects of G, and adN encapsulates all of the left RG-modulestructure on N .

Definition 3.22. Define a covariant functor

(−)ad : RG-bimod−→ (mod-RG)G (3.16)

as follows. Let N be an RG-bimodule. For x ∈Ob(G), let

N ad(x)=N(x,−). (3.17)

For g a map in G, let

N ad(g)=N(g ,−). (3.18)

Explicitly, N ad(x) : Gop → R-mod is given by N ad(x)(y) = N(x, y) and N ad(x)(h) =N(idx,h) for h : y→ z a map in Gop.

Definition 3.23. Define a covariant functor

ad(−) : RG-bimod−→ (RG-mod)Gop

(3.19)

Vesta Coufal 9

as follows. Let N be an RG-bimodule. For x ∈Ob(Gop), let

adN(x)=N(−,x). (3.20)

For g a map in Gop, let

adN(g)=N(−,g). (3.21)

Explicitly, adN(x) : G→ R-mod is given by adN(x)(y)=N(y,x) and adN(x)(h)=N(h,idx) for h : y→ z a map in G.

Example 3.24. Apply the ad functors to the RG-bimodule RG. For instance, if x ∈Ob(G),then adRG(x) = RG(x,−) = RGx. Hence, adRG(x) : G → R-mod, with adRG(x)(y) =RG(x, y) and adRG(x)(h) = h ◦ (−) for h : y → z a map in G. Also, for g : x→ x′ a mapin Gop, adRG(g)= RG(−,g) : RG(x,−)→ RG(x′,−) is the natural transformation of leftRG-modules given by adRG(g)y = (−)◦ g : RG(x, y)→ RG(x′, y).

Next, if N is an RG-bimodule and M is an RG-module, we define HomRG(N ,M),HomRG(M,N), N ⊗RG Ml and Mr ⊗RG N in such a way that they are also RG-modules,as one would expect. Let Ml (resp., Mr) denote a left (resp., right) RG-module.

Definition 3.25. Let N be an RG-bimodule. HomRG(Ml,N) is defined to be the right RG-module given by the composition

Gop adNRG-mod

HomRG(Ml ,−)R-mod. (3.22)

HomRG(N ,Ml) is defined to be the left RG-module given by the composition

Gop adNRG-mod

HomRG(−,Ml)R-mod. (3.23)

HomRG(Mr ,N) is defined to be the left RG-module given by the composition

GN ad

mod-RGHomRG(Mr ,−)

R-mod. (3.24)

HomRG(N ,Mr) is defined to be the right RG-module given by the composition

GN ad

mod-RGHomRG(−,Mr )

R-mod. (3.25)

Definition 3.26. Let N be an RG-bimodule. Define N ⊗RG Ml to be the left RG-modulegiven by the composition

GN ad

mod-RG(−)⊗RGMl

R-mod. (3.26)

Define Mr ⊗RG N to be the right RG-module given by the composition

Gop adNRG-mod

Mr⊗RG(−)R-mod. (3.27)


Applying the above definitions to the RG-bimodule RG, we get the results for Homand tensor product which we would expect from algebra. These next three propositionsjustify viewing RG as “the free rank-one” RG-module. Notice that it is not, however, afree RG-module. The proofs are straightforward and left to the reader.

Proposition 3.27. Given an RG-module M, HomRG(RG,M)∼=M as RG-modules.

Proposition 3.28. Given a left RG-module M, RG⊗RGM ∼=M as left RG-modules.

Proposition 3.29. Given right RG-module M, M⊗RG RG∼=M as right RG-modules.

In particular, we can now define the dual of an RG-module.

Definition 3.30. Let M be a left (right) RG-module. The dual of M is the right (left) RG-module M∗ =HomRG(M,RG).

Proposition 3.31. LetM andN be RG-modules. Then there is a natural equivalence (M⊕N)∗ ∼=M∗ ⊕N∗.

3.1.4. Chain complexes.

Definition 3.32. An RG-chain complex is a (covariant) functor C� : G→ Ch(R), whereCh(R) is the category of chain complexes over the ring R.

Lemma 3.33. The following are equivalent:(i) C� is an RG-chain complex;

(ii) there exist a family {Cn} of RG-modules together with a family of natural transfor-mations {dn : Cn→ Cn−1}, called differentials, such that dn−1 ◦dn = 0.

Using the second characterization of RG-chain complexes, we can now define finitelygenerated projective chain complexes, chain maps and chain homotopies in the usualmanner.

Definition 3.34. An RG-chain complex P� is said to be a finitely generated projective ifeach Pn is a finitely generated projective RG-module and P� is bounded (i.e., Pn = 0 forall but a finite number of n). Let �(RG) denote the subcategory of finitely generatedprojective RG-chain complexes.

Definition 3.35. An RG-chain map f : C�→D� is a family { fn : Cn→Dn} of natural trans-formations such that d′n ◦ fn = fn−1 ◦ dn for all n, where the dn are the differentials of C�and the d′n are the differentials of D�.

Definition 3.36. Two RG-chain maps f : C�→ D� and g : C�→ D� are RG-chain homo-topic, denoted by f ∼ch g, if there exists a family {sn : Cn→Dn−1} of natural transforma-tions such that

fn− gn = d′n+1 ◦ sn + sn−1 ◦dn. (3.28)

Definition 3.37. Two RG-chain complexes C� and D� are chain homotopy equivalent ifthere exist RG-chain maps f : C�→ D� and g : D�→ C� such that f ◦ g ∼ch idD� and g ◦f ∼ch idC�. In this case, f is said to be a chain homotopy equivalence.

Vesta Coufal 11

3.1.5. Everything α-twisted. For the remainder of the paper, let α : G→ G be a functor.We can use α to create an “α-twisted” version of many of our algebraic objects.

Definition 3.38. Define an RG-bimodule αRG :G×Gop → R-mod by

αRG(x, y)= RG(y,α(x))

(3.29)

for x, y ∈Ob(G), and

αRG(g,h)= α(g)◦ (−)◦h (3.30)

for g a map in G and h a map in Gop. This is the RG-bimodule RG, but with the leftmodule structure twisted by α.

Definition 3.39. Let M and N be RG-modules. An α-linear homomorphism M → N isdefined to be a natural transformation η : M → N ◦ α. A chain map f : C�→ D� of RG-chain complexes is called α-linear if for each n, fn is α-linear.

Lemma 3.40. Given left RG-modules P and Q, there is an isomorphism

HomRG(P,Q ◦α)∼=HomRG(P,αRG⊗RG Q

). (3.31)

Definition 3.41. Let M be an RG-module. The α-dual of M is

Mα =HomRG(M,αRG

). (3.32)

Proposition 3.42. Let P and Q be RG-modules and N an RG-bimodule. Then there is anatural equivalence of RG-modules

HomRG(P⊕Q,N)∼=HomRG(P,N)⊕HomRG(Q,N). (3.33)

Corollary 3.43. Let P and Q be left RG-modules. Then there is a natural equivalence

(P⊕Q)α ∼= Pα⊕Qα. (3.34)

3.2. Generalized Hattori-Stallings trace. In this section, we define an α-twisted Hattori-Stallings trace for RG-modules. One can define a more general Hattori-Stallings trace forRG-modules, in the same manner as the classical definition given in Section 2.3. However,as we will not need this more general form, we will concern ourselves only with the specialα-twisted case. We also extend the trace to RG-chain complexes.

3.2.1. Definition and commutativity. Given leftRG-modulesN and P, define anR-modulehomomorphism

φP = φP,N : Pα⊗RG N −→HomRG(P,N ◦α) (3.35)

by letting: φP(τ⊗n) : P→N ◦α be the natural transformation given by

φP(τ ⊗n)y(p)=N(τy(p))(n), (3.36)

where τ ∈ Pα(x), m∈N(x), p ∈ P(y), and x, y ∈Ob(G).


Proposition 3.44. If P is a finitely generated projective RG-module, then φP is an isomor-phism.

The proof will use the following three lemmas.

Lemma 3.45. Given x ∈Ob(G), then φRGxis an isomorphism.

Proof. Write φ for φRGx. Define

ψ : HomRG(RGx,N ◦α)−→ RG

αx ⊗RG N (3.37)

by

η −→ α⊗ηx(

idx), (3.38)

where η : RGx → N ◦ α is a natural transformation. Here, α∈ Pα(x) is the natural trans-formation induced by α, that is, αy( f )= α( f ) for y ∈Ob(G) and f ∈ RG(x, y).

It is easy to show that φ ◦ψ = id and ψ ◦φ= id. �

Lemma 3.46. If P and Q are left RG-modules, then φP⊕Q = φP ⊕φQ.

Proof. Consider the following diagram:

(P⊕Q)α⊗RG NφP⊕Q

∼=

HomRG(P⊕Q,N ◦α)

∼=(Pα⊕Qα

)⊗RG N

∼=(Pα⊗RG N

)⊕ (Qα⊗RG N)

φP⊕φQHomRG(P,N ◦α)⊕HomRG(Q,N ◦α)

(3.39)

The vertical isomorphisms are as in Propositions 3.8 and 3.9 and Corollary 3.43. Usingthose isomorphism, one can see that the diagram commutes. �

Lemma 3.47. Let P and Q be left RG-modules and let N = P⊕Q. If φN is an isomorphism,then φP is an isomorphism also.

Proof. By the previous lemma, φN = φP ⊕φQ. The result follows immediately. �

Proof of Proposition 3.44. The proof is in two steps.Step 1. Suppose that P is a finitely generated free RG-module. Then P is naturally equiv-alent to RGB =

⊕b∈�RGβ(b) for some Ob(G)-set B. By Lemma 3.46, φP =

⊕b∈�φRGβ(b)

,and by Lemma 3.45, it is an isomorphism.Step 2. Suppose that P is a finitely generated projective RG-modules and so P is a directsummand of a finitely generated free RG-module. Combining Step 1 and Lemma 3.47 wesee that φP is an isomorphism.

�

Vesta Coufal 13

For P a left RG-module, define an R-module homomorphism

Pα⊗RG P −→ αRG/[RG,αRG

](3.40)

by τ ⊗ p → τx(p) where τ ∈ Pα(x) and p ∈ P(x).

Definition 3.48. Let P be a finitely generated projective left RG-module. The Hattori-Stallings trace, denoted by tr, is the composition

HomRG(P,P ◦α)

tr

Pα⊗RG P∼=

αRG/[RG,αRG

]

HH0(RG;αRG

)(3.41)

where the isomorphism is the map φP and the unadorned arrow is the homomorphismdescribed above.

Proposition 3.49 (commutativity). Let P and Q be finitely generated projective left RG-modules. If f ∈HomRG(P,Q ◦α) and g ∈HomRG(Q,P), then

tr( f ◦ g)= tr(g ◦α◦ f ). (3.42)

Proof. The result follows from commutativity of three diagrams.The first diagram is

HomRG(P,Q ◦α)×HomRG(Q,P)(Pα⊗RG Q

)× (Q∗ ⊗RG P)

B

HomRG(P,P ◦α) Pα⊗RG PφP

(3.43)

where B is given by (η⊗ p)× (τ⊗ q)→ (α◦η)⊗Q(τy(p))(q), the unlabelled vertical mapis given by ( f ,g)→ g ◦α◦ f and the unlabelled horizontal map is φPα,Q×φQ,P .

The second diagram is gotten by transposing the products in the first diagram.The third diagram is

Qα⊗RG Q

(Q∗ ⊗RG P

)× (Pα⊗RG Q)

B′

(Pα⊗RG Q

)× (Q∗ ⊗RG P)

B

HH0(RG;αRG

)

Pα⊗RG P

(3.44)


where the unlabelled arrow is transposition, B′ is analogous to B, and the other maps aredefined in the obvious ways. �

3.2.2. For connected groupoids. Consider the following setup. LetG be a connected group-oid, that is, one for which there exists a map between any two objects. Let α :G→G be afunctor and let P be a finitely generated projective left RG-module. Choose an object ∗of G and choose a map τ :∗→ α(∗) in G.

Let RG(∗) be the subcategory of RG with a single object, ∗, and with maps givenby the maps in RG from ∗ to ∗. Then the inclusion RG(∗)→ RG is an equivalence ofcategories. The proof amounts to choosing a map μx : ∗ → x for each x ∈ Ob(G). Foreach x, we fix a choice of μx.

The functor α induces a functor ατ : RG(∗) → RG(∗) which maps the object ∗ toitself. If g : ∗→ ∗, let ατ(g) = τ−1 ◦ α(g) ◦ τ. In the obvious way, the RG-module P in-duces a finitely generated projective left RG(∗)-module, denoted P(∗). A natural trans-formation β ∈ HomRG(P,P ◦ α) induces a natural transformation βτ = P(τ−1) ◦ β∗ ∈HomRG(∗)(P(∗),P(∗)◦ατ).

Lemma 3.50. There is an isomorphism of groups

A :HH0(RG(∗);ατRG(∗)

)−→HH0(RG;αRG

)(3.45)

given by A(m)= τ ◦m for m∈HH0(RG(∗);ατRG(∗)).

Proposition 3.51. The Hattori-Stallings trace of βτ and β are equivalent, that is,

A(

tr(βτ))= tr(β). (3.46)

Proof. Given η ∈ Pα(x) for some x ∈ Ob(G), define η : P(∗) → RG(∗,∗) ∈ P(∗)ατ byη(p)= τ−1 ◦η∗(p)◦μx, where p ∈ P(∗). This gives us a map Pα→ P(∗)ατ .

Define a map B : Pα ⊗RG P → P(∗)ατ ⊗RG(∗)P(∗) by η⊗ p → η⊗ P(μ−1x )(p), where

η ∈ Pα(x) and p ∈ P(x) for some x ∈ Ob(G). Define a map C : HomRG(P,P ◦ α) →HomRG(∗)(P(∗),P(∗)◦ατ) by γ → γτ = P(τ−1)◦ γ∗ for γ ∈HomRG(P,P ◦α).

Commutativity of the following two diagrams implies that A(tr(βτ))= tr(β).

HomRG(∗)(P(∗),P(∗)◦ατ

)P(∗)ατ ⊗RG(∗) P(∗)

φP(∗)

[3pt]HomRG(P,P ◦α)

C

Pα⊗RG PφP

B

P(∗)ατ ⊗RG(∗) P(∗) HH0(RG(∗);ατRG(∗)

)

A

[3pt]Pα⊗RG P

B

HH0(RG;αRG)

(3.47)

�

Notice that A(tr(βτ)) is independent of the choices of maps μx.

Vesta Coufal 15

3.2.3. For chain complexes. We begin with the general case.

Definition 3.52. Let P� be a finitely generated projective RG-chain complex. Define theHattori-Stallings trace

Tr : Hom�(RG)(P�,P �◦α)−→HH0(RG;αRG

)(3.48)

by

f −→∑

i

(−1)i tr(fi), (3.49)

where f : P�→ P �◦α is given by the family { fi ∈HomRG(Pi,Pi ◦α)}.Commutativity follows from commutativity of the Hattori-Stallings trace for RG-

modules.

Proposition 3.53 (commutativity). Let P� and Q� be finitely generated projective RG-chain complexes, and let f ∈Hom�(RG)(P�,Q �◦α) and g ∈Hom�(RG)(Q�,P�). Then

Tr( f ◦ g)= Tr(g ◦α◦ f ). (3.50)

The Hattori-Stallings trace is also invariant up to chain homotopy.

Proposition 3.54. Let P� be a finitely generated projective RG-chain complex. If f : P�→P �◦α and g : P�→ P �◦α are chain homotopic, then Tr( f )= Tr(g).

Proof. Let {sn : Pn→ Pn+1 ◦α} be a chain homotopy from f to g . Then

Tr( f )−Tr(g)=∑

i

(−1)i tr(fi− gi

)

=∑

i

(−1)i tr(di+1 ◦α◦ si + si−1 ◦di

)

=∑

i

(−1)i[

tr(si ◦di+1) + tr(si−1 ◦di

)].

(3.51)

The last equality comes from applying commutativity. Rearranging the terms in the lastsum gives Tr( f )−Tr(g)= 0. �

Now suppose that C� is an RG-chain complex which is chain homotopy equivalentto a finitely generated projective RG-chain complex. Suppose further that φ : C�→ C � ◦αis a chain map. Choose a finitely generated projective RG-chain complex P�, choose achain homotopy equivalence f : C�→ P�, and choose a lift ψ : P�→ P �◦α of φ. We get thediagram

P�ψ

P �◦α

C�f

φC �◦αf (3.52)

which commutes up to chain homotopy.


Definition 3.55. The Hattori-Stallings trace of φ : C�→ C �◦α is defined to be the trace ofψ : P�→ P �◦α:

Tr(φ)= Tr(ψ). (3.53)

We must show that Tr is independent of the choices we made. First, suppose that φ′ isanother lift of φ. Then ψ ∼ch f ◦φ ◦ f −1 ∼ch ψ′ and by Proposition 3.54, Tr(ψ)= Tr(ψ′).Second, suppose that Q� is another finitely generated projective RG-chain complex andg : C�→Q� is a chain homotopy equivalence. Then

Tr(g ◦φ ◦ g−1)= Tr

(g ◦ f ◦ f −1 ◦φ ◦ f −1 ◦ f ◦ g−1)

= Tr(f ◦ g−1 ◦ g ◦ f −1 ◦ f ◦φ ◦ f −1)

= Tr(f ◦φ ◦ f −1).

(3.54)

4. Base-point-free Lefschetz-Nielsen invariants

In this section, we present our base-point-free refinements of the classical geometric andalgebraic Lefschetz-Nielsen invariants. We begin by defining the fundamental groupoid,and describing the way in which we think of the universal cover.

4.1. Fundamental groupoid. An important example of a groupoid is the fundamentalgroupoid. Let X be a topological space.

Definition 4.1. The fundamental groupoid ΠX is the category whose objects are thepoints in X , whose maps are the homotopy classes rel endpoints of paths in X . Com-position is given by concatenation of paths. To be precise, if f and g are paths in X suchthat f(1)= g(0), then

[g]◦ [ f ]= [ f · g]. (4.1)

For each morphism, an inverse is given by traversing a representative path backwards.

This groupoid deserves to be called the fundamental groupoid since for a given pointx ∈ X , the subcategory of ΠX generated by x is π1(X ,x). The subcategory generated by xis the category with one object, x, and whose morphism set is ΠX(x,x). In a sense, then,the fundamental groupoid is a way of encoding in one object the fundamental groupswith all possible choices of base point.

Let f : X → X be a continuous map. Then f induces a functor Π f : ΠX →ΠX givenby Π f (x)= f (x) and Π f (g)= f ◦ g where x ∈ X and g is a path in X .

4.2. Universal cover. Let X be a path connected, locally path connected, semilocally sim-ply connected space. For each x ∈ X , one can describe the universal cover [5, page 64] ofX as the space

Xx = (X ,x)(I ,0)/ ∼, (4.2)

Vesta Coufal 17

where I is the closed unit interval and ∼ is the equivalence relation given by homotopyrel endpoints. The set (X ,x)(I ,0) is given the compact-open topology, and Xx is given thequotient topology. The projection map p : Xx → X is given by p([γ])= γ(1).

Recall ΠX , the fundamental groupoid of X . Let Top be the category of topologicalspaces.

Definition 4.2. The universal cover functor

U : ΠX −→ Top (4.3)

is defined by U(x)= Xx for x ∈Ob(ΠX). For g : x→ y a map in ΠX , define U(g) : Xx →Xy by U(g)[γ]= [g−1 · γ], where [γ]∈ Xx.

4.3. The geometric invariant. Fix a compact, path-connected n-dimensional manifoldX and a continuous endomorphism f : X → X such that Fix( f ) is finite.

Let Π be the fundamental groupoid of X . The map f induces a functor ϕ=Π f : Π→Π defined by ϕ(x)= f (x), where x ∈Ob(Π). For g : x→ y a map in Π let ϕ(g)= f ◦ g .

Let Fix(ϕ) be the subcategory of Π whose set of objects is Fix( f ), and whose maps arethe maps g : x→ y in Π (x, y ∈ Fix( f )) such that f ◦ g = g . The category Fix(ϕ) decom-poses into a finite number of connected components; denote them by F1, . . . ,Fr .

Define an ZΠ-bimodule ϕZΠ : Π×Πop → Ab given by (x, y) → ZΠ(y,ϕ(x)), wherex, y ∈ Ob(Π). For g : x → x′ a map in Π and h : y → y′ a map in Πop, let ϕZΠ(g ,h) =ϕ(g)◦ (−)◦h. By definition,

HH0(ZΠ;ϕZΠ

)= ϕZΠ/[ZΠ,ϕZΠ

]

=⊕

x∈Ob(Π)

ZΠ(x,ϕ(x)

)/Q, (4.4)

where Q is generated by elements of the form σ − ϕ(g) ◦ σ ◦ g−1 for maps σ : x→ ϕ(x)and g : x→ y in Π.

Define

Φ :{Fk}rk=1 −→HH0

(ZΠ;ϕZΠ

)(4.5)

by choosing an object x in Fk and mapping Fk to idx : x→ x = ϕ(x). One can check thatthis is a well-defined injection.

Also, let

i(f ,Fk

)=∑

x∈Ob(Fk)

i( f ,x)∈ Z, (4.6)

where i( f ,x) is the fixed point index.

Definition 4.3. The geometric Lefschetz invariant of f : X → X is

Lgeo( f )=∑

k

i(f ,Fk

)Φ(Fk)∈HH0

(ZΠ;ϕZΠ

). (4.7)


Theorem 4.4. The classical geometric Lefschetz invariant and the base-point-free geometricLefschetz invariant correspond under an isomorphism

A : Zπφ −→HH0(ZΠ;ϕZΠ

). (4.8)

The isomorphism A is not canonical; it depends on choosing a path from ∗ to f (∗).On the other hand, HH0(ZΠ;ϕZΠ) is canonical.

Proof. Recall that in the classical definition, we have chosen a base point∗ and a base pathτ. The fundamental group π1(X ,∗) is denoted by π, the map on π induced by f : X → Xand the base path τ is denoted by φ, and the injection {Fi}si=1 → πφ is denoted by Φ.Step 1. After appropriate reordering of the fixed point classes F1, . . . ,Fs, s = r and Fi =Ob(Fi). This can be seen as follows. If x and y are equivalent in Fix( f ), then there existsa path ν from x to y in X such that ν · ( f ◦ ν)−1 � ∗. But this is equivalent to sayingthat ν is a map in Fix(ϕ) from x to y, and hence that x and y are in the same connectedcomponent of Fix(ϕ).Step 2. Define an isomorphism of abelian groups

A : Zπφ −→HH0(ZG;ϕZG

)(4.9)

by A(ω)= ω · τ = τ ◦ω, where [ω]∈ π.To see that A is well defined, suppose that [ω] and [ω1] are equivalent in Zπφ. By

definition, there exists g ∈ π such that ω1 = g · ω · τ · ( f ◦ g)−1 · τ−1. Hence, τ ◦ ω1 =ϕ(g−1)◦ τ ◦ω ◦ g = τ ◦ g in HH0(ZG;ϕZG), and A is well-defined.

To see that A is an epimorphism, suppose that σ : x→ ϕ(x)∈HH0(ZG;ϕZG). Choosea path μ in X from ∗ to x, that is, a map μ : ∗ → x in G. Then σ = ϕ(μ−1) ◦ σ ◦ μ inHH0(ZG;ϕZG), and μ · σ · ( f ◦ μ)−1 · τ−1 gives an element in π which is mapped to σ byA.

The last thing to check is that A is a monomorphism. Suppose [ω] and [ω1] are el-ements of π such that τ ◦ ω = τ ◦ ω1. Then there exists g ∈ Ob(G) such that τ ◦ ω1 =ϕ(g−1) ◦ τ ◦ ω ◦ g. It follows that ω1 = g · ω · τ · ( f ◦ g)−1 · τ−1 and hence that [ω1] isequivalent to [ω] in Zπφ.Step 3. Let F be a fixed point class, and F the corresponding connected component ofFix(ϕ). For any choice of x ∈ F and path μ from ∗ to x, we have that A(Φ(F)) = A(μ ·( f ◦μ)−1 · τ−1)= ϕ(μ−1)◦μ= idx in HH0(ZG;ϕZG).

Therefore, the image of

Lgeo( f ,∗,τ)=s∑

k=1

i(f ,Fk

)Φ(Fk)∈ Zπφ (4.10)

is equivalent to

Lgeo( f )=r∑

k=1

i(f ,Fk

)Φ(Fk)∈HH0

(ZG;ϕZG

). (4.11)

�

Vesta Coufal 19

4.4. The algebraic invariant. Let X be a finite CW complex and f : X → X a continuousmap. Let Π = ΠX be the fundamental groupoid of X and let ϕ : Π→ Π be the functorinduced by f , as above.

The map f induces a natural transformation f : U → U ◦ϕ. Given an object x in Π,

fx : Xx → X f (x) is defined by [γ]→ [ f ◦ γ], where [γ]∈ Xx. One can check naturality.There is a functor S : Top→ Ch(Z) given by taking the singular chain complex of a

space. If f : X → Y is a continuous map, then S( f ) : S(X)→ S(Y) is given by σ → f ◦ σ ,where σ : Δn→ X . Here, Δn is the standard n-simplex.

Let C� be the ZΠ-chain complex given by the composition

ΠU−−→ Top

S−→ Ch(Z). (4.12)

The map f induces a natural transformation f∗ : SU → SUϕ. Given an object x in Π,

let f∗(x) : S(Xx)→ S(X f (x)) be given by σ → fx ◦ σ , where σ : Δn → Xx. Naturality of f∗follows from naturality of f . Hence, f∗ is a ϕ-linear chain map C�→ C�. As usual, f∗ is

given by a family of ϕ-linear natural transformations fn : Cn→ Cn.The singular chain complex of a finite CW complex is chain homotopy equivalent to

a finitely generated projective ZΠ chain complex. Hence, the Hattori-Stallings trace of f∗is defined, and we can define the algebraic Lefschetz invariant as follows.

Definition 4.5. The algebraic Lefschetz invariant of f : X → X is

Lalg( f )= Tr(f∗)=

∑

k≥0

(−1)k tr(fk)∈HH0

(ZΠ;ϕZΠ

). (4.13)

As an immediate corollary of Proposition 3.51 we get the following theorem.

Theorem 4.6. The classical algebraic Lefschetz invariant and the base point free algebraicLefschetz invariant correspond under the isomorphism

A : Zπφ −→HH0(ZΠ;ϕZΠ

). (4.14)

References

[1] H. Bass, Euler characteristics and characters of discrete groups, Inventiones Mathematicae 35(1976), no. 1, 155–196.

[2] , Traces and Euler characteristics, Homological Group Theory (Proc. Sympos., Durham,1977), London Math. Soc. Lecture Note Ser., vol. 36, Cambridge University Press, Cambridge,1979, pp. 1–26.

[3] R. F. Brown, The Lefschetz Fixed Point Theorem, Scott, Foresman, Illinois, 1971.[4] R. Geoghegan, Nielsen fixed point theory, Handbook of Geometric Topology, North-Holland,

Amsterdam, 2002, pp. 499–521.[5] A. Hatcher, Algebraic Topology, Cambridge University Press, Cambridge, 2002.[6] B. J. Jiang, Lectures on Nielsen Fixed Point Theory, Contemporary Mathematics, vol. 14, Ameri-

can Mathematical Society, Rhode Island, 1983.[7] W. Luck, Transformation Groups and Algebraic K-Theory, Lecture Notes in Mathematics,

vol. 1408, Mathematica Gottingensis, Springer, Berlin, 1989.


[8] , The universal functorial Lefschetz invariant, Fundamenta Mathematicae 161 (1999),no. 1-2, 167–215.

[9] J. Stallings, Centerless groups—an algebraic formulation of Gottlieb’s theorem, Topology. An Inter-national Journal of Mathematics 4 (1965), no. 2, 129–134.

Vesta Coufal: Department of Mathematics, Fort Lewis College, Durango, CO 81301, USAE-mail address: coufal [email protected]

THE ANOSOV THEOREM FOR INFRANILMANIFOLDS WITHAN ODD-ORDER ABELIAN HOLONOMY GROUP

K. DEKIMPE, B. DE ROCK, AND H. POUSEELE

Received 9 September 2004; Revised 18 February 2005; Accepted 21 July 2005

We prove that N( f )= |L( f )| for any continuous map f of a given infranilmanifold withAbelian holonomy group of odd order. This theorem is the analogue of a theorem ofAnosov for continuous maps on nilmanifolds. We will also show that although theirfundamental groups are solvable, the infranilmanifolds we consider are in general notsolvmanifolds, and hence they cannot be treated using the techniques developed for solv-manifolds.

Copyright © 2006 K. Dekimpe et al. This is an open access article distributed under theCreative Commons Attribution License, which permits unrestricted use, distribution,and reproduction in any medium, provided the original work is properly cited.

1. Introduction

Let M be a smooth closed manifold and let f : M →M be a continuous self-map of M.In fixed point theory, two numbers are associated with f to provide information on itsfixed points: the Lefschetz number L( f ) and the Nielsen number N( f ). Inspired by thefact that N( f ) gives more information than L( f ), but unfortunately N( f ) is not readilycomputable from its definition (while L( f ) is much easier to calculate), in literature, aconsiderable amount of work has been done on investigating the relation between bothnumbers. In [1] Anosov proved that N( f )= |L( f )| for all continuous maps f : M →Mif M is a nilmanifold, but he also observed that there exists a continuous map f : K → Kof the Klein bottle K such that N( f ) �= |L( f )|.

There are two possible ways of trying to generalize this theorem of Anosov. Firstly, onecan search classes of maps for which the relation holds for a specific type of manifold. Forinstance, Kwasik and Lee proved in [10] that the Anosov theorem holds for homotopicperiodic maps of infranilmanifolds and in [14] Malfait did the same for virtually unipo-tent maps of infranilmanifolds. Secondly, one can look for classes of manifolds, otherthan nilmanifolds, for which the relation holds for all continuous maps of the given man-ifold, as was established by Keppelmann and McCord for exponential solvmanifolds (see[8]).


2 The Anosov theorem for infranilmanifolds

In this article we will work on the class of infranilmanifolds. After the preliminaries wewill first describe a class of maps for which the Anosov theorem holds and thereafter wewill follow the second approach and work with infranilmanifolds with odd-order Abelianholonomy group. The main result of this paper is that the Anosov theorem always holdsfor these kinds of infranilmanifolds. This result cannot be extended to the case of even-order Abelian holonomy groups, since Anosov already constructed a counterexample forthe Klein bottle, which has Z2 as holonomy group. A detailed investigation of the case ofeven-order holonomy is much more delicate and will be dealt with in an other paper.

Throughout the paper we will illustrate all concepts by means of examples. In fact thewhole collection of examples together forms one big example. Moreover, by means of thisexample, we will also show that the manifolds we study are in general not solvmanifoldsand therefore cannot be treated by the techniques developed for solvmanifolds.

2. Preliminaries

Let G be a connected, simply connected, nilpotent Lie group. An affine endomorphismof G is an element (g,ϕ) of the semigroup G� Endo(G) with g ∈G the translational partand ϕ ∈ Endo(G) (= the semigroup of all endomorphisms of G) the linear part. Theproduct of two affine endomorphisms is given by (g ,ϕ)(h,μ) = (g · ϕ(h),ϕμ) and (g ,ϕ)maps an element x ∈ G to g ·ϕ(x). If the linear part ϕ belongs to Aut(G), then (g ,ϕ) isan invertible affine transformation of G. We write Aff (G)=G� Aut(G) for the group ofinvertible affine transformations of G.

Example 2.1. One of the best known examples of a connected and simply connectednilpotent Lie group is the Heisenberg group

H =⎧⎪⎨⎪⎩

⎛⎜⎝

1 y z0 1 x0 0 1

⎞⎟⎠ | x, y,z ∈R

⎫⎪⎬⎪⎭. (2.1)

For further use, we will use h(x, y,z) to denote the element(

1 y (1/3)z0 1 x0 0 1

). (The reason for

introducing a 3 in the upper right corner lies in the use of this example later on.) Thereader easily computes that

h(x1, y1,z1

)h(x2, y2,z2

)= h(x1 + x2, y1 + y2,z1 + z2 + 3x2y1). (2.2)

Let us fix the following elements for use throughout the paper: a= h(1,0,0), b = h(0,1,0)and c = h(0,0,1). The group N generated by the elements a, b, c has a presentation of theform

N = ⟨a,b,c | [b,a]= c3, [c,a]= [c,b]= 1⟩. (2.3)

(We use the convention that [b,a] = b−1a−1ba.) Obviously the group N consists exactlyof all elements h(x, y,z), for which x, y,z ∈ Z.

K. Dekimpe et al. 3

For any connected, simply connected nilpotent Lie group G with Lie algebra g, it isknown that the exponential map exp : g→G is bijective and we denote by log the inverseof exp.

Example 2.2. The Lie algebra of H , is the Lie algebra of matrices of the form

h=⎧⎪⎨⎪⎩

⎛⎜⎝

0 y z0 0 x0 0 0

⎞⎟⎠ | x, y,z ∈R

⎫⎪⎬⎪⎭. (2.4)

The exponential map is given by

exp : h−→H :

⎛⎜⎝

0 y z0 0 x0 0 0

⎞⎟⎠�−→

⎛⎜⎜⎝

1 y z+xy

20 1 x0 0 1

⎞⎟⎟⎠ . (2.5)

Hence

log :H −→ h :

⎛⎜⎝

1 y z0 1 x0 0 1

⎞⎟⎠�−→

⎛⎜⎜⎝

0 y z− xy

20 0 x0 0 0

⎞⎟⎟⎠ . (2.6)

For later use, we fix the following basis of h:

C =

⎛⎜⎜⎝

0 013

0 0 00 0 0

⎞⎟⎟⎠= log(c), B =

⎛⎜⎝

0 1 00 0 00 0 0

⎞⎟⎠= log(b),

A=⎛⎜⎝

0 0 00 0 10 0 0

⎞⎟⎠= log(a).

(2.7)

For any endomorphism ϕ of the Lie group G to itself there exists a unique endomor-phism ϕ∗ of the Lie algebra g (namely the differential of ϕ), making the following diagramcommutative:

Gϕ

log

G

log

gϕ∗

exp

g

exp (2.8)

Conversely, every endomorphism ϕ∗ of g appears as the differential of an endomorphismof G.


Example 2.3. Let H and h be as before. With respect to the basis C, B and A (in thisorder!), any endomorphism ϕ∗ is given by a matrix of the form

⎛⎜⎝k1l2− k2l1 l3 k3

0 l2 k2

0 l1 k1

⎞⎟⎠ . (2.9)

This follows from the fact that 3C = [B,A] and hence 3ϕ∗(C) = [ϕ∗(B),ϕ∗(A)]. Con-versely, any such a matrix represents an endomorphism of g. The corresponding endo-morphism ϕ of H satisfies

ϕ(h(x, y,z)

)= exp(ϕ∗(

log(h(x, y,z)

)))

= h(k1x+ l1y,k2x+ l2y,3k3x+ 3l3y +

3(k1x+ l1y

)(k2x+ l2y

)

2

+(k1l2− k2l1

)(z− 3xy

2

)).

(2.10)

As one sees, although the map ϕ∗ is linear and thus easy to describe, the correspondingϕ is much more complicated. In order to be able to continue presenting examples, we willuse a matrix representation of the semigroup H � Endo(H). Given an endomorphism ϕof H , let us denote by Mϕ the 4× 4-matrix

Mϕ =(P 00 1

), (2.11)

where P denotes the 3× 3-matrix, representing ϕ∗ with respect to the basis C, B, A (againin this fixed order). Define the map

ψ :H � Endo(H)−→M4(R) :(h(x, y,z),ϕ

)�−→

⎛⎜⎜⎜⎜⎜⎝

1 −3x2

3y2

−3xy2

+ z

0 1 0 y

0 0 1 x

0 0 0 1

⎞⎟⎟⎟⎟⎟⎠·Mϕ.

(2.12)

We leave it to the reader to verify that ψ defines a faithful representation of the semigroupH � Endo(H) into the semigroupM4(R) (respectively of the group Aff(H) into the groupGl(4,R)).

Remark 2.4. An analogous matrix representation can be obtained for any G� Endo(G)in case G is two-step nilpotent. (Recall that a group G is said to be k-step nilpotent ifthe k + 1’th term of the lower central series γk+1(G)= 1, where γ1(G)= G and γi+1(G)=[G,γi(G)]. For example, the Heisenberg group is 2-step nilpotent.) This is proved in [3]for the group Aff(G), but the details in that paper can easily be adjusted to the case of thesemigroup G� Endo(G).

K. Dekimpe et al. 5

2.1. Infranilmanifolds and continuous maps. In this section we quickly recall the no-tion of almost-crystallographic groups and infranilmanifolds. We refer the reader to [4]for more details.

An almost-crystallographic group is a subgroup E of Aff(G), such that its subgroupof pure translations N = E∩G, is a uniform lattice (by which we mean a discrete andcocompact subgroup) ofG and moreover,N is of finite index in E. Therefore the quotientgroup F = E/N is finite and is called the holonomy group of E. Note that the group F isisomorphic to the image of E under the natural projection Aff(G)→ Aut(G), and henceF can be viewed as a subgroup of Aut(G) and of Aff(G).

Any almost-crystallographic group acts properly discontinuously on (the correspond-ing) G and the orbit space E\G is compact. Recall that an action of a group E on a locallycompact space X is said to be properly discontinuous, if for every compact subset C of X ,the set {γ ∈ E | γC∩C �= ∅} is finite. When E is a torsion free almost-crystallographicgroup, it is referred to as an almost-Bieberbach group and the orbit space M = E\G iscalled an infranilmanifold. In this case E equals the fundamental group π1(M) of theinfranilmanifold, and we will also talk about F as being the holonomy group of M.

Any almost-crystallographic group determines a faithful representationT :F→Aut(G),which is induced by the natural projection p : Aff(G)=G� Aut(G)→ Aut(G), and whichis referred to as the holonomy representation.

Remark 2.5. As isomorphic crystallographic subgroups are conjugated inside Aff(G) (seeTheorem 2.7 below or [13]), it follows that the holonmy representation of an almost-crystallographic group is completely determined from the algebraic structure of E up toconjugation by an element of Aff(G).

Let g denote the Lie algebra of G. By taking differentials, the holonomy representationalso induces a faithful representation

T∗ : F −→ Aut(g) : x �−→ T∗(x) := d(T(x)). (2.13)

Example 2.6. Let ϕ be the automorphism of H , whose differential ϕ∗ is given by the

matrix(1 −3/2 0

0 −1 10 −1 0

). Let α= (h(0,0,1/3),ϕ)∈ Aff(H). Then the group E generated by a, b,

c and α has a presentation of the form

E =⟨a,b,c,α |[b,a]= c3 [c,a]= 1 [c,b]= 1

αa= bα αb = a−1b−1α αc = cα α3 = c

⟩. (2.14)

(This is easily checked using the matrix representation (2.12).) E is an almost-crystallo-graphic group with translation subgroup N = H ∩ E = 〈a,b,c〉 and a holonomy groupF = E/N ∼= Z3 of order three. (See also [4, page 164, type 13].) We have that

T∗(F)=

⎧⎪⎪⎪⎨⎪⎪⎪⎩I3,

⎛⎜⎜⎜⎝

1 −32

0

0 −1 10 −1 0

⎞⎟⎟⎟⎠ ,

⎛⎜⎜⎜⎝

1 −32

0

0 −1 10 −1 0

⎞⎟⎟⎟⎠

2

=

⎛⎜⎜⎜⎝

1 0 −32

0 0 −10 1 −1

⎞⎟⎟⎟⎠

⎫⎪⎪⎪⎬⎪⎪⎪⎭. (2.15)

(Of course In will denote the n×n-identity matrix.)


As E is torsion-free, it is an almost-Bieberbach group and it determines an infranil-manifold M = E\H .

Essential for our purposes is the following result due to K. B. Lee (see [11]).

Theorem 2.7. Let E,E′ ⊂ Aff(G) be two almost-crystallographic groups. Then for any ho-momorphism θ : E→ E′, there exists a g = (d,D)∈ G� Endo(G) such that θ(α) · g = g ·αfor all α∈ E.

Important for us is the following corollary of this theorem (we refer to [11] for adetailed proof).

Corollary 2.8. Let M = E\G be an infranilmanifold and f :M→M a continuous map ofM. Then f is homotopic to a map h : M →M induced by an affine endomorphism (d,D) :G→G.

We say that (d,D) is a homotopy lift of f . Note that one can find the homotopy lift ofa given f , by using Theorem 2.7 for the homomorphism f∗ : π1(M)→ π1(M) induced byf . In fact, using this method one can characterize all continuous maps, up to homotopy,of a given infranilmanifold M.

Example 2.9. Let E be the almost Bieberbach group of the previous example, then thereis a homomorphism θ1 : E→ E, which is determined by the images of the generators asfollows:

θ1(a)= b2c3, θ1(b)= a2c3, θ1(c)= c−4, θ1(α)= c−2α2. (2.16)

Using the matrix representation (2.12) it is easy to check that θ1 really determines anendomorphism of E and that this endomorphism is induced by the affine endomorphism(h(0,0,0),D1), where

D1,∗ =⎛⎜⎝−4 3 30 0 20 2 0

⎞⎟⎠ . (2.17)

Another example is given by the morphism θ2 determined by

θ2(a)= a4b4c20, θ2(b)= a−4c−10, θ2(c)= c16, θ2(α)= c5α, (2.18)

and induced by (h(0,0,0),D2), where

D2,∗ =⎛⎜⎝

16 −10 −40 0 40 −4 4

⎞⎟⎠ . (2.19)

2.2. Lefschetz and Nielsen numbers on infranilmanifolds. Let M be a compact mani-fold and assume f : M →M is a continuous map. The Lefschetz number L( f ) is definedby

L( f )=∑

i

(−1)iTrace(f∗ :Hi(M,Q)−→Hi(M,Q)

). (2.20)

K. Dekimpe et al. 7

The set Fix( f ) of fixed points of f is partitioned into equivalence classes, referred to asfixed point classes, by the relation: x, y ∈ Fix( f ) are f -equivalent if and only if there is apath w from x to y such that w and f w are (rel. endpoints) homotopic. To each class oneassigns an integer index. A fixed point class is said to be essential if its index is nonzero.The Nielsen number of f is the number of essential fixed point classes of f . The relationbetween L( f ) andN( f ) is given by the property that L( f ) is exactly the sum of the indicesof all fixed point classes. For more details we refer to [2, 7] or [9].

In this paper, we examine the relation N( f )= |L( f )| for continuous maps f :M→Mon an infranilmanifoldM. Since L( f ) andN( f ) are homotopy invariants, one can restrictto those maps which are induced by an affine endomorphism of the covering Lie groupG.

In fact, this is exploited completely in the following theorem of K. B. Lee (see [11]),which will play a crucial role throughout the rest of this paper.

Theorem 2.10. Let f : M →M be a continuous map of an infranilmanifold M and letT : F → Aut(G) be the associated holonomy representation. Let (d,D)∈ G� Endo(G) be ahomotopy lift of f . Then

N( f )= L( f )⇐⇒ det(In−T∗(x)D∗)≥ 0, ∀x ∈ F, and respectively,

N( f )=−L( f )⇐⇒ det(In−T∗(x)D∗)≤ 0, ∀x ∈ F.(2.21)

Remark 2.11. Recently J. B. Lee and K. B. Lee generalized (see [12]) this theorem byproving that the following formulas for L( f ) and N( f ) hold on infranilmanifolds. Usingthe notations from above:

L( f )= 1|F|

∑

x∈Fdet

(In−T∗(x)D∗

),

N( f )= 1|F|

∑

x∈F

∣∣det(In−T∗(x)D∗

)∣∣.

(2.22)

3. A class of maps for which the Anosov theorem holds

With Theorem 2.10 in mind, we can describe a class of maps on infranilmanifolds, forwhich the Anosov theorem always holds. Note that we do not claim that such maps existon all infranilmanifolds.

Proposition 3.1. Let M be an infranilmanifold with holonomy group F and associatedholonomy representation T : F → Aut(G). Let f :M→M be a continuous map and (d,D)a homotopy lift of f .

Suppose that for all x ∈ F, x �= 1 : T∗(x)D∗ �=D∗T∗(x). Then

∀x ∈ F : det(In−D∗

)= det(In−T∗(x)D∗

), (3.1)

and hence N( f )= |L( f )|.Proof. Let 1 �= x ∈ F. Since (d,D) is obtained from Theorem 2.7, we know that there existsan y ∈ F such that T(y)∗D∗ = D∗T(x)∗. Indeed, if x is a pre-image of x ∈ E = π1(M),


then y is the natural projection of f∗(x), where f∗ denote the morphism induced by fon π1(M).

Because of the condition on T∗ and D∗ we know that x �= y. Then

det(In−D∗

)= det(T∗(x)−D∗T∗(x)

)det

(T∗(x−1))

= det(T∗(x)−T∗(y)D∗

)det

(T∗(x−1))

= det(In−T∗

(x−1y

)D∗).

(3.2)

Since x �= y and T∗ is faithful, we have that T∗(x−1y) �= In. Moreover, for any other1 �= x′ ∈ F, with x �= x′ and T∗(y′)D∗ = D∗T∗(x′), we have that x−1y �= x′−1y′. Indeed,suppose that there exists an x′ ∈ F, x �= x′, such that x−1y = x′−1y′. Then

T∗(x−1y

)D∗ = T∗

(x′−1y′

)D∗ ⇐⇒ T∗

(x−1)D∗T∗(x)= T∗

(x′−1)D∗T∗(x′)

⇐⇒D∗T∗(xx′−1)= T∗

(xx′−1)D∗.

(3.3)

This last equality is only satisfied when xx′−1 = 1. This proves the proposition because anyx ∈ F determines an unique element x−1y ∈ F, and thus all elements of F are obtained.The last conclusion easily follows from Theorem 2.10. �

Example 3.2. Let M = E\H be the infra-nilmanifold from before and suppose that f1 :M →M is a continuous map inducing the endomorphism θ1 on E = π1(M). We knowalready that f∗ = θ1 is induced by (1,D1) and it is easy to check that

ϕ∗D1,∗ =

⎛⎜⎜⎜⎝

1 −32

0

0 −1 10 −1 0

⎞⎟⎟⎟⎠

⎛⎜⎝−4 3 30 0 20 2 0

⎞⎟⎠=

⎛⎜⎝−4 3 30 0 20 2 0

⎞⎟⎠

⎛⎜⎜⎜⎝

1 0 −32

0 0 −10 1 −1

⎞⎟⎟⎟⎠=D1,∗ϕ2

∗ (3.4)

which implies that the map f (or D1,∗) satisfies the criteria of the theorem, and indeedwe have that

det(I3−D1,∗

)= det(I3−ϕ∗D1,∗

)= det(I3−ϕ2

∗D1,∗)=−15. (3.5)

4. Infranilmanifolds with Abelian holonomy group of odd order

In this section, we concentrate on the infranilmanifolds with an odd-order Abelian ho-lonomy group F and show that the Anosov theorem can be generalized to this class ofmanifolds.

Let T : F → Aut(G) denote the holonomy representation as before, then, for any x ∈ F,we have that T∗(x) is of finite order, since F is finite, and so the eigenvalues T∗(x) areroots of unity. Moreover, since the order of T∗(x) has to be odd, we know that the onlyeigenvalues of T∗(x) are 1 or not real. The usefulness of this observation follows from thenext lemma concerning commuting matrices.

K. Dekimpe et al. 9

Lemma 4.1. Let B,C ∈Mn(R) be two real matrices such that BC = CB and suppose thatB has only nonreal eigenvalues. Then the (algebraic) multiplicity of any real eigenvalue of Cmust be even which implies that det(In−C)≥ 0.

Proof. We prove this lemma by induction on n. Note that n is even because B only hasnon real eigenvalues.

Suppose n = 2 and λ is a real eigenvalue of C with eigenvector v such that Cv = λv.Then Bv is also an eigenvector of C, since CBv = BCv = λBv. Moreover, v and Bv arelinearly independent over R. Otherwise there would exist a μ∈R such that Bv = μv con-tradicting the fact that B has no real eigenvalues. So the dimension of the eigenspace of λis 2 and therefore the multiplicity of λ must be 2.

Suppose the lemma holds for r × r matrices with r even and r < n. We then have toshow that the lemma holds for n× n matrices. Again, let λ be a real eigenvalue of C andv an eigenvector of C such that Cv = λv. Then, for any m ∈ N, we have that Bmv is aneigenvector of C. Indeed, CBmv = BmCv = λBmv. Let S be the subspace of Rn generatedby all vectors Bmv with m ∈N. Then, for any s ∈ S, we have that Cs = λs, so S is part ofthe eigenspace of λ and secondly Bs ∈ S, which implies that S is a B-invariant subspaceof Rn. Let {v1, . . . ,vk} be a basis for S, then we can complete this basis with vk+1, . . . ,vn toobtain a basis for Rn. Writing (the matrices of the linear transformations determined by)B and C with respect to this new basis, implies the existence of a matrix P ∈Gl(n,R) suchthat

PCP−1 =(λIk C2

0 C3

), PBP−1 =

(B1 B2

0 B3

)(4.1)

with B1 a real k× k matrix; B2,C2 real k× (n− k) matrices; and B3,C3 real (n− k)× (n−k) matrices. Of course, the eigenvalues of B1 and B3 are also not real and B3C3 = C3B3.Therefore, k has to be even and we can proceed by induction on B3 and C3 to concludethat the real eigenvalues of C indeed have even multiplicities.

To prove the second claim of the lemma, we suppose that λ1, . . . ,λr are the real eigen-values of C with even multiplicities m1, . . . ,mr and that μ1,μ1, . . . ,μt,μt are the complexeigenvalues of C with multiplicities n1, . . . ,nt. Then

det(In−C

)

= (1− λ1)m1 ···(1− λr

)mr(1−μ1

)n1(1−μ1

)n1 ···(1−μt)nt(1−μt

)nt

= (1− λ1)m1 ···(1− λr

)mr((

1−μ1)(

1−μ1))n1 ···((1−μt

)(1−μt

))nt

= (1− λ1)m1 ···(1− λr

)mr∣∣1−μ1

∣∣2n1 ···∣∣1−μt∣∣2nt .

(4.2)

This last expression is clearly nonnegative since the mi are even. �

We are now ready to prove the main theorem of this paper.

Theorem 4.2. Let M be an n-dimensional infranilmanifold with Abelian holonomy groupF of odd order. Then, for any continuous map f :M→M, N( f )= |L( f )|.


Proof. Let T : F → Aut(G) be the associated holonomy representation and suppose that(d,D) is a homotopy lift of f . To apply Theorem 2.10, we have to calculate the deter-minants det(In − T∗(x)D∗) for any x ∈ F. If D∗ does not commute with T(x)∗ for all1 �= x ∈ F, we can use Proposition 3.1 to obtain that N( f )= |L( f )|.

Now assume that there exists an x0 ∈ F, x0 �= 1, such that T∗(x0)D∗ =D∗T∗(x0). SinceT∗(x0) is of finite odd order, the eigenvalues of T∗(x0) are 1 or non real and T∗(x0) isdiagonalizable (over C). This implies that there exists a P ∈Gl(n,R) such that

PT∗(x0)P−1 =

(In1 00 A2

), (4.3)

with n1 the multiplicity of the eigenvalue 1 and A2 an (n−n1)× (n−n1)-matrix havingnon real eigenvalues. Note that we do not exclude the case where n1 = 0 (i.e., the casewhere 1 is not an eigenvalue of T∗(x0)). Since PD∗P−1 now commutes with PT∗(x0)P−1,we must have that

PD∗P−1 =(D1 00 D2

), (4.4)

with D1 an n1 × n1-matrix and D2 an (n− n1)× (n− n1)-matrix commuting with A2.Moreover, since F is Abelian, all T∗(x) commute with T∗(x0), and hence

∀x ∈ F : PT∗(x)P−1 =(T′1(x) 0

0 T′2(x)

), (4.5)

with T′1 : F →Gl(n1,R) and T′2 : F →Gl(n−n1,R). So we obtain for any x ∈ F

det(In−T∗(x)D∗)= det(In−PT∗(x)P−1PD∗P−1)

= det(In1 −T′1(x)D1

)det

(In−n1 −T′2(x)D2

).

(4.6)

On the second factor of the above expression we can apply Lemma 4.1 sinceA2 commuteswith T′2(x)D2, for any x ∈ F, and A2 only has non real eigenvalues. So the second factoris always positive or zero. (In case n1 = 0, there is no “first factor” and the proof finisheshere.)

To calculate the first factor, we define F1 = F/kerT′1 and consider the faithful repre-sentation T1∗ : F1 → Gl(n1,R) : x �→ T′1(x). One can easily verify that T1∗ is well defined.Note that |F1| < |F| since x0 ∈ ker(T′1) and so we can proceed by induction on the or-der. This induction process ends when F1 = 1 or when for any x1 ∈ F1 : T1∗(x1)D1 �=D1T1∗(x1). �

Example 4.3. Let M = E\H as before and let f2 : M →M be a continuous map inducingthe endomorphism θ2 on E = π1(M). Then we have that

det(I3−D2,∗

)= det(I3−ϕ2

∗D2,∗)=−195, det

(I3−ϕ∗D2,∗

)=−375. (4.7)

Although these determinants are no longer all equal, they still have the same sign, imply-ing N( f )= |L( f )|. (In fact here N( f )=−L( f ).)

K. Dekimpe et al. 11

Finally, we would like to remark that although the fundamental group of an infranil-manifold with an Abelian holonomy group is always solvable (in fact polycyclic), thesemanifolds do not need to be solvmanifolds in general and so the Nielsen theory on thesemanifolds cannot be treated by the techniques developed for solvmanifolds (as in e.g.[5, 6, 8]).

Example 4.4. The almost-Bieberbach group E = 〈a,b,c,α〉 is not the fundamental groupof a solvmanifold. Indeed, suppose that E is the fundamental group of a solvmanifold,then it is known that the manifold admits a fibering over a torus with a nilmanifold asfibre. On the level of the fundamental group, this implies that there exists a short exactsequence

1−→ Γ−→ E −→ A−→ 1, (4.8)

where Γ is a finitely generated torsion free nilpotent group and A is a free Abelian groupof finite rank. However, it is easy to see that [E,E] is of finite index in E, and therefore,the only free Abelian quotient of E is the trivial group. Therefore, there does not exist anormal nilpotent group Γ⊆ E, with E/Γ free Abelian. This shows that E is not the funda-mental group of a solvmanifold.

References

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[2] R. F. Brown, The Lefschetz Fixed Point Theorem, Scott, Foresman, Illinois, 1971.[3] K. Dekimpe, The construction of affine structures on virtually nilpotent groups, Manuscripta

Mathematica 87 (1995), no. 1, 71–88.[4] , Almost-Bieberbach Groups: Affine and Polynomial Structures, Lecture Notes in Mathe-

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[9] T.-H. Kiang, The Theory of Fixed Point Classes, Springer, Berlin; Science Press, Beijing, 1989.[10] S. Kwasik and K. B. Lee, The Nielsen numbers of homotopically periodic maps of infranilmanifolds,

Journal of the London Mathematical Society. Second Series 38 (1988), no. 3, 544–554.[11] K. B. Lee, Maps on infra-nilmanifolds, Pacific Journal of Mathematics 168 (1995), no. 1, 157–

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[14] W. Malfait, The Nielsen numbers of virtually unipotent maps on infra-nilmanifolds, Forum Math-ematicum 13 (2001), no. 2, 227–237.

K. Dekimpe: Department of Mathematics, Katholieke Universiteit Leuven Campus Kortrijk,Universitaire Campus, Etienne Sabbelaan 53, 8500 Kortrijk, BelgiumE-mail address: [email protected]

B. De Rock: Department of Mathematics, Katholieke Universiteit Leuven Campus Kortrijk,Universitaire Campus, Etienne Sabbelaan 53, 8500 Kortrijk, BelgiumE-mail address: [email protected]

H. Pouseele: Department of Mathematics, Katholieke Universiteit Leuven Campus Kortrijk,Universitaire Campus, Etienne Sabbelaan 53, 8500 Kortrijk, BelgiumE-mail address: [email protected]

WECKEN TYPE PROBLEMS FOR SELF-MAPS OFTHE KLEIN BOTTLE

D. L. GONCALVES AND M. R. KELLY

Received 6 October 2004; Revised 1 March 2005; Accepted 21 July 2005

We consider various problems regarding roots and coincidence points for maps into theKlein bottle K . The root problem where the target is K and the domain is a compactsurface with non-positive Euler characteristic is studied. Results similar to those whenthe target is the torus are obtained. The Wecken property for coincidences from K to Kis established, and we also obtain the following 1-parameter result. Families fn,g : K →K which are coincidence free but any homotopy between fn and fm, n �= m, creates acoincidence with g. This is done for any pair of maps such that the Nielsen coincidencenumber is zero. Finally, we exhibit one such family where g is the constant map and if weallow for homotopies of g, then we can find a coincidence free pair of homotopies.

Copyright © 2006 D. L. Goncalves and M. R. Kelly. This is an open access article distrib-uted under the Creative Commons Attribution License, which permits unrestricted use,distribution, and reproduction in any medium, provided the original work is properlycited.

1. Introduction

Given a pair of maps ( f ,g) : X → Y denote by Coin( f ,g) the set {x ∈ X | f (x) = g(x)}.Assume X and Y to be compact manifolds of the same dimension, in which case this set isgenerically a finite set of points. Now suppose that ( f1,g1), ( f2,g2) are homotopic as a pairof maps, and that #Coin( f1,g1) = #Coin( f2,g2) = MC[ f1,g1], where MC[ f ,g] denotesthe minimal number of coincidence points occurring among all pairs ( f ,g′) homotopicto ( f ,g).

A natural question is the following: Is it possible to find a pair of homotopies, Hfrom f1 to f2 and G from g1 to g2, such that #Coin(H(·, t),G(·, t)) = MC[ f1,g1] forall t ∈ [0,1]? In this paper we will refer to this as the 1-parameter minimal coincidenceproblem and will often shorten this to the minimal coincidence problem. A variation ofthe above question is to consider the situation where in one of the two coordinates thesame function appears, and the homotopy between them is constant. We refer to this asthe restricted minimal coincidence problem. If we specialize the restricted problem to the


2 Wecken type problems for self-maps of the Klein bottle

case where X = Y , both g1 and g2 are the identity and the homotopy G remains con-stant this is called the fixed point problem and has been considered in a number of papers[7, 10, 15, 16]. The last partially generalized to coincidence in [14]. If we specialize to thecase where both g1 and g2 are the constant map and the homotopy G remains constantthis is called the root problem.

In a previous paper the authors studied the coincidence problem when the target Yis the 2-dimensional torus [13] (and for the most part the domain as well.) The primaryfeature used was that because the torus has a multiplication the two coincidence problemsare equivalent and can be reduced to considering a root problem, where computations arenot as difficult.

The purpose of this present paper is to study these minimal problems for the casewhere the target is the Klein bottle. Here we are able to take advantage of the multipli-cation on the torus only after passing to a two-fold cover. As a by-product of our calcu-lations we obtain the Wecken property for coincidences of self maps of the Klein bottle.For the fixed point problem this was first established by Brouwer [5].

The results obtained in this paper are organized as follows. In Section 2 we consider theroot problem and show that in the root free case we can always construct an infinite familyof maps no two of which can be joined by a root free homotopy. Here the domain is anarbitrary surface and this result is analogous to that for the torus in [13]. In Section 4 weshow that the root problem has an affirmative solution when both the domain and targetare the Klein bottle and the end maps are not root free. The result is given in Theorem 4.2.

The main body of this paper is Section 3, which gives a study of coincidence for selfmaps of the Klein bottle K . We show that the Wecken property holds (Theorem 3.8)and we also consider the restricted minimal problem defined above. We establish inTheorem 3.10 the following: in any homotopy class of pairs of maps which contains acoincidence free pair the existence of an infinite family of coincidence free pairs eachhaving the same second map, but each homotopy between two distinct members of thefamily which is constant on the second factor must have a coincidence point.

Finally, in Section 5 we consider the relationship between the minimal coincidenceproblem and the restricted minimal problem. In particular, we show that the root prob-lem and the minimal coincidence problem where the second map is the constant map,are not equivalent. The result is given in Corollary 5.2. The proof relies on the fact thatthe second map is the constant map, is the leg which remains unchanged in the restrictedcoincidence problem and the pair of maps is root free. In general the relations betweenthe two problems is not known. For example, we do not know if the minimal coincidenceproblem and the fixed point problem for maps in the Klein bottle are equivalent.

Lastly, we point out that no results are given on these coincidence problems for pairsof maps which have Nielsen coincidence number different from zero. For the torus onehas an affirmative answer [13]. But this seems difficult to extend to the Klein bottle.

2. Root free maps into the Klein bottle

In [12, Theorem 2.2] it was shown that in the setting of orientable surfaces one couldalways construct countable families of root free maps for which no two members of thefamily can be joined by a root free homotopy. In this section we present an analogous

D. L. Goncalves and M. R. Kelly 3

result when the target space is the Klein bottle K . For this result the only restriction onthe domain is that the surface have non-positive Euler characteristic.

Let y0 ∈ K be the base point. Let F be the free group on the two generators a, b, andlet B = a2b2, the relation which defines the fundamental group of the Klein bottle. Givena reduced word w ∈ F and an integer n let w(a,n) be the word obtained by replacing eacha by Bna and each a−1 by a−1B−n.

Lemma 2.1. Consider a cyclically reduced word w ∈ F. If w �= bk for some k, then w(a,n)and w(a,m) are not conjugate for m,n different positive integers.

Proof. This can be proved in exactly the same manner as was [12, Lemma 2.1]. �

Lemma 2.2. Let U be a word, not necessarily reduced, in F. If U = 1 as a reduced word, thenfor any integer n, U(a,n)= 1.

Proof. Just follow the cancellation pattern for U , cancelling B, B−1 pairs along the way.�

Let S be a closed surface with non-positive Euler characteristic. Let e1, . . . ,ek be gen-erators for the fundamental group π1(S) with defining relation given by; [e1,e2]×···×[eh−1,eh]= 1 if F is orientable (here h is even) or by e2

1×···× e2h = 1 if F is non-orientable

(h≥ 2).

Theorem 2.3. Suppose f0 : S→ K is such that y /∈ f0(S) for some y ∈ K . Then there exista countable family of maps fn, each homotopic to f0, such that y /∈ fn(S) and for any twomaps fm, fn in the family with m �= n each homotopy between them has a root at y.

Proof. First identify π1(K − y) with the free group F generated by a and b. Consider thehomomorphism f# : π1(S)→ π1(K) induced by f0 which we express by f#(ei)=wi, whereeach wi is a word in F. The defining relation for S implies the equation [w1,w2]×···×[wh−1,wh]= 1 when S is orientable or w2

1 ×···×w2h = 1 when S is non-orientable.

In the special case that wi = 1 for each i, and since h≥ 2, we define a family of maps byfn(e1)= Bn, fn(e2)= B−n and fn(ei)= 1 for each i > 2. These are well defined maps fromS into K − y, and since B = 1 in K , each is homotopic to any other as maps into K . Onthe other hand, when n �=m the words Bn and Bm are not conjugate in F. Hence, fn andfm are not homotopic as maps into K − y.

Now suppose that at least one of the wi’s is non-trivial. Without loss of generality wecan assume that one such word wj is cyclically reduced and is not a power of b. Define afamily of maps by fn(ei)=wi(a,n), which by Lemma 2.2, is a well defined family of mapsinto K − y. Since wi(a,n)=wi in K , each is homotopic to any other member as maps intoK . But by [12, Lemma 2.1] if S is orientable, or Lemma 2.1 if S is non-orientable, fn andfm are not homotopic as maps into K − y when n �=m and both are positive. �

3. Coincidence free maps from the Klein bottleinto the Klein bottle

In this section we consider the situation where the domain and target are the Klein bottleand the pair of maps ( f ,g) is coincidence free. The purpose of the section is to give two


results regarding such pairs of maps. In the first section we address the Wecken problemfor coincidence free pairs. To do so we set up the notation and preliminary results neededfor the result, given in Theorem 3.8, that the Wecken property holds for coincidences onthe Klein bottle. In the second section we restrict our attention to those homotopies forwhich one of the two factors is kept constant; either the map f at each level or g at eachlevel. We then consider the restricted 1-parameter problem and obtain the result statedin Theorem 3.10.

3.1. The Wecken property. Fix generators α,β for π1(K ,1′) so that the relation αβαβ−1 =1 holds. Let p : T → K be the double cover by the torus, and let a,b generate π1(T ,1),where p(a) = α and p(b) = β2. As an abuse of notation both a and b will also representsimple closed curves meeting at the basepoint of the torus.

To prove this result we first recall the homotopy classification of self-maps of the Kleinbottle.

Lemma 3.1. Let f : K → K be given. Then f# has one of two forms. Either Type I: f#(α)= αrf#(β)= αsβ2q+1 or Type II: f#(α)= 1 f#(β)= αsβ2q.

Proof. The Klein bottle relation αβαβ−1 = 1 viewed as αβ = βα−1 allows for any word tobe converted to a word of the form αaβb. In the process the exponent sum on β remainsthe same. As a result, and since K is a K(π,1), all self-maps are represented by a memberof the family given by

α�−→ αrβu, β �−→ αsβt. (3.1)

Apply this map to the Klein bottle relation to get αrβuαsβtαrβuβ−tα−s = 1. When ap-plying the relation to put this word in normal form we see that the exponent on β is 2u.Hence, u= 0. Also, if t is even, then βtαr = αrβt. The equation above reduces to α2r = 1,and so r = 0. �

Remark 3.2. Following classical notation the map f is orientation-true, see [6] or [9],exactly when it is Type I. In those papers all other maps are classified as either Type II orIII. To simplify notation in this paper we will not use Type III. We simply note that theonly self-map of K that is of Type III is the constant map, which we will consider underthe case of Type II.

As a notation we will index maps as a triple (r,s, t), where r,s are as above and t is theβ exponent of f#(β). When t is even we must have that r = 0. For a given pair ( f1, f2) ofmaps there is a formula for the Nielsen coincidence number of the pair.

Theorem 3.3 [8]. Let f1, f2 : K → K be as above, with fi = (ri,si, ti). Then

N(f1, f2

)= ∣∣t1− t2∣∣max

{∣∣r1∣∣,∣∣r2∣∣}. (3.2)

For various calculations in this paper we will be considering lifts of maps to the torus.We list here for reference the form of these lifts.


In the case that f is Type I we can lift the map by lifting both the domain and the range.

Here we get f : T → T , and with the notation above, f (a)= ar , f (b)= b2q+1. Let θ : T →T denote the deck transformation corresponding to the cover p. Then the associated lift

f ′ = θ f satisfies f ′(a)= a−r , f ′(b)= b2q+1, where a,b in the target mean the translatedloops based at θ(1).

When f is not orientation-true it factors through the torus. So we have a map f : K →T where f (α) = 1 and f (β) = asbq. In this case the associated lift f ′ is f ′(α) = 1 and

f ′(β)= a−sbq. We can also lift in the domain as well. When we do so we now have that f

is given by f (a)= 1 and f (b)= a2sb2q, and that f ′(a)= 1 and f ′(b)= a−2sb2q. As before,

for f and f ′ the loops in the target are based at θ(1).At this point we digress a moment to compare a result for maps on the torus to its

counterpart for the Klein bottle. A class of spaces called Jiang type spaces, which includesthe torus, has the property that when the Nielsen number of a pair of maps is zero thenthe Reidemeister number is infinite, and when the Nielsen number is nonzero the two areequal (see, e.g., [19].) The next proposition shows that this does not hold for the Kleinbottle, even in the fixed point case.

Proposition 3.4. Let f1, f2 : K → K be as above, with fi = (ri,si, ti). Then the coincidenceReidemeister number for the pair is infinite exactly when either N( f1, f2)= 0 or |r1| = |r2|.Proof. This follows directly from the matrices which arise when lifting f1, f2 to maps ofthe torus. �

In order to study coincidences of pairs of maps on the Klein bottle we will first lift mapsto the torus and then use a multiplication on the torus to reduce to the root problem fora deviation map h given by h(x) = f1(x) f2(x)−1. We now vary h by a homotopy for the

root problem, and then recover equivariant maps on the torus by keeping f2 fixed and

obtaining a new f1 from this formula.As an immediate consequence of the definition of h and Theorem 3.3 we have the

following lemma. Its proof is left to the reader.

Lemma 3.5. IfN( f1, f2)= 0, then the map h : T → T has the property that a�→ ar1−r2 , whereri = 0 when fi is of Type II. Furthermore, r1− r2 is non-zero only when both maps are of TypeI and t1 = t2.

As a consequence we see that the Wecken problem for coincidence in the caseN( f1, f2)= 0 and r1 = r2 is now easy to solve. Since the loop a is mapped by h to a single point,h(θ(a)) must also be a single point. As a result h maps T into a 1-complex determinedby h(b). After a small deformation we arrange that h is a root free map at both 1 and

θ(1). Thus, both pairs of lifts h f2, f2 and h f2,θ f2 are coincidence free, and we concludethat ( f1, f2) can be deformed to a coincidence free pair. Details are given in the proof ofProposition 3.4.

The case when r1 �= r2 is more subtle. To analyze this case, and also to deal with the 1-parameter problem, we will need to see how the various lifts, and hence h, act on a certain1-complex in T . To define this complex let T be the identification of the unit square[0,1]2 with a = {0}× [0,1] and b = [0,1]×{0}. Let σa = a and σb = [0,1/2]×{0}. Our


1-complex L is the union of a,b and θ(a). The action of an equivariant map is determinedby the action on σa and on σb.

Given f : K → K which is assumed to be given efficiently in terms of the generatorsα,β, a model for a lift of f on σa∪ σb is given by the following lemma

Lemma 3.6. If f is Type I, then f (σa)= ar , f (σb)= asbqσb. If f is Type II, then

f(σa)= 1, f

(σb)= asbq. (3.3)

We now check the action of h : T→T on L by first computing h(θz)= f1(θz)( f2(θz))−1.We express points in complex (x, y) coordinates where x measures the b or β direction,and y the a or α direction, depending on location in T or in K . The action of θ is givenby (x, y) �→ (eπix, y), and multiplication is the product of the coordinates. The inverse isobtained by complex conjugation in each coordinate.

One feature of multiplication that we exploit is its relation with the deck transforma-tion θ. Namely, under products we see that

θ(AB)= θ(A)θ(B)(eπi,1

). (3.4)

Case 1. Both f1, f2 are Type I. (t1, t2 odd.)

Then reduce h(θz) to θ f1(z)(θ f2(z))−1. This is equal to θ(h(z) f2(z))(θ f2(z))−1, or

θh(z)θ f2(z)(eπi,1)(θ f2(z))−1 = θh(z)(eπi,1).

Case 2. Both f1, f2 are Type II. (t1, t2 even.)

Then reduce to f1(z)( f2(z))−1 = (h(z) f2(z))( f2(z))−1 = h(z).

Case 3. t1 odd, t2 even.Reduce to θ(h(z) f2(z))( f2(z))−1. Which is θh(z)θ f2(z)(eπi,1)( f2(z))−1.

Using coordinates for f2(z)= (ub,ua), this reduces to θh(z)(1,u−2a ).

Case 4. t1 even, t2 odd.Reduce to (h(z) f2(z))(θ f2(z))−1 which is h(z)(eπi,u2

a).We now revisit these four cases using the given information of h(z) on the loop σa and

the path σb as given in Lemma 3.6.

Case 1. In view of Lemma 3.5 we divide this case into two subcases.

Subcase 1(i). r1 �= r2. Here h(σa)= (1,ar1−r2 ), and since q1 = q2, h(σb)= (1,as1−s2 ).So we get h(θσa) = θ(1,ar1−r2 )(eπi,1) = (1,ar2−r1 ), and similarly, h(θσb) = θ(1,as1−s2 )

×(eπi,1)= (1,as2−s1 ).We see from these equations that on the complex L the action of h is by a power

of a. Moreover, consider the loop in L formed by (in order) σa, [0,1/2]× {1},{1/2} ×[1,0],σ−1

b . (Or equivalently σaσbθ(σa)σ−1b .) The action of h on this loop is given by the

word h(σa)h(σb)h(θσa)h(σb)−1. But this is (1,ar1−r2 )(1,as1−s2 )(1,ar2−r1 )(1,as2−s1 ) = (1,1).A similar calculation on the other loop in L shows that the exponents cancel to zero aswell. As a result we can arrange that h(T) is contained in a 1-complex determined by h(σa)


and so this representative gives us a root free map. Moreover, keeping f2 fixed, from the

construction we see that the corresponding f1 is an equivariant map. So we have a coin-cidence free pair.

Subcase 1(ii). r1 = r2. Then σa �→ 1, σb �→ (bq1−q2 ,as1−s2 ) by h. So h(θσa)= θ(1,1)(eπi,1)=(1,1) and h(θσb)= θ(bq1−q2 ,as1−s2 )(eπi,1)= (bq1−q2 ,as2−s1 ).

Here h as defined on L extends to T by mapping into a 1-complex, this time deter-mined by h(σb). This will also happen in the remaining three cases.

Case 2. Here a�→ 1, σb �→ (bq1−q2 ,as1−s2 ) h(θσa)= h(σa)(1,1)=(1,1) and h(θσb)=h(σb)(1,1)= (bq1−q2 ,as1−s2 ).

Case 3. a�→ 1, σb �→ (bq1−q2σb,as1−s2 ) h(θσa)= θ(1,1)(1,u−a 2)= (eπi,1), as (ub,ua)= (1,1)in this case, and h(θσb)= θ((bq1−q2σb,as1−s2 )(1,a−2s2 )= (eπibq1−q2σb,a−s1−s2 ).

Case 4. a �→ 1, σb �→ (bq1−q2σ−1b as1−s2 ) h(θσa) = (h(σa)(1,1))(eπi,1) = (eπi,1) and h(θσb)

= (bq1−q2σ−1b ,as1−s2 )(eπi,a2s2 )= (eπibq1−q2σ−1

b ,as1+s2 ).

Each of the above calculations can be repeated with f2 replaced by θ f2. For example,in the Case 1(i) it can be shown that h(σa) = (eπi,ar1+r2 ), h(σb) = (eπi,a−s1+s2 ), h(θσa) =(eπi,a−r1−r2 ), and h(θσb)= (eπi,as1−s2 ). Similar formulas arise in all the other cases.

As a result, we see that the deviation map h has image in a 1-complex. In particular,the image could be taken to be in the “lines” in the torus determined by h(b). As a conse-quence we can now show that the Klein bottle has the Wecken property for coincidences.

Proposition 3.7. Let f1, f2 : K → K be such that N( f1, f2) = 0. Then we can deform thepair to one that is coincidence free.

Proof. Choose representatives and lifts f1, f2 as above. Let y be a point on the torus.Consider the map hy : T → T constructed just as h, but now given on L by the dataσa �→ yh(σa), σb �→ yh(σb). In all cases the image of hy lies in a 1-complex, and for a suit-able choice of y the image lies in T − (1∪ θ(1)).

Now, keeping f2 fixed we construct from hy an equivariant map f ′1 such that each of

the pairs ( f ′1 , f2) and ( f ′1 ,θ( f2)) is coincidence free. Hence, ( f ′1 , f2) is coincidence free

where f ′1 is the projection of f ′1 . �

Theorem 3.8. Given any pair f1, f2 : K → K we can deform the pair to one that has exactlyN( f1, f2) coincidence points.

Proof. When the Nielsen number is nonzero we show that the “linear” model has exactlyN( f1, f2) coincidence points. This model is obtained by lifting to the torus where we havea piecewise linear map. We present the case where f1 is of Type I and f2 is of Type II. Thedetails of the other cases are similar and left to the reader.

By Lemma 3.6 we have that f1(σa)= ar1 , f1(σb)= as1bq1σb, f2(σa)= 1, f2(σb)= as2bq2 .

The map f2 extends to a linear map of the torus given by (x, y) �→ (2q2x,2s2x)(mod1),

where the x-factor corresponds to the b direction. The map f1 can be represented by themap (x, y) �→ ((2q1 + 1)x,r1y± s1x)(mod1). The choice of ± depends on the value x, +

if 0 < x < 0.5, − if 0.5 < x < 1. Finding coincidence points for the pair ( f1, f2) reduces


to solving the equations (2q1 + 1− 2q2)x = 0(mod1) and (r1y± s1x)− 2s2x = 0(mod1).The first has exactly |(2q1 + 1− 2q2)| solutions and for each of these solutions the secondequation has |r1| solution provided that r1 �= 0.

In the case under consideration the coincidence set for the lifts projects one-to-oneand onto the coincidence set for ( f1, f2) and so by Theorem 3.3 the result is proved. �

3.2. The 1-parameter problem. For the problem of deforming a pair of maps to one thathas the least number of coincidence points it is known, in the setting of closed manifolds,that it suffices to deform either one of the two maps [2]. For the 1-parameter Weckenproblem this is not known in general, but does hold when the target is a topologicalgroup.

The following proposition shows that when considering the restricted coincidenceproblem (where the second factor g is unchanged) the solution does not depend on thechoice of map in the homotopy class.

Proposition 3.9. Let ( f ,g) :M→N be a pair of maps which satisfies Coin( f ,g)=MC( f ,g) and g1 a map homotopic to g. Then:

(a) the restricted minimal coincidence problem has a positive solution for ( f ,g) if andonly if it has a solution for a pair ( f ′,g), where f ′ is homotopic to f ;

(b) there exists f1 homotopic to f such that coin( f1,g1)=MC( f ,g) and the restrictedminimal coincidence problem has a positive solution for ( f ,g) if and only if it has asolution for the pair ( f1,g1).

Proof. In both parts (a) and (b) it suffices to assume that the Wecken problem has apositive solution. For the part (a) we have that the pair ( f ,g) can be connected to thepair ( f ′,g) by a Wecken homotopy. So given any pair ( f ′′,g) where f ′′ is homotopic tof ′ (so homotopic to f ) we consider the homotopy which is the Wecken homotopy from( f ′,g) to ( f ,g) followed by a Wecken homotopy from ( f ,g) to ( f ′′,g), which exists byhypothesis, and the result follows.

For part (b) consider the fibre pair (N ×N ,N ×N −Δ) → N , by projection on thesecond coordinate. We argue as in [11, Proposition 1.5], letH :M× I →N be a homotopybetween g and g′. This homotopy restricted to M ×{0} has a lift given by ( f ,g), whichmapsM− coin( f ,g) intoN ×N −Δ. Define f1 as the first coordinate function of the mapH restricted to M×{1}, where H is the lifting of H given by the homotopy property ofthe pair. So H(M− coin( f ,g)×{t})⊂N ×N −Δ for all t ∈ [0,1] and f1,g1 is a minimalpair. Now let f ′1 be any map homotopic to f1 (so homotopic to f ) such that the pair ( f ′1 ,g)is minimal. By using the procedure above we can produce a map f ′ which is homotopicto f and the pair ( f ′,g) is minimal. So by hypothesis there exists a Wecken homotopyL connecting ( f ,g) to ( f ′,g). Now we use the procedure above to produce a Weckenhomotopy connecting ( f ′1 ,g1) to ( f1,g1). Define L1 : M× I × I → N given by L1(x,s, t) =H(x,s) where H is the homotopy between g and g′. This map restricted to M ×{0}× Iadmits a lift given by L. Define L1 as the first coordinate function of the lift L1 of L1

restricted to M×{1}× I . This is a Wecken homotopy connecting ( f1,g1) to ( f ′1 ,g1) andthe result follows. �

The main purpose of this section is to prove the following theorem.


Theorem 3.10. Let ([ f ],[g]) be a pair of homotopy classes of self-maps of K such thatN( f ,g)= 0. Then given a map g′ ∈ [g] there is a countable family of maps fn, where eachfn ∈ [ f ] and coin( fn,g′)=∅, such that for any two pairs ( fm,g′), ( fn,g′) withm �= n thereis no homotopy H between fm, fn with the property that (H( , t),g′) is coincidence free for allt ∈ [0,1].

In order to proceed with the proof of this theorem we will first need to constructa suitable family of maps. In view of Proposition 3.9 we choose g′ as the linear model

determined by the images of σa and σb. For the following proof we will use f2 to denotethe lift of g′ to the torus. Families in the homotopy class of [ f ] will constructed for eachcase, by first defining the maps on σa∪ σb, and then using the formulas of Cases 1–4 we

define families of deviation maps hl. We set f1(x) = hl(x) f2(x) keeping in mind that f1represents an arbitrary member of some family of maps. Finally, we project to get familiesof maps on K . Choices will be made so that N( f1, f2) = 0, for each possible f1. To getcoincidence free pairs we will need to make a slight perturbation, which will be done onthe torus.

We first consider the situation in Case 1. If a segment of σa ∪ σb is mapped by h tothe loop a, then hθ of the segment is mapped to (θa)(eπi,1). But this is just the loopa−1. Similarly, if a segment of σa ∪ σb traces out the loop b, then hθ of the segment ismapped to b. As a result, under the hypothesis of Case 1 we have the following methodof substitution.

Lemma 3.11. Suppose that both f1 and f2 are of Type I. Let W be a word in the letters a,band let W be the word obtained by replacing each occurrence of a with a−1 keeping b withunchanged. If h(σl), l ∈ {a,b}, contains W , then h(θσl) contains W in its place.

We now define families of pairs of maps in a given homotopy class of pairs. We firstmake an adjustment so that all maps are coincidence free.

Let ε : T → T be a homeomorphism near the identity and such that ε(a) and ε(b) donot contain 1. Our deviation maps will be defined on generators a,b and will have imagesin ε(a)∪ ε(b), and will extend to the interior of T with image in the same 1-complex.Clearly, any such map will have no root at 1. By abuse of notation in the following we willwrite a, b instead of ε(a), ε(b). For instance, h(a) = ab means the image of a traces outε(a) then ε(b).

Let B denote the commutator aba−1b−1 in the free group on generators a,b. Given aword ω, let Bω = ωBω−1.

Let h be as presented in Case 1. For integers m,x, y we define a family of maps hm,x,y

according to the following:

σa �−→ BmaxBmaya

r1−r2 , σb �−→ as1−s2 . (3.5)

Keeping f2 fixed each hm,x,y determines a family of lifts f1, where each pair is homopticto the original model.

The following gives a condition that ensures our pair is coincidence free.


Lemma 3.12. If x + y = r1 − r2 + s1 − s2 − 1, then hm,x,y extends to all of T with image ina∪ b.

Proof. We check the condition for cancellation around the loop σaσb(θσa)σ−1b . The other

loop in L is the same. Set r = r1− r2 and s= s1− s2. Then this is equivalent to the equation

BmaxBmaya

r = as(BmaxBmayar)−1

a−s = asarB−ma−y B−ma−x a−s = as+rBma−1−yBma−1−x a−s, (3.6)

due to the fact that B−1 = Ba−1 . Rewrite the right-hand side as

Bmar+s−1−yBmar+s−1−x ar+sa−s. (3.7)

Equate with the left-hand side to obtain x + y = r + s− 1 which gives the desired result.�

Proof of Theorem 3.10

Case 1(i). Suppose two pairs from the construction above which also satisfy the conclu-sion of Lemma 3.12 are joined by a coincidence free homotopy. Lift the homotopy to thetorus to get a coincidence free homotopy between the corresponding pairs. Choose thelift so that it produces the deviation map of the form σa �→ ar , σb �→ as.

As a result we have a homotopy ht between two deviation maps h0 = hm1,x1,y1 and h1 =hm2,x2,y2 with no root at 1 for each level of the homotopy. Hence, h0 and h1 are homotopicas maps into T − 1 (recall that a,b below are actually ε(a),ε(b).) Since σa is a loop itsimage under each level of the homotopy is a loop in T − 1. This implies that there existsa word φ in the free group such that

Bm1ax1B

m1ay1ar = φ

(Bm2ax2B

m2ay2ar

)φ−1, (3.8)

or

Bm1ax1B

m1ay1B

−m2ay2 B−m2

ax2 = φ(Bm2ax2B

m2ay2ar

)φ−1a−r

(Bm2ax2B

m2ay2

)−1,

Bm1ax1B

m1ay1B

−m2ay2 B−m2

ax2 = [φ,(Bm2ax2B

m2ay2ar

)].

(3.9)

So by [1, Theorem 9.1 part (b)], this equation has no solution if 2|m1−m2| > 4 (withh = g = 1 and l = 4.) Therefore if we take the sequence of integers of the form 3n weobtain the result.

Case 1(ii). In order to facilitate computations we take as our base pair ( f1, f2) obtainedfrom the deviation map given by σa �→ 1 σb �→ X , where X is the path in a∪ b correspond-ing to the word akbl, where k = s1 − s2 and l = q1 − q2. This pair is homotopic to thatgiven by the deviation map which sends σb to the path (bq1−q2 ,as1−s2 ). Our family of maps

hn : T → T given by σa �→ 1, σb �→ BnX determine equivariant pairs f1, f2 as in Case 1(i),


and thus pairs of maps on the Klein bottle. The construction ensures that hn and hmdetermine the same homotopy class of pairs on K . Also, since each maps σa to 1 each hnmaps into T − 1 resulting in coincidence free pairs.

The existence of a coincidence free homotopy between any two pairs lifts and mul-tiplies to a root free at 1 homotopy between hn and hm. This implies that BnXBnX =φBmXBmXφ−1. We claim that this only happens when n=m.

Given a word α (in the free group on letters a,b) define the integer t(α) to be theminimal number of transitions between the letters a± and b± among all words conjugateto α. For example, t(ab3a−1bab−1a)= 5. Clearly, t() is an invariant of a conjugacy class.

For the words of the form BnX(BnX) the calculation of t() is straightforward. UsingLemma 3.11 one can show that t(BnX(BnX)) = 8n− c, where c is a constant which de-pends on k, l and takes values in {−3,1,5}, with one exception. When k = 1, l = 0 we gett(Bna(Bna))= 0, which is a result of the fact that B = a−1B−1a.

Hence we are finished with the proof except when k = 1, l = 0. Now to handle this onespecial case we go back to a different choice.

Of a family of maps. In place of using B = aba−1b−1 we use B = ba−1b−1a. Followingthrough the exact same proof, and now with this change the value of t() becomes 8n− 1.

Case 2. Is similar and in fact this is the easy case because the action of θ on the devi-ation map is given by h(θσb) = h(σb). We consider a family of maps hn : T → T just asin Case 1(ii), determined by σa �→ 1, σb �→ BnX , where X denotes the path (bq1−q2 ,as1−s2 )deformed into as1−s2bq1−q2 as a path in a∪ b.

A Wecken homotopy between hn and hm reduces to the algebraic conclusion that

BnXBnX = φBmXBmXφ−1. (3.10)

A straightforward calculation of t(BnXBnX) shows that this is impossible when n �=m.

Case 4. In this case we deform (bq1−q2σb,as1−s2 ) into the path Xσb contained in a∪ b,where X corresponds to the word akbl with k = s1 − s2, l = q1 − q2. We parametrize insuch a way so that X(eπi,a2s2 ) deforms into a∪ b by a Wecken homotopy, producing thepath σ−1

b akbl+1a2s2 . That is when X traces out a letter only the eπi factor acts on the letter.This action replaces the letter x by σ−1

b xσb and the path σb by σ−1b b.

Our family of maps will be hn sending σa �→ 1, σb �→ BnX . Then h(θσb) = σ−1b Bnak

×bl+1a2s2 .In this case a Wecken homotopy between the pairs corresponding to hn and hm implies

that BnakblBnakbl+1a2s2 and BmakblBmakbl+1a2s2 are conjugate in the free group. The in-variant t() can be computed to show that this is only possible when n=m.

Case 3. This case is just like Case 4 and the details are left to the reader. �

4. Maps which are not root free

In Section 2 we studied the 1-parameter root problem for root free maps from an arbi-trary closed surface into K . We now consider this problem in the presence of roots, and


to do so we will need to restrict the domain to K as well so that we may take advantage ofsome known results regarding roots. In this setting being root free (up to homotopy) isequivalent to having absolute degree equal to zero. To see this, [4, Theorem 2] tells us thatthe Nielsen root number is the same as the absolute degree. (See also [3, Theorem A.3].)By consideration of Theorem 3.3 and Lemma 3.1 in this paper we see that the Nielsenroot number is zero exactly when the loop α is mapped to a point. The reader can checkthat such maps can always be deformed to be root free.

The result we obtain below for roots is the same as that for the torus [13]. We first needthe following lemma in the case when the absolute degree is 1.

Lemma 4.1. Let f : K → K be a map of absolute degree 1. Suppose that y1, . . . , yl is a finiteset of points in K such that f −1(yi) is a single point for each i, and further, the local degree off at f −1(yi) is independent of i. Then we can deform f relative to f −1(y1), . . . , f −1(yl) to ahomeomorphism f ′ such that the homotopyH between f and f ′ satisfies #H(·, t)−1(yi)= 1for all t ∈ [0,1] and i= 1, . . . , l.

Proof. For simplicity we assume i= 1 and set y = y1. The proof for i > 1 is identical. LetN be a small neighborhood of y with N \ y foliated by circles γt, 0 < t ≤ 1. Let M be asmall neighborhood of f −1(y) with M \ f −1(y) foliated by δt. Since f −1(y) is a singlepoint we can deform f to a map g such that g−1(y)= f −1(y), g(δt)⊂ γt for each t, andg(K \M)⊂ (K \N). As the local degree at f −1(y) is ±1 we can assume that g is one-to-one on δ1. Moreover, there is such a homotopy between g and f which has a single pointin the preimage of y at each level.

Since g has absolute degree 1, by excision we see that the relative map g : (K \M,∂M)→(K \N ,∂N) also has absolute degree 1. It then follows from [18, Theorem 1.1] that thisrelative map g is homotopic, rel boundary, to a homeomorphism. Finish by extending toall of K using the constant homotopy from M to N to obtain the desired f ′. �

Theorem 4.2. Let f1, f2 : K → K be homotopic maps each having absolute degree n, withn �= 0. If for some y ∈ K , # f −1

i (y) = n for i = 1,2, then there is a homotopy H between f1and f2 such that #H(·, t)−1(y)= n for all t ∈ [0,1].

Proof. We first note that it follows from [4, Theorem 2] that fi#(π1(K)) is a subgroupof π1(K) of index n. Let p : K ′ → K be the n-fold covering which corresponds to the

subgroup fi#(π1(K)) and let fi : K → K ′ denote the lift of fi. Certainly fi# is surjective,and since K ′ is either the Torus or the Klein bottle it follows again from [4, Theorem 2]

that this map is also injective. Therefore, fi is a homotopy equivalence. So K ′ is a Klein

Bottle and fi has absolute degree one. Moreover, fi is homotopic to a homeomorphismand is a Type I map of the form α�→ α, β�→ αsβ.

Let y1, . . . , yn be the preimage of the point y of the covering map. Then fi# maps the

set f −1i (y) onto these n points. So, fi viewed as a map on the compliment of these points

is also homotopic to a homeomorphism, and we see that the local degree at each point in

f −1i (y) has the same value in±1. By Lemma 4.1 there are homotopies Fi with ends fi and

a homeomorphism hi. Since h1,h2 are homotopic, they are also isotopic. Putting together


F1,F2 and the isotopy yields a homotopy H between f1 and f2 such that the compositionp ◦ H provides the required homotopy. �

5. The coincidence free case revisited

In this section we revisit the two stated variations of the 1-parameter coincidence prob-lem. One being the situation where we allow homotopies of both maps and the otherbeing the restricted problem where one map is held constant. The later was addressed inSection 3 of this paper.

The purpose of this section is to show that these two problems are different for coin-cidences between pairs of maps from the Klein bottle to itself. In general such an analysisis quite difficult. To facilitate calculation we set one of the maps to be the constant mapat a point c. This will also be the fixed map for the restricted Wecken problem, and as aresult this restricted problem reduces to a root problem.

Denote by e1,e2 ∈ π1(K) generators which satisfy the relation e21e

22 = 1. Given a pair of

maps f and g which are coincidence free, denote by ( f ,g)# : π1(K)→ π1(K ×K −Δ) thehomomorphism in the fundamental group induced by the map ( f ,g) : K → K ×K −Δ.

Proposition 5.1. Given f1, f2 : K → K − c consider the two homomorphisms ( f1,c)# and( f2,c)#. The pairs ( f1,c) and ( f2,c) can be connected by a Wecken homotopy if and only ifthe two homomorphisms are conjugate.

Proof. Assuming the existence of a Wecken homotopy it is well known that the inducedhomomorphisms in the fundamental group are conjugate (the homotopy is not neces-sarily base point preserving.) Conversely, since the spaces K and K ×K −Δ are K(π,1)’sit then follows that the pairs ( f1,c) and ( f2,c) can be joined by a coincidence free homo-topy. �

By abuse of notation let e1,e2 ∈ π1(K − c) denote a basis for the free group where theseelements project into e1,e2 ∈ π1(K) respectively, and let B = e2

1e22.

Corollary 5.2. Let f1, f2 : K → K − c be two maps such that in the fundamental groupthey induce the following homomorphisms:

f1#(e1)= e1, f1#

(e2)= (e1

)−1,

f2#(e1)= e1B

−1, f2#(e2)= B(e1

)−1.

(5.1)

Then the pairs ( f1,c) and ( f2,c) can be connected by a Wecken homotopy but f1 can not beconnected to f2 by a homotopy which is root free at c.

Proof. That f1 can not be connected to f2 by a root free at c homotopy follows from thefact that f1#(e1) and f2#(e1) are not conjugate as elements of π1(K − c). This is straight-forward in K − c because the words e1 and e1B−1 do not have the same exponent sumsin the free group π1(K − c). To see that ( f1,c) and ( f2,c) can be connected by a Weckenhomotopy, by Proposition 5.1 it suffices to see that the two homomorphisms, ( f1,c)# and( f2,c)# are conjugate. Using the relations given in the paper of Scott [17] we have that


ρ21ρ11ρ−121 = ρ11B−1, and so ρ21ρ

−111 ρ

−121 = Bρ−1

11 . In this notation we have

(f1,c)

#

(e1)= ρ11,

(f1,c)

#

(e2)= (ρ11

)−1,

(f2,c)

#

(e1)= ρ11B

−1 (f2,c)

#

(e2)= B(ρ11

)−1.

(5.2)

This implies that ρ21( f1,c)#ρ−121 = ( f2,c)# and the result follows. �

Acknowledgment

The second author would like to thank the IME at Universidade de” Sao Paulo for itshospitality during the preparation of this manuscript.

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D. L. Goncalves: Departamento de Matematica, IME-USP, Caixa Postal 66281,Ag. Cidade de Sao Paulo CEP: 05315-970, Sao Paulo, SP, BrasilE-mail address: [email protected]

M. R. Kelly: Department of Mathematics and Computer Science, Loyola University,6363 St. Charles Avenue, New Orleans, LA 70118, USAE-mail address: [email protected]

ALGEBRAIC PERIODS OF SELF-MAPS OF A RATIONALEXTERIOR SPACE OF RANK 2

GRZEGORZ GRAFF

Received 29 November 2004; Revised 27 January 2005; Accepted 21 July 2005

The paper presents a complete description of the set of algebraic periods for self-maps ofa rational exterior space which has rank 2.

Copyright © 2006 Grzegorz Graff. This is an open access article distributed under theCreative Commons Attribution License, which permits unrestricted use, distribution,and reproduction in any medium, provided the original work is properly cited.

1. Introduction

A natural numberm is called a minimal period of a map f if f m has a fixed point which isnot fixed by any earlier iterates. One important device for studying minimal periods arethe integers im( f )=∑k/m μ(m/k)L( f k), where L( f k) denotes the Lefschetz number of f k

and μ is the classical Mobius function. If im( f )�= 0, then we say that m is an algebraicperiod of f . In many cases the fact that m is an algebraic period provides informationabout the existence of minimal periods that are less then or equal to m. For example, letus consider f , a self-map of a compact manifold. If f is a transversal map and odd mis an algebraic period, then m is a minimal period (cf. [10, 12]). If f is a nonconstantholomorphic map, then there exists M > 0 such that for each prime number m >M, mis a minimal period of f if and only if m is an algebraic period of f (cf. [3]). Furtherrelations between algebraic and minimal periods may be found in [8].

Sometimes the structure of the set of algebraic periods is a property of the space andmay be deduced from the form of its homology groups. In [11] there is a descriptionof algebraic periods for self-maps of a space M with three nonzero (reduced) homologygroups, each of which is equal to Q, in [6] the authors consider a space M with nonzerohomology groups H0(M;Q) =Q, H1(M;Q) =Q⊕Q. The main difficulty in giving theoverall description in the latter case is that for a map f∗ induced by f on homology, foreach m there are complex eigenvalues for which m is not an algebraic period. Rationalexterior spaces are a wide class of spaces (e.g., Lie groups) which do not have this disad-vantage, namely under the natural assumption of essentiality of f there is a constant mX

and computable set TM , such that if m >mX , m�∈ TM , then m is an algebraic period off (cf. [5]). The aim of this paper is to provide a full characterization of algebraic periods


2 Algebraic periods for maps of rational exterior spaces

in the case when homology spaces of X are small dimensional, namely when X is of therank 2. Our work is based on [1, 9], where the description of the so-called “homotopicalminimal periods” of self-maps of, respectively the two-, and three-dimensional torus aregiven using Nielsen numbers. We follow the algebraical framework of [9], the final de-scription is similar to the one obtained in [1]. The differences result from the fact that thecoefficients im( f ) are a sum of Lefschetz numbers, which unlike Nielsen numbers, do nothave to be positive.

2. Rational exterior spaces

For a given space X and an integer r ≥ 0 let Hr(X ;Q) be the rth singular cohomologyspace with rational coefficients. Let H∗(X ;Q) =⊕s

r=0Hr(X ;Q) be the cohomology al-

gebra with multiplication given by the cup product. An element x ∈Hr(X ;Q) is decom-posable if there are pairs (xi, yi) ∈ Hpi(X ;Q)×Hqi(X ;Q) with pi,qi > 0, pi + qi = r > 0so that x =∑xi ∪ yi. Let Ar(X) = Hr(X)/Dr(X), where Dr is the linear subspace of alldecomposable elements.

Definition 2.1. By A( f ) we denote the induced homomorphism on A(X)=⊕sr=0A

r(X).Zeros of the characteristic polynomial ofA( f ) onA(X) will be called quotient eigenvaluesof f . By rankX we will denote the dimension of A(X) overQ.

Definition 2.2. A connected topological space X is called a rational exterior space if thereare some homogeneous elements xi ∈ Hodd(X ;Q), i = 1, . . . ,k, such that the inclusionsxi↩H∗(X ;Q) give rise to a ring isomorphism ΛQ(x1, . . . ,xk)=H∗(X ;Q).

Finite H-spaces including all finite dimensional Lie groups and some real Stiefel man-ifolds are the most common examples of rational exterior spaces. The two dimensionaltorus T2, a product of two n-dimensional sphere Sn× Sn, and the unitary group U(2) areexamples of rational exterior spaces of rank 2.

The Lefschetz number of self-maps of a rational exterior space can be expressed interms of quotient eigenvalues.

Theorem 2.3 (cf. [7]). Let f be a self-map of a rational exterior space, and let λ1, . . . ,λk bethe quotient eigenvalues of f . LetA denote the matrix ofA( f ). Then L( f m)= det(I −Am)=∏k

i=1(1− λmi ).

Remark 2.4. A basis of the space A(X) may be chosen in such a way that the matrix A isintegral (cf. [7]).

3. The set of algebraic periods of self-maps of rational exterior space of rank 2

Let μ denote the Mobius function, that is, the arithmetical function defined by the threefollowing properties: μ(1) = 1, μ(k) = (−1)r if k is a product of r different primes, andμ(k)= 0 otherwise. Let APer( f )= {m∈N : im( f )�= 0}, where im( f )=∑k/m μ(m/k)L( f k).We will study the form of APer( f ) for f : X → X and X a rational exterior space of rank 2.We assume that X is not simple which means that there exists r ≥ 1 such that dimAr = 2,otherwise, that is, if there are i, j ≥ 1 such that dimAi = dimAj = 1, we get the case with

Grzegorz Graff 3

Table 3.1. The set of algebraic periods APer( f ) for the set R.

No. (t,d) APer( f )

10 (−2,1) {1,2}20 (−1,0) {1,2}30 (0,0) {1}40 (0,1) {1,2,4}50 (1,1) {1,2,3,6}60 (−1,1) {1,3}

integer quotient eigenvalues (cf. [7]) for which the description of APer( f ) easily followsfrom the case under consideration.

By Theorem 2.3 we see that A is a 2× 2 matrix and that the Lefschetz numbers L( f m)are expressed by its two quotient eigenvalues (in short we will call them eigenvalues):λ1,λ2 : L( f m) = (1− λm1 )(1− λm2 ). The characteristic polynomial of A has integer co-efficients by Remark 2.4 and is given by the formula: WA(x) = x2 − tx + d, where t =λ1 + λ2 is the trace of A and d = λ1λ2 is its determinant. The characterization of the setAPer( f ) will be given in terms of these two parameters: t and d. Let us define the setR= {(−2,1),(−1,0),(0,0),(0,1),(1,1),(−1,1)}.Theorem 3.1. Let f be a self-map of a rational exterior space X of rank 2, which is notsimple. Then APer( f ) is one of the three mutually exclusive types:

(E) APer( f ) is empty if and only if 1 is an eigenvalue of A, which is equivalent tot−d = 1.

(F) APer( f ) is nonempty but finite if and only if all the eigenvalues of A are either zeroor roots of unity not equal to 1, which is equivalent to (t,d) ∈ R. The algebraic periodsfor the set R are given in Table 3.1.

(G) APer( f ) is infinite. Assume that (t,d) is not covered by the types (E) and (F),then,

(1) for (t,d)= (−2,2), APer( f )=N \ {2,3};(2) for (t,d)= (−1,2), APer( f )=N \ {3};(3) for (t,d)= (0,2), APer( f )=N \ {4};(4) for t =−d and (t,d)�= (−2,2), APer( f )=N \ {2};(5) for t+d =−1, APer( f )=N \ {n∈N : n≡ 0 (mod4)};(6) if (t,d) is not covered by any of the cases 1–5, then APer( f )=N.

Remark 3.2. The letters E, F, G are chosen to represent empty, finite and generic case,respectively, which corresponds to the notation used in [9].

The rest of the paper consists of the proof of Theorem 3.1 and is organized in thefollowing way: in the first part we describe the conditions equivalent to the fact that m∈{1,2,3} is not an algebraic period. In the second part we analyze the situation whenm> 3and none of eigenvalues is a root of unity. This is done by considering two cases: we willstudy the behaviour of im( f ) separately for real and complex eigenvalues. In the thirdstage we consider the case when m> 3 and one of eigenvalues is a root of unity.


3.1. Algebraic periods in {1,2,3}(A) Conditions for 1�∈ APer( f ). We have: i1( f )= L( f )= (1− λ1)(1− λ2)= 0. This mayhappen if and only if one of the eigenvalues is equal to 1, that is, t−d = 1.

(B) Conditions for 2 �∈ APer( f ). We have: i2( f ) = L( f 2)− L( f ) = 0, which is equiv-alent to: (1 − λ2

1)(1 − λ22) − (1 − λ1)(1 − λ2) = 0. This gives: (1 − λ1)(1 − λ2)[(1 + λ1)

(1 + λ2)− 1]= 0, so again t−d = 1 or:

λ1λ2 + λ1 + λ2 = 0, (3.1)

which gives d+ t = 0. The conditions for 2�∈ APer( f ) are: t−d = 1 or t =−d.

(C) Conditions for 3�∈ APer( f ). We have: i3( f )= L( f 3)−L( f )= 0, which is equivalentto: (1− λ3

1)(1− λ32)− (1− λ1)(1− λ2) = 0. We obtain the following equation: (1− λ1)

(1− λ2)[(1 + λ1 + λ21)(1 + λ2 + λ2

2)− 1] = 0. Again t − d = 1 if one of the eigenvalues isequal to 1, otherwise:

λ1 + λ2 + λ1λ2 + λ21 + λ2

2 + λ1λ2(λ1 + λ2

)+(λ1λ2

)2 = 0. (3.2)

In parameters t and d this gives:

t2 + t−d+dt+d2 = 0. (3.3)

The last equality may be written as:

(d− 1− t

2

)2

+34

(1 + t)2 = 1, (3.4)

which leads to the following alternatives.If t = 0, then d ∈ {0,1}, which corresponds to characteristic polynomials x2 = 0 (λ1 =

λ2 = 0) and x2 + 1= 0 (λ1,2 =±i).If t =−1, then d ∈ {0,2}, which corresponds to characteristic polynomials x2 + x = 0

(λ1 = 0, λ2 =−1) and x2 + x+ 2= 0 (λ1,2 =−(1/2)± i(√7/2)).If t = −2, then d ∈ {1,2}, which corresponds to characteristic polynomials x2 + 2x +

1= 0 (λ1,2 =−1) and x2 + 2x+ 2= 0 (λ1,2 =−1± i).The conditions for 3�∈ APer( f ) are: t− d = 1 or (t,d)∈ {(0,0),(0,1),(−1,0),(−1,2),

(−2,1),(−2,2)}.

3.2. Algebraic periods in the set m > 3 in the case when none of the two eigenvaluesis a root of unity. Let for the rest of the paper |λ1| =max{|λ1|,|λ2|}. We will need thefollowing lemma.

Lemma 3.3. If for some m and each n|m, n�=m we have |L( f m)|/|L( f n)| > 2√m− 1, then

m is an algebraic period.

Grzegorz Graff 5

Proof. Let |L( f s)| =max{|L( f l)| : l|m, l�=m}. We have

∣∣im( f )∣∣=

∣∣∣∣∣∑

l|mμ(m

l

)L(f l)∣∣∣∣∣≥

∣∣L(f m)∣∣−

∣∣∣∣∣∑

l|m, l�=mμ(m

l

)L(f l)∣∣∣∣∣

≥ ∣∣L( f m)∣∣− (2√m− 1)∣∣L

(f s)∣∣.

(3.5)

The last inequality is a consequence of the fact that the number of different divisors ofm is not greater than 2

√m (cf. [2]), by the assumption we get |im( f )| > 0, which is the

desired assertion. �

Now, using the algebraic arguments of [9] in a case of two eigenvalues, we find thebound for the ratio |L( f m)|/|L( f n)|. We have

∣∣L(f m)∣∣

∣∣L(f n)∣∣ =

∣∣1− λm1∣∣∣∣1− λm2

∣∣∣∣1− λn1

∣∣∣∣1− λn2∣∣ ≥

∣∣λ1∣∣m− 1∣∣λ1∣∣n + 1

∣∣λ2∣∣m− 1∣∣λ2∣∣n + 1

. (3.6)

Let us consider two cases.

Case 1. λ1, λ2 are complex conjugates, then |λ1| = |λ2|. Notice that |λ1| =√d, so if we ex-

clude three pairs (t,d)∈ {(0,1),(−1,1),(1,1)}, which correspond to some roots of unity,we obtain: |λ1| > 1.4.

Let n|m, for Lefschetz numbers in this case we have∣∣L(f m)∣∣

∣∣L(f n)∣∣ ≥

(∣∣λ1∣∣m/2− 1

)(∣∣λ2∣∣m/2− 1

)=(∣∣λ1

∣∣m/2− 1)2. (3.7)

Case 2. λ1, λ2 are real. Then |λ1| = (|t|+√t2− 4d)/2. If (t,d) = (0,0) then we immedi-

ately have APer( f ) = {1}. Cases t = 0, d = −1 and t = ±1, d = 0 and t = ±2, d = 1 givesome roots of unity. In the rest of the cases: |λ1| ≥ 1.4.

In order to obtain the estimation for Lefschetz numbers we use the following inequal-ity for the moduli of eigenvalues (cf. [9, Lemma 5.2]).

Lemma 3.4. Let λi�=±1, i= 1,2, then

∣∣1−∣∣λ2∣∣∣∣≥ 1

1 +∣∣λ1

∣∣ . (3.8)

Proof. |(±1− λ1)(±1− λ2)| = |WA(±1)| ≥ 1, because both eigenvalues are different from±1. We obtain |1±λ2|≥1/|1±λ1|≥1/(1+|λ1|), which gives the needed inequality. �

We have by Lemma 3.4: |λ2| − 1≥ (|λ1|+ 1)−1 for |λ2| > 1 and 1− |λ2| ≥ (|λ1|+ 1)−1

for |λ2| < 1.Let h(x)= (xm− 1)/(xn + 1), notice that h(x) is an increasing and−h(x) is a decreasing

function for m> n > 0 and x > 0.Taking into account the two facts mentioned above we obtain:

∣∣1− λm2∣∣

∣∣1− λn2∣∣ ≥min

⎧⎪⎨⎪⎩

[1 +

(∣∣λ1∣∣+ 1

)−1]m− 1

[1 +

(∣∣λ1∣∣+ 1

)−1]n

+ 1,1−

[1− (∣∣λ1

∣∣+ 1)−1

]m

1 +[

1− (∣∣λ1∣∣+ 1

)−1]n

⎫⎪⎬⎪⎭. (3.9)


As n|m we get

∣∣L(f m)∣∣

∣∣L(f n)∣∣ ≥

(∣∣λ1∣∣m/2− 1

)min

{[1 +

(∣∣λ1∣∣+ 1

)−1]m/2− 1,1−

[1− (∣∣λ1

∣∣+ 1)−1

]m/2}.

(3.10)

Let fC(|λ1|,m), fR(|λ1|,m) be the functions equal to the right-hand side of the formu-las (3.7) and (3.10), respectively. We define functions fC(|λ1|,m)= fC(|λ1|,m)− (2

√m−

1) and fR(|λ1|,m)= fR(|λ1|,m)− (2√m− 1). Notice that the inequalities:

fC(∣∣λ1

∣∣,m)> 0, (3.11)

fR(∣∣λ1

∣∣,m)> 0, (3.12)

imply that |L( f m)|/|L( f n)| > 2√m− 1 for n|m.

It is not difficult to verify the following statement by calculation and estimation ofappropriate partial derivatives.

Remark 3.5. fC(·,m) and fC(|λ1|,·) are increasing functions for |λ1| > 1.4, m≥ 4.fR(·,m) and fR(|λ1|,·) are increasing functions for |λ1| > 1.4, m≥ 6 and for |λ1| ≥ 3,

m≥ 4.

If one of the inequalities (3.11), (3.12) is satisfied for given values |λ01| andm0, then, by

Remark 3.5, it is valid for each |λ1| > |λ01| and m>m0 and by Lemma 3.3 all such m>m0

are algebraic periods.

Lemma 3.6. Let us assume that both eigenvalues are complex(a) if m≥ 7, then m is an algebraic period,(b) if |λ1| ≥ 2 and m≥ 4, then m is an algebraic period.

Proof. We take the minimal modulus of the eigenvalue which may appear and put itin the formula (3.11): (a) fC(1.4,7) > 0.75, (b) fC(2,4) = 6, which gives the result byRemark 3.5. �

Lemma 3.7. Let us assume that both eigenvalues are real(a) if m≥ 12, then m is an algebraic period,(b) if |λ1| ≥ 3 and m≥ 6, then m is an algebraic period.

Proof. We put in the formula (3.12) the minimal modulus of the greater eigenvalue: (a)fR(1.4,12) > 0.59, (b) fR(3,6) > 17.47, which implies the result by Remark 3.5. �

Remark 3.8. We must only check the cases when |λ1| < 3 and 4 ≤m ≤ 11. Notice thatfor the coefficients t, d of the characteristic polynomial WA(x) we have the followingestimates: |t| ≤ 2|λ1|, |d| ≤ |λ1|2. This gives the bound: |t| < 6, |d| < 9, thus there areat most 11× 17× 8 = 1496 cases which should be checked. This is done by numericalcomputation. If we exclude (t,d) = (0,0) and the pairs which give the eigenvalues beingroots of unity, we find in the range under consideration that only for (t,d)= (0,2), m= 4is not an algebraic period.

Grzegorz Graff 7

3.3. Algebraic periods in the set m > 3 in the case when one of the two eigenvalues isa root of unity. If both eigenvalues are real, then one of them is equal ±1. If they arecomplex conjugates, then λ1λ2 = λ1λ1 = 1, thus d = 1. On the other hand 0≤ |λ1 + λ2| ≤|λ1|+ |λ2| = 2, thus |t| ≤ 2. This gives three pairs of complex eigenvalues:±i (t = 0,d = 1)and (1/2)± i(√3/2) (t = 1,d = 1) and−(1/2)± i(√3/2) (t =−1,d = 1). Each of these fivecases we consider separately.

(1) 1 is one of eigenvalues (t−d = 1). Then L( f m)= 0 for allm and consequently im( f )=0 for all m. Thus APer( f )=∅.

(2) −1 is one of eigenvalues (t+d =−1). We have to consider the subcases.(2a) If d =−1, then t = 0, so we are in case 1.(2b) If d = 0, then t = −1, so WA(x) = x2 + x and the second eigenvalue is equal to

0. L( f m)= 1− (−1)m, thus L( f m)= 0 for m even and L( f m)= 2 for m odd. Weget: im( f ) =∑k:2|k|mμ(m/k)L( f k) +

∑k:2�k|mμ(m/k)L( f k) = 2

∑k:2�k|mμ(m/k). It

is easy to find (see the calculation of im( f ) in (2d)) that i1( f ) = 2, i2( f ) = −2,im( f )= 0 for m≥ 3. As a consequence: APer( f )= {1,2}.

(2c) If d = 1, then t =−2, so WA(x)= x2 + 2x+ 1 and the second eigenvalue is equalto −1. L( f m) = (1− (−1)m)2, thus L( f m) = 0 for m even and L( f m) = 4 for modd. We check in the same way as above that i1( f )= 4, i2( f )=−4, im( f )= 0 form≥ 3, so APer( f )= {1,2}.

(2d) If d ∈ Z \ {−1,0,1}, then for each m : |L( f m)| = |(1− (−1)m)||1− λm1 |. Noticethat in the case under consideration {1,2,3} ⊂ APer( f ), which follows fromSection 3.1.

As |d| = |λ1||λ2| and −1 is one of eigenvalues we obtain for k odd : |L( f k)| ≥ 2(|λk1| −1) = 2(|d|k − 1), |L( f k)| ≤ 2(|λk1|+ 1) = 2(|d|k + 1). Thus, for m odd, estimating in thesame way as in Lemma 3.3, we get:

∣∣im( f )∣∣≥ 2

(|d|m− 1)− (2√m− 1

)2(|d|m/3 + 1

). (3.13)

The right-hand side of the above formula is greater then zero for |d| ≥ 2, m> 3, so allodd m> 3 are algebraic periods.

If m> 3 is even, then m= 2nq, where q is odd. By the fact that L( f r)= 0 if 2|r, we getL( f 2iq)= 0, for 1≤ i≤ n, thus

im( f )=∑

l|2nqμ(

2nq

l

)L(f l)=

∑

l|qμ(

2nq

l

)L(f l). (3.14)

As μ is multiplicative and μ(2n)=−1 for n= 1 and μ(2n)= 0 for n > 1, we get

im( f )=⎧⎨⎩−iq( f ) if n= 1,

0 if n > 1.(3.15)

This leads to the conclusion that even m is an algebraic period if and only if m = 2qwhere q is odd. Finally in the case (2d) we obtain

APer( f )=N \ {n∈N : n≡ 0 (mod4)}. (3.16)


Before we consider complex cases let us state the following fact (cf. [4]). Let g∗, gen-erated by g on homology, have as its only eigenvalues ε1, . . . ,εφ(d) which are all the dthprimitive roots of unity (φ(d) denotes the Euler function). Then the Lefschetz numbers

of iterations of g are the sum of powers of these roots: L(gm) =∑φ(d)i=1 ε

mi . We have the

formula for im(g) in such a case:

im(g)=

⎧⎪⎪⎨⎪⎪⎩

0 if m�|d,∑

k|mμ(d

k

)μ(m

k

)φ(d)φ(d/k)

if m | d. (3.17)

Let now λ1,2 be complex conjugates eigenvalues, then

L(f m)= 1− λm1 − λm2 +

(λ1λ2

)m = 2− (λm1 + λm2). (3.18)

We may rewrite formula for L( f m) in the following way: L( f m) = 2− L(gm), where g isdescribed above. As

∑k|mμ(m/k)2= 2 for m= 1 and 0 for m> 1; we get

im( f )=⎧⎨⎩

2− im(g) if m= 1,

−im(g) if m> 1.(3.19)

(3) λ1,2 =±i (t = 0,d = 1) are all primitive roots of unity of degree 4. Thus, applyingformula (3.17) and (3.19), we get i1( f )= 2, i2( f )= 2, i3( f )= 0, i4( f )=−4, and im( f )=0 for m> 4. Summing it up: APer( f )= {1,2,4}.

(4) λ1,2 = −1/2± i(√

3/2) (t = 1,d = 1) are all the primitive roots of unity of de-gree 6. Again by formulas (3.17) and (3.19) we calculate the values of im( f ) and get:i1( f )= 1, i2( f )= 2, i3( f )= 3, i4( f )= 0, i5( f )= 0, i6( f )=−6 and im( f )= 0 for m > 6,so APer( f )= {1,2,3,6}.

(5) λ1,2 = (1/2)± i(√3/2) (t =−1,d = 1) are all the primitive roots of unity of degree3. By (3.17) and (3.19) we have: i1( f )= 3, i2( f )= 0, i3( f )=−3, im( f )= 0 for m> 3, soAPer( f )= {1,3}.

Acknowledgment

Research supported by KBN Grant no. 2 P03A 04522.

References

[1] L. Alseda, S. Baldwin, J. Llibre, R. Swanson, and W. Szlenk, Minimal sets of periods for torus mapsvia Nielsen numbers, Pacific Journal of Mathematics 169 (1995), no. 1, 1–32.

[2] K. Chandrasekharan, Introduction to Analytic Number Theory, Die Grundlehren der mathema-tischen Wissenschaften, vol. 148, Springer, New York, 1968.

[3] N. Fagella and J. Llibre, Periodic points of holomorphic maps via Lefschetz numbers, Transactionsof the American Mathematical Society 352 (2000), no. 10, 4711–4730.

[4] G. Graff, Existence of δm-periodic points for smooth maps of compact manifold, Hokkaido Mathe-matical Journal 29 (2000), no. 1, 11–21.

[5] , Minimal periods of maps of rational exterior spaces, Fundamenta Mathematicae 163(2000), no. 2, 99–115.

Grzegorz Graff 9

[6] A. Guillamon, X. Jarque, J. Llibre, J. Ortega, and J. Torregrosa, Periods for transversal maps viaLefschetz numbers for periodic points, Transactions of the American Mathematical Society 347(1995), no. 12, 4779–4806.

[7] D. Haibao, The Lefschetz numbers of iterated maps, Topology and its Applications 67 (1995),no. 1, 71–79.

[8] J. Jezierski and M. Marzantowicz, Homotopy Methods in Topological Fixed and Periodic PointTheory, Springer, Dordrech, 2005.

[9] B. Jiang and J. Llibre, Minimal sets of periods for torus maps, Discrete and Continuous DynamicalSystems 4 (1998), no. 2, 301–320.

[10] J. Llibre, Lefschetz numbers for periodic points, Nielsen Theory and Dynamical Systems (SouthHadley, Mass, 1992), Contemp. Math., vol. 152, American Mathematical Society, Rhode Island,1993, pp. 215–227.

[11] J. Llibre, J. Paranos, and J. A. Rodrıguez, Periods for transversal maps on compact manifolds witha given homology, Houston Journal of Mathematics 24 (1998), no. 3, 397–407.

[12] W. Marzantowicz and P. M. Przygodzki, Finding periodic points of a map by use of a k-adic ex-pansion, Discrete and Continuous Dynamical Systems 5 (1999), no. 3, 495–514.

Grzegorz Graff: Department of Algebra, Faculty of Applied Physics and Mathematics,Gdansk University of Technology, ul G. Narutowicza 11/12, 80-952, Gdansk, PolandE-mail address: [email protected]

NIELSEN NUMBER OF A COVERING MAP

JERZY JEZIERSKI

Received 23 November 2004; Revised 13 May 2005; Accepted 24 July 2005

We consider a finite regular covering pH : XH → X over a compact polyhedron and a map

f : X → X admitting a lift f : XH → XH . We show some formulae expressing the Nielsennumber N( f ) as a linear combination of the Nielsen numbers of its lifts.

Copyright © 2006 Jerzy Jezierski. This is an open access article distributed under the Cre-ative Commons Attribution License, which permits unrestricted use, distribution, andreproduction in any medium, provided the original work is properly cited.

1. Introduction

Let X be a finite polyhedron and let H be a normal subgroup of π1(X). We fix a coveringpH : XH → X corresponding to the subgroup H , that is, p#(π1(XH))=H .

We assume moreover that the subgroup H has finite rank, that is, the covering pH isfinite. Let f : X → X be a map satisfying f (H)⊂H . Then f admits a lift

XHf

pH

XH

pH

Xf

X

(1.1)

Is it possible to find a formula expressing the Nielsen number N( f ) by the numbers

N( f ) where f runs the set of all lifts? Such a formula seems very desirable since thedifficulty of computing the Nielsen number often depends on the size of the fundamental

group. Since π1X ⊂ π1X , the computation of N( f ) may be simpler. We will translate thisproblem to algebra. The main result of the paper is Theorem 4.2 expressing N( f ) as a

linear combination of {N( fi)}, where the lifts are representing all the H-Reidemeisterclasses of f .

The discussed problem is analogous to the question about “the Nielsen number prod-uct formula” raised by Brown in 1967 [1]. A locally trivial fibre bundle p : E→ B and a


2 Nielsen number of a covering map

fibre map f : E→ E were given and the question was how to express N( f ) by N( f ) andN( fb), where f : B→ B denoted the induced map of the base space and fb was the restric-tion to the fibre over a fixed point b ∈ Fix( f ). This problem was intensively investigatedin 70ties and finally solved in 1980 by You [4]. At first sufficient conditions for the “prod-uct formula” were formulated: N( f ) = N( f )N( fb) assuming that N( fb) is the same forall fixed points b ∈ Fix( f ). Later it turned out that in general it is better to expect theformula

N( f )=N( fb1

)+ ···+N

(fbs), (1.2)

where b1, . . . ,bs represent all the Nielsen classes of f . One may find an analogy betweenthe last formula and the formulae of the present paper. There are also other analogies: inboth cases the obstructions to the above equalities lie in the subgroups {α∈ π1X ; f#α=α} ⊂ π1X .

2. Preliminaries

We recall the basic definitions [2, 3]. Let f : X → X be a self-map of a compact polyhe-dron. Let Fix( f )= {x ∈ X ; f (x)= x} denote the fixed point set of f . We define the Nielsenrelation on Fix( f ) putting x ∼ y if there is a path ω : [0,1] → X such that ω(0) = x,ω(1) = y and the paths ω, f ω are fixed end point homotopic. This relation splits theset Fix( f ) into the finite number of classes Fix( f ) = A1 ∪ ··· ∪As. A class A ⊂ Fix( f )is called essential if its fixed point index ind( f ;A)�= 0. The number of essential classes iscalled the Nielsen number and is denoted byN( f ). This number has two important prop-erties. It is a homotopy invariant and is the lower bound of the number of fixed points:N( f )≤ #Fix(g) for every map g homotopic to f .

Similarly we define the Nielsen relation modulo a normal subgroup H ⊂ π1X . We as-sume that the map f preserves the subgroupH , that is, f#H ⊂H . We say that then x ∼H yifω = f ωmodH for a pathω joining the fixed points x and y. This yieldsH-Nielsen classesand H-Nielsen number NH( f ). For the details see [4].

Let us notice that each Nielsen class modH splits into the finite sum of ordinaryNielsen classes (i.e., classes modulo the trivial subgroup):A= A1∪···∪As. On the otherhand NH( f )≤N( f ).

We consider a regular finite covering p : XH → X as described above.Let

�XH ={γ : XH −→ XH ; pHγ = pH

}(2.1)

denote the group of natural transformations of this covering and let

liftH( f )={f : XH −→ XH ; pH f = f pH

}(2.2)

denote the set of all lifts.

Jerzy Jezierski 3

We start by recalling classical results giving the correspondence between the coveringsand the fundamental groups of a space.

Lemma 2.1. There is a bijection �XH = p−1H (x0)= π1(X)/H which can be described as fol-

lows:

γ ∼ γ(x0)∼ pH(γ). (2.3)

We fix a point x0 ∈ p−1H (x0). For a natural transformation γ ∈ �XH , γ(x0) ∈ p−1

H (x0) is apoint and γ is a path in XH joining the points x0 and γ(x0). The bijection is not canonical. Itdepends on the choice of x0 and x0.

Let us notice that for any two lifts f , f ′ ∈ liftH( f ) there exists a unique γ ∈ �XH satis-

fying f ′ = γ f . More precisely, for a fixed lift f , the correspondence

�XH α−→ α f ∈ liftH( f ) (2.4)

is a bijection. This correspondence is not canonical. It depends on the choice of f .The group �XH is acting on liftH( f ) by the formula

α◦ f = α · f ·α−1 (2.5)

and the orbits of this action are called Reidemeister classes modH and their set is denoted�H( f ). Then one can easily check [3]

(1) pH(Fix( f )) ⊂ Fix( f ) is either exactly one H-Nielsen class of the map f or is

empty (for any f ∈ liftH( f ))

(2) Fix( f )=⋃ f pH(Fix( f )) where the summation runs the set liftH( f )

(3) if pH(Fix( f ))∩ pH(Fix( f ′)) �= ∅ then f , f ′ represent the same Reidemeisterclass in �H( f )

(4) if f , f ′ represent the same Reidemeister class then pH(Fix( f ))= pH(Fix( f ′)).

Thus Fix( f )=⋃ f pH(Fix( f )) is the disjoint sum where the summation is over a sub-

set containing exactly one lift f from each H-Reidemeister class. This gives the natu-ral inclusion from the set of Nielsen classes modulo H into the set of H-Reidemeisterclasses

�H( f )−→�H( f ). (2.6)

The H-Nielsen class A is sent into the H-Reidemeister class represented by a lift f satis-

fying pH(Fix( f ))= A. By (1) and (2) such lift exists, by (3) the definition is correct and(4) implies that this map is injective.


3. Lemmas

For a lift f ∈ liftH( f ), a fixed point x0 ∈ Fix( f ) and an element β ∈ π1(X ;x0) we definethe subgroups

�( f )={γ ∈ �XH ; f γ = γ f

}

C(f#,x0;β

)= {α∈ π1(X ;x0

); αβ = β f#(α)

}

CH(f#,x0;β

)= {[α]H ∈ π1(X ;x0

)/H(x0); αβ = β f#(α) modulo H

}.

(3.1)

If β = 1 we will write simply C( f#,x0) or CH( f#,x0).We notice that the canonical projection j : π1(X ;x0) → π1(X ;x0)/H(x0) induces the

homomorphism j : C( f#,x0;β)→ CH( f#,x0;β).

Lemma 3.1. Let f be a lift of f and let A be a Nielsen class of f . Then pH(A) ⊂ Fix( f ) isa Nielsen class of f . On the other hand if A⊂ Fix( f ) is a Nielsen class of f then p−1

H (A)∩Fix( f ) splits into the finite sum of Nielsen classes of f .

Proof. It is evident that pH(A) is contained in a Nielsen class A⊂ Fix( f ). Now we showthat A ⊂ pH(A). Let us fix a point x0 ∈ A and let x0 = pH(x0). Let x1 ∈ A. We have toshow that x1 ∈ pH(A). Let ω : I → X establish the Nielsen relation between the pointsω(0) = x0 and ω(1) = x1 and let h(t,s) denote the homotopy between ω = h(·,0) andf ω = h(·,1). Then the path ω lifts to a path ω : I → XH , ω(0)= x0. Let us denote ω(1)=x1. It remains to show that x1 ∈ A. The homotopy h lifts to h : I × I → XH , h(0,s) = x0.

Then the paths h(·,1) and f ω as the lifts of f ω starting from x0 are equal. Now f (x1)=f (ω(1))= h(1,1)= h(1,0)= ω(1)= x1. Thus x1 ∈ Fix( f ) and the homotopy h gives theNielsen relation between x0 and x1 hence x1 ∈ A.

Now the second part of the lemma is obvious. �

Lemma 3.2. Let A⊂ Fix( f ) be a Nielsen class of f . Let us denote A= pH(A). Then(1) pH : A→ A is a covering where the fibre is in bijection with the image j#(C( f#,x))⊂

π1(X ;x)/H(x) for x ∈ A,(2) the cardinality of the fibre (i.e., #(p−1

H (x)∩ A)) does not depend on x ∈ A and we willdenote it by JA,

(3) if A′ is another Nielsen class of f satisfying pH(A′)= pH(A) then the cardinalities ofp−1H (x)∩ A and p−1

H (x)∩ A′ are the same for each point x ∈A.

Proof. (1) Since pH is a local homeomorphism, the projection pH : A→ A is the covering.(2) We will show a bijection φ : j(C( f#;x0))→ p−1

H (x0)∩ A (for a fixed point x0 ∈A).Let α∈ C( f#). Let us fix a point x0 ∈ p−1

H (x0). Let α : I → X denote the lift of α startingfrom α(0)= x0. We define φ([α]H)= α(1). We show that

(2a) The definition is correct. Let [α]H = [α′]H . Then α ≡ α′modH hence α(1) =α′(1). Now we show that α(1) ∈ A. Since α ∈ C( f#), there exists a homotopy h between

the loops h(·,0)= α and h(·,1)= f α. The homotopy lifts to h : I × I → XH , h(0,s)= x0.

Then x1 = h(1,s) is also a fixed point of f and moreover h is the homotopy between the

paths ω and f ω. Thus x0, x1 ∈ Fix( f ) are Nielsen related hence x1 ∈ A.

Jerzy Jezierski 5

(2b) φ is onto. Let x1 ∈ p−1H (x0)∩ A. Now x0, x1 ∈ Fix( f ) are Nielsen related. Let ω :

I → XH establish this relation ( f ω ∼ ω). Now

f(pHω

)= pH f ω ∼ pHω (3.2)

hence pHω ∈ C( f#;x0). Moreover φ[pHω]H = ω(1)= x1.(2c) φ is injective. Let [α]H , [α′]H ∈ j(C( f#)) and let α, α′ : I → XH be their lifts starting

from α(0) = α′(0) = x0. Suppose that φ[α]H = φ[α′]H . This means α(1) = α′(1) ∈ XH .Thus pH(α∗α′−1)= α∗α′−1 ∈H which implies [α]H = [α′]H .

(3) Let x0 ∈ pH(A) = pH(A′). Then by the above #(p−1(x0) ∩ A) = j#(C( f#)) =#(p−1(x0)∩ A′). �

Lemma 3.3. The restriction of the covering map pH : Fix( f )→ pH(Fix( f )) is a covering.The fibre over each point is in a bijection with the set

�( f )={γ ∈ �XH ; f γ = γ f

}. (3.3)

Proof. Since the fibre of the covering pH is discrete, the restriction pH : Fix( f ) →pH(Fix( f )) is a locally trivial bundle. Let us fix points x0 ∈ pH(Fix( f )), x0 ∈ p−1

H (x0)∩Fix( f ). We recall that

α : p−1H

(x0)−→ �XH , (3.4)

where αx ∈ �XH is characterized by αx(x0) = x, is a bijection. We will show that

α(p−1H (x0)∩Fix( f ))=�( f ).

Let f (x)= x for an x ∈ p−1H (x0). Then

f αx(x0)= f (x)= x = αx

(x0)= αx f

(x0)

(3.5)

which implies f αx = αx f hence αx ∈�( f ).

Now we assume that f αx = αx f . Then in particular f αx(x0) = αx f (x0) which gives

f (x)= αx(x0), f (x)= x hence x ∈ Fix( f ). �

We will denote by IAH the cardinality of the subgroup #�( f ) for the H-Nielsen class

AH = pH(Fix( f )). We will also write IAi = IAH for any Nielsen class Ai of f contained inA.

Lemma 3.4. Let A0 ⊂ Fix( f ) be a Nielsen class and let A0 ⊂ Fix( f ) be a Nielsen class con-tained in p−1

H (A0). Then, by Lemma 3.1 A0 = pH(A0) and moreover

ind(f ; p−1

H

(A0))= IA0 · ind

(f ;A0

)

ind(f ;A0

)= JA0 · ind(f ;A0

).

(3.6)


Proof. Since the index is the homotopy invariant we may assume that Fix( f ) is finite. Now

for any fixed points x0 ∈ Fix( f ), x0 ∈ Fix( f ) satisfying pH(x0)= x0 we have ind( f0; x0)=ind( f0;x0) since the projection pH is a local homeomorphism. Thus

ind(f ; p−1

H

(A0))=

∑

x∈A0

ind(f ; p−1

H (x))=

∑

x∈A0

IA0 · ind( f ;x)

= IA0

∑

x∈A0

ind( f ;x)= IA0 · ind(f ;A0

).

(3.7)

Similarly we prove the second equality:

ind(f ;A0

)=∑

x∈A0

ind(f ; p−1

H (x)∩ A0

)=∑

x∈A0

∑

x∈p−1H (x)∩A0

ind(f ; x

)

=∑

x∈A0

JA0 · ind( f ;x)= JA0 ·(∑

x∈A0

ind( f ;x)

)= JA0 · ind

(f ;A0

).

(3.8)

�

To get a formula expressing N( f ) by the numbers N( f ) we will need the assumptionthat the numbers JA = JA′ for any two H-Nielsen related classes A,A′ ⊂ Fix( f ). The nextlemma gives a sufficient condition for such equality.

Lemma 3.5. Let x0 ∈ p(Fix( f )). If the subgroups H(x0),C( f ,x0) ⊂ π1(X ,x0) commute,that is, h · α = α · h, for any h ∈ H(x0), α ∈ C( f ,x0), then JA = JA′ for all Nielsen classes

A,A′ ⊂ p(Fix( f )).

Proof. Let x1 ∈ p(Fix( f )) be another point. The points x0,x1 ∈ p(Fix( f )) are H-Nielsenrelated, that is, there is a path ω : [0,1] → X satisfying ω(0) = x0, ω(1) = x1 such thatω∗ f (ω−1)∈H(x0). We will show that the conjugation

π1(X ,x0

) α−→ ω−1∗α∗ω ∈ π1(X ,x1

)(3.9)

sends C( f ,x0) onto C( f ,x1). Let α∈ C( f ,x0). We will show that ω−1∗α∗ω ∈ C( f ,x1).In fact f (ω−1 ∗ α∗ ω) = ω−1 ∗ α∗ ω ⇔ (ω∗ f ω−1)∗ α = α∗ (ω∗ f ω−1) but the lastequality holds since ω∗ f ω−1 ∈H(x0) and α∈ C( f ,x0). �

Remark 3.6. The assumption of the above lemma is satisfied if at least one of the groupsH(x0), C( f ,x0) belongs to the center of π1(X ;x0).

Remark 3.7. Let us notice that if the subgroupsH(x0),C( f ,x0)⊂ π1(X ,x0) commute then

so do the corresponding subgroups at any other point x1 ∈ pH(Fix( f )).

Proof. Let us fix a path ω : [0,1]→ X . We will show that the conjugation

π1(X ,x0

) α−→ ω−1∗α∗ω ∈ π1(X ,x1

)(3.10)

sends C( f ,x0) onto C( f ,x1). Let α∈ C( f ,x0). We will show that ω−1∗α∗ω ∈ C( f ,x1).But the last means f (ω−1∗ α∗ω) = ω−1∗ α∗ω hence f (ω−1)∗ f α∗ f ω = ω−1∗ α∗ω ⇔ f (ω−1)∗ α∗ f ω = ω−1 ∗ α∗ω ⇔ (ω∗ f ω−1)∗ α = α∗ (ω∗ f ω−1) and the last

Jerzy Jezierski 7

holds since (ω∗ f ω−1) ∈H(x0) and α ∈ C( f ,x0). Now it remains to notice that the el-ements of H(x1), C( f ;x1) are of the form ω−1 ∗ γ∗ω and ω−1 ∗ α∗ω respectively forsome γ ∈H(x0) and α∈ C( f ,x0). �

Now we will express the numbers IA, JA in terms of the homotopy group homomor-

phism f# : π1(X ,x0)→ π1(X ,x0) for a fixed point x0 ∈ Fix( f ). Let f : XH → XH be a lift

satisfying x0 ∈ p−1H (x0)∩Fix( f ). We also fix the isomorphism

π1(X ;x0

)/H(x0) α−→ γα ∈ �XH , (3.11)

where γα(x0)= α(1) and α denotes the lift of α starting from α(0)= x0.

We will describe the subgroup corresponding to C( f ) by this isomorphism and then

we will do the same for the other lifts f ′ ∈ liftH( f ).

Lemma 3.8.

f γα = γ f α f . (3.12)

Proof.

f γα(x0)= f α(1)= γ f α

(x0)= γ f α f

(x0), (3.13)

where the middle equality holds since f α is a lift of the path f α from the point x0. �

Corollary 3.9. There is a bijection between

�( f )={γ ∈ �XH ; f γ = γ f

},

CH( f )= {α∈ π1(X ;x0

)/H(x0); fH#(α)= α}.

(3.14)

Thus

IA/JA = #�( f )/# j(C( f )

)= #(CH( f )/ j

(C( f )

)). (3.15)

Let us emphasize that C( f ), CH( f ) are the subgroups of π1(X ;x0) or π1(X ;x0)/H(x0)respectively where the base point is the chosen fixed point. Now will take another fixedpoint x1 ∈ Fix( f ) and we will denote C′( f )= {α′ ∈ π1(X ;x1); f#α= α} and similarly wedefine C′H( f ). We will express the cardinality of these subgroups in terms of the groupπ1(X ;x0).

Lemma 3.10. Let η : [0,1]→ X be a path from x0 to x1. This path gives rise to the isomor-phism Pη : π1(X ;x1)→ π1(X ;x0) by the formula Pη(α)= ηαη−1. Let δ = η · ( f η)−1. Then

Pη(C′( f )

)= {α∈ π1(X ;x0

); αδ = δ f#(α)

}

Pη(C′H( f )

)= {[α]∈ π1(X ;x0

)/H(x0); αδ = δ f#(α) modulo H

}.

(3.16)


Proof. We notice that δ is a loop based at x0 representing the Reidemeister class of thepoint x1 in �( f )= π1(X ;x0)/�.

We will denote the right-hand side of the above equalities by C( f ;δ) and CH( f ;δ)respectively. Let α′ ∈ π1(X ;x1). We denote α = Pη(α′) = ηα′η−1. We will show that α ∈C( f ;δ)⇔ α′ ∈ C′( f ).

In fact α ∈ C( f ;δ) ⇔ αδ = δ · f α ⇔ (ηα′η−1)(η · f η−1) = (η · f η−1)( f η · f α′ ·( f η)−1)⇔ ηα′ · ( f η)−1 = η · f α′ · ( f η)−1 ⇔ α′ = f α′.

Similarly we prove the second equality. �

Thus we get the following formulae for the numbers IA, JA.

Corollary 3.11. Let δ ∈ π1(X ;x0) represent the Reidemeister class A∈�( f ). Then IA =#CH( f ; j(δ)), JA = # j(C( f ;δ)).

4. Main theorem

Lemma 4.1. Let A⊂ pH(Fix( f )) be a Nielsen class of f . Then p−1H A contains exactly IA/JA

fixed point classes of f .

Proof. Since the projection of each Nielsen class A⊂ p−1H (A)∩ Fix( f ) is onto A (Lemma

3.1), it is enough to check how many Nielsen classes of f cut p−1H (a) for a fixed point

a ∈ A. But by Lemma 3.3 p−1H (a)∩ Fix( f ) contains IA points and by Lemma 3.2 each

class in this set has exactly JA common points with p−1H (a). Thus exactly IA/JA Nielsen

classes of f are cutting p−1H (a)∩Fix( f ). �

Let f : X → X be a self-map of a compact polyhedron admitting a lift f : XH → XH . Wewill need the following auxiliary assumption:

for any Nielsen classes A,A′ ∈ Fix( f ) representing the same class modulothe subgroup H the numbers JA = JA′ .

We fix lifts f1, . . . , fs representing all H-Nielsen classes of f , that is,

Fix( f )= pH(

Fix(f1))∪···∪ pH

(Fix

(fs))

(4.1)

is the mutually disjoint sum. Let Ii, Ji denote the numbers corresponding to a (Nielsen

class of f ) A⊂ pH(Fix( fi)). By the remark after Lemma 3.3 and by the above assumption

these numbers do not depend on the choice of the class A⊂ pH(Fix( fi)). We also noticethat Lemmas 3.3, 3.2 imply

Ii = #�(fi)= #

{γ ∈ �XH ; γ fi = fiγ

}

Ji = # j(C(f#;x

))= # j({γ ∈ π1

(X ,xi

); f#γ = γ

}) (4.2)

for an xi ∈ Ai.

Jerzy Jezierski 9

Theorem 4.2. Let X be a compact polyhedron, PH : XH → X a finite regular covering and let

f : X → X be a self-map admitting a lift f : XH → XH . We assume that for each two Nielsenclasses A,A′ ⊂ Fix( f ), which represent the same Nielsen class modulo the subgroup H , thenumbers JA = JA′ . Then

N( f )=s∑

i=1

(Ji/Ii

) ·N( fi), (4.3)

where Ii, Ji denote the numbers defined above and the lifts fi represent all H-Reidemeisterclasses of f , corresponding to nonempty H-Nielsen classes.

Proof. Let us denote Ai = pH(Fix( fi)). Then Ai is the disjoint sum of Nielsen classes of

f . Let us fix one of them A ⊂ Ai. By Lemma 3.1 p−1H A∩ Fix( fi) splits into IA/JA Nielsen

classes in Fix( fi). By Lemma 3.4 A is essential iff one (hence all) Nielsen classes in p−1H A⊂

Fix fi is essential. Summing over all essential classes of f in Ai = pA(Fix( fi)) we get

the number of essential Nielsen classes of f in Ai

=∑

A

(JA/IA

) · (number of essential Nielsen classes of fi in p−1H A

), (4.4)

where the summation runs the set of all essential Nielsen classes contained in Ai.But JA = Ji, IA = Ii for all A⊂Ai hence

(the number of essential Nielsen classes of f in Ai

)= Ji/Ii ·N(fi). (4.5)

Summing over all lifts { fi} representing non-empty H-Nielsen classes of f we get

N( f )=∑

i

(Ji/Ii

) ·N( fi)

(4.6)

sinceN( f ) equals the number of essential Nielsen classes in Fix( f )=⋃si=1 pH Fix( fi). �

Corollary 4.3. If moreover, under the assumptions of Theorem 4.2, C = Ji/Ii does not de-pend on i then

N( f )= C ·s∑

i=1

N(fi). (4.7)

5. Examples

In all examples given below the auxiliary assumption JA = JA′ holds, since the assump-tions of Lemma 3.5 are satisfied (in 1, 2, 3 and 5 the fundamental groups are commutativeand in 4 the subgroup C( f ,x0) is trivial).


(1) If π1X is finite and p : X → X is the universal covering (i.e.,H = 0) then X is simply

connected hence for any lift f : X → X

N( f )=⎧⎨⎩

1 for L( f )�= 0

0 for L( f )= 0.(5.1)

But L( f ) �= 0 if and only if the Nielsen class p(Fix( f )) ⊂ Fix( f ) is essential (Lemma3.4). Thus

N( f )= number of essential classes=N( f1)

+ ···+N(fs), (5.2)

where the lifts f1, . . . , fs represent all Reidemeister classes of f .(2) Consider the commutative diagram

S1pl

pk

S1

pk

S1pl

S1

(5.3)

Where pk(z) = zk, pl(z) = zl, k, l ≥ 2. The map pk is regarded as k-fold regular cover-ing map. Then each natural transformation map of this covering is of the form α(z) =exp(2πp/k) · z for p = 0, . . . ,k− 1 hence is homotopic to the identity map. Now all thelifts of the map pl are maps of degree l hence their Nielsen numbers equal l − 1. Onthe other hand the Reidemeister relation of the map pl : S1 → S1 modulo the subgroupH = impk# is given by

α∼ β ⇐⇒ β = α+ p(l− 1)∈ k ·Z for a p ∈ Z⇐⇒ β = α+ p(l− 1) + qk for some p,q ∈ Z⇐⇒ α= β modulo g.c.d. (l− 1,k).

(5.4)

Thus #�H(pl)= g.c.d.(l− 1,k). Now the sum

∑

p′l

N(p′l)= (g.c.d.(l− 1,k)

) · (l− 1), (5.5)

(where the summation runs the set having exactly one common element with each H-Reidemeister class) equals N(pl)= l− 1 iff the numbers k, l− 1 are relatively prime.

Notice that in our notation I = g.c.d.(l− 1,k) while J = 1.(3) Let us consider the action of the cyclic group Z8 on S3 = {(z,z′) ∈ C×C; |z|2 +

|z′|2 = 1} given by the cyclic homeomorphism

S3 (z,z′)−→ (exp(2πi/8) · z,exp(2πi/8) · z′)∈ S3. (5.6)

The quotient space is the lens space which we will denote L8. We will also consider thequotient space of S3 by the action of the subgroup 2Z4 ⊂ Z8. Now the quotient group is

Jerzy Jezierski 11

also a lens space which we will denote by L4. Let us notice that there is a natural 2-foldcovering pH : L4 → L8

L4 = S3/Z4 [z,z′]−→ [z,z′]∈ S3/Z8 = L8. (5.7)

The group of natural transformations �L of this covering contains two elements: theidentity and the map A[z,z′]= [exp(2πi/8) · z, exp(2πi/8) · z′]. Now we define the map

f : L8 → L8 putting f [z,z′]= [z7/|z|6,z′7/|z|′6]. This map admits the lift f : L4 → L4 given

by the same formula and the lift A f . We notice that each of the maps f , f , A f is a map ofa closed oriented manifold of degree 49. SinceH1(L;Q)=H2(L;Q)= 0 for all lens spaces,the Lefschetz number of each of these three maps equals; L( f ) = 1− 49 = −48 �= 0. Onthe other hand since the lens spaces are Jiang [3], all involved Reidemeister classes areessential hence the Nielsen number equals the Reidemeister number in each case.

Now

�( f )= coker(id−7 · id)= coker(−6 · id)= coker(2 · id)= Z2. (5.8)

Similarly �( f ) = Z2 and �(A · f ) = �( f ) = Z2 since A is homotopic to the identity.Thus

R( f )= 2�= 2 + 2= R( f ) +R(A · f ). (5.9)

Since all the classes are essential, the same inequality holds for the Nielsen numbers.(4) If the group {α∈ π1(X ;x)/H(x); f#α= α} is trivial for each x ∈ Fix( f ) lying in an

essential Nielsen class of f then all the numbers Ii = Ji = 1 and the sum formula holds.(5) If π1X/H is abelian then the rank of the groups

C(fH#)= {α∈ π1(X ,x)/H(x); f#α= α

}= ker(

id− f#)

: π1(X ,x)/H(x)−→ π1(X ,x)/H(x)(5.10)

does not depend on x ∈ Fix( f ) hence I is constant. If moreover π1X is abelian then alsothe group C( f#)= ker(id− f#) does not depend on x ∈ Fix( f ). Then we get

N( f )= J/I ·(N(f1)

+ ···+N(fs)). (5.11)

References

[1] R. F. Brown, The Nielsen number of a fibre map, Annals of Mathematics. Second Series 85 (1967),483–493.

[2] , The Lefschetz Fixed Point Theorem, Scott, Foresman, Illinois, 1971.[3] B. J. Jiang, Lectures on Nielsen Fixed Point Theory, Contemporary Mathematics, vol. 14, Ameri-

can Mathematical Society, Rhode Island, 1983.[4] C. Y. You, Fixed point classes of a fiber map, Pacific Journal of Mathematics 100 (1982), no. 1,

217–241.

Jerzy Jezierski: Department of Mathematics, University of Agriculture, Nowoursynowska 159,02 766 Warszawa, PolandE-mail address: jezierski [email protected]

GEOMETRIC AND HOMOTOPY THEORETIC METHODS INNIELSEN COINCIDENCE THEORY

ULRICH KOSCHORKE

Received 30 November 2004; Accepted 21 July 2005

In classical fixed point and coincidence theory, the notion of Nielsen numbers has provedto be extremely fruitful. Here we extend it to pairs ( f1, f2) of maps between manifoldsof arbitrary dimensions. This leads to estimates of the minimum numbers MCC( f1, f2)(and MC( f1, f2), resp.) of path components (and of points, resp.) in the coincidence setsof those pairs of maps which are ( f1, f2). Furthermore we deduce finiteness conditionsfor MC( f1, f2). As an application, we compute both minimum numbers explicitly in fourconcrete geometric sample situations. The Nielsen decomposition of a coincidence set isinduced by the decomposition of a certain path space E( f1, f2) into path components.Its higher-dimensional topology captures further crucial geometric coincidence data. Ananaloguous approach can be used to define also Nielsen numbers of certain link maps.

Copyright © 2006 Ulrich Koschorke. This is an open access article distributed under theCreative Commons Attribution License, which permits unrestricted use, distribution,and reproduction in any medium, provided the original work is properly cited.

1. Introduction and discussion of results

Throughout this paper f1, f2 :M→N denote two (continuous) maps between the smoothconnected manifolds M and N without boundary, of strictly positive dimensions m andn, respectively, M being compact.

We would like to measure how small (or simple in some sense) the coincidence locus

C(f1, f2

):= {x ∈M | f1(x)= f2(x)

}(1.1)

can be made by deforming f1 and f2 via homotopies. Classically one considers theminimum number of coincidence points

MC(f1, f2

):=min

{#C(f ′1 , f ′2

) | f ′1 ∼ f1, f ′2 ∼ f2}

(1.2)

(cf. [1], (1.1)). It coincides with the minimum number min{#C( f ′1 , f2) | f ′1 ∼ f1} whereonly f1 is modified by a homotopy (cf. [2]). In particular, in topological fixed point theory


2 Methods in Nielsen coincidence theory

(where M =N and f2 is the identity map) this minimum number is the principal objectof study (cf. [3, page 9]).

In higher codimensions, however, the coincidence locus is generically a manifold ofdimension m− n > 0, and MC( f1, f2) is often infinite (see, e.g., Examples 1.4 and 1.6below). Thus it seems more meaningful to study the minimum number of coincidencecomponents

MCC(f1, f2

):=min

{#π0

(C(f ′1 , f ′2

)) | f ′1 ∼ f1, f ′2 ∼ f2}

, (1.3)

where #π0(C( f ′1 , f ′2 )) denotes the (generically finite) number of path components of theindicated coincidence subspace of M.

Question 1.1. How big are MCC( f1, f2) and MC( f1, f2)? In particular, when do these in-variants vanish, that is, when can the maps f1 and f2 be deformed away from one another?

In this paper, we discuss lower bounds for MCC( f1, f2) and geometric obstructions toMC( f1, f2) being trivial or finite.

A careful investigation of the differential topology of generic coincidence submanifoldsyields the normal bordism classes (cf. (4.6) and (4.7))

ω(f1, f2

)∈Ωm−n(M;ϕ),

ω(f1, f2

)∈Ωm−n(E(f1, f2

); ϕ) (1.4)

as well as a sharper (“nonstabilized”) version

ω#( f1, f2)∈Ω#( f1, f2

)(1.5)

of ω( f1, f2) (cf. Remark 4.2). Here the path space

E(f1, f2

):= {(x,θ)∈M×NI | θ(0)= f1(x), θ(1)= f2(x)

}(1.6)

(cf. Section 2), also known as (a kind of) homotopy equalizer of f1 and f2, plays a crucialrole. In general it has a very rich topology involving both M and the loop space of N .Already the set π0(E( f1, f2)) of path components can be huge—it corresponds bijectivelyto the Reidemeister set

R(f1, f2

)= π1(N)/Reidemeister equivalence (1.7)

(cf. [1, 3.1] and our Proposition 2.1 below) which is of central importance in classi-cal Nielsen theory. Thus it is only natural to define a “Nielsen number” N( f1, f2) (anda sharper version N#( f1, f2), resp.) to be the number of those (“essential”) path com-ponents which contribute nontrivially to the bordism class ω( f1, f2) (and to ω#( f1, f2),resp.), compare Definition 4.1 and Remark 4.2.

Ulrich Koschorke 3

Theorem 1.2. (i) The integers N( f1, f2) and N#( f1, f2) depend only on the homotopyclasses of f1 and f2; (ii) N( f1, f2) = N( f2, f1) and N#( f1, f2) = N#( f2, f1); (iii) 0 ≤ N( f1,f2)≤N#( f1, f2)≤MCC( f1, f2)≤MC( f1, f2); if n�= 2, then also MCC( f1, f2)≤ #R( f1, f2);(iv) if m= n, then N( f1, f2)=N#( f1, f2) coincides with the classical Nielsen number (cf. [1,Definition 3.6]).

Remark 1.3. In various situations, some of the estimates spelled out in part (iii) of thistheorem are known to be sharp (compare also [12]). For example, in the self-coincidencesetting (where f1 = f2) we have always MCC( f1, f2) ≤ 1 (since here C( f1, f2) =M). Inthe “root setting” (where f2 maps to a constant value ∗ ∈ N) all Nielsen classes are si-multaneously essential or inessential (since our ω-invariants are always compatible withhomotopies of ( f1, f2) and hence, in this particular case, with the action of π1(N ,∗), cf.the discussion in [12] following (1.10)). Therefore in both settings MCC( f1, f2) is equalto the Nielsen number N( f1, f2) provided ω( f1, f2)�= 0 (and n�= 2 if f2 ≡∗).

Further geometric and homotopy theoretic considerations allow us to determine theNielsen and minimum numbers explicitly in several concrete sample situations (forproofs see Section 6 below).

Example 1.4. Given integers q > 1 and r, let N = CP(q) be q-dimensional complex pro-jective space, letM = S(⊗r

CλC) be the total space of the unit circle bundle of the rth tensorpower of the canonical complex line bundle, and let f : M → N denote the fiber projec-tion. Then

N( f , f )=N#( f , f )=MCC( f , f )=⎧⎨⎩

0 if q ≡−1(r), q ≡ 1(2),

1 else;

MC( f , f )=

⎧⎪⎪⎪⎨⎪⎪⎪⎩

0 if q ≡−1(r), q ≡ 1(2),

1 if q ≡−1(r), q ≡ 0(2),

∞ if q �≡−1(r).

(1.8)

As was shown above (cf. Remark 1.3), in any self-coincidence situation (where f1 = f2)MCC( f1, f2) must be 0 or 1 and it remains only to decide which value occurs. In theprevious example this can be settled by the normal bordism class ω( f , f ) ∈ Ω1(M;ϕ),a weak form of ω( f , f ) which, however, captures a delicate (“second order”) Z2-aspectas well as the dual of the classical first order obstruction. Already in this simple casestandard methods of singular (co)homology theory yield only a necessary condition forMCC( f1, f2) to vanish (cf. [7, 2.2]). In higher codimensions m− n the advantage of thenormal bordism approach can be truely dramatic.

Example 1.5. Given natural numbers k < r, let M = Vr,k (and N = Gr,k, resp.) be theStiefel manifold of orthonormal k-frames (and the Grassmannian of k-planes, resp.) inRr . Let f :M→N map a frame to the plane it spans.

Assume r ≥ 2k ≥ 2. Then

N( f , f )=N#( f , f )=MCC( f , f )=MC( f , f )=⎧⎨⎩

0 if ω( f , f )= 0,

1 else.(1.9)


Here the normal bordism obstructionω( f , f )∈Ωm−n(M;ϕ) (cf. (4.7)) contains preciselyas much information as its “highest order component”

2χ(Gr,k

) · [SO(k)]∈Ωfr

m−n ∼= πSm−n, (1.10)

where [SO(k)] denotes the framed bordism class of the Lie group SO(k), equipped witha left invariant parallelization; the Euler number χ(Gr,k) is easily calculated: it vanishes

if k �≡ r ≡ 0(2) and equals(

[r/2][k/2]

)otherwise. Without loosing its geometric flavor, our

original question translates here—via the Pontryagin-Thom isomorphism—into deepproblems of homotopy theory (compare the discussion in the introduction of [11]). For-tunately powerful methods are available in homotopy theory which imply, for example,that MCC( f , f ) =MC( f , f ) = 0 if k is even or k = 7 or 9 or χ(Gr,k) ≡ 0(12); however,if k = 1 and r ≡ 1(2), or if k = 3 and r �≡ 1(12) is odd, or if k = 5 and r ≡ 5(6), thenMCC( f , f )=MC( f , f )= 1.

These results seem to be entirely out of the reach of the methods of singular(co)homology theory since we would have to deal here with obstructions of order m−n+ 1= k(k− 1)/2 + 1.

Example 1.6. Let N be the torus (S1)n and let ι1, . . . , ιn denote the canonical generators ofH1((S1)n;Z). If the homomorphism

f1∗ − f2∗ :H1(M;Z)−→H1

((S1)n;Z

)(1.11)

has an infinite cokernel (or, equivalently, the rank of its image is strictly smaller than n),then

N(f1, f2

)=N#( f1, f2)=MCC

(f1, f2

)=MC(f1, f2

)= 0. (1.12)

On the other hand, if the cup product

n∏

j=1

(f ∗1 − f ∗2

)(ι j)∈Hn(M;Z) (1.13)

is nontrivial, then MC( f1, f2)=∞wheneverm> n; if in addition n�= 2, then MCC( f1, f2)equals the (finite) cardinality of the cokernel of f1∗ − f2∗ (cf.(1.11)).

In the special case when N is the unit circle S1 we have: MCC( f1, f2) =MC( f1, f2) =0 if f1 is homotopic to f2; otherwise MCC( f1, f2) = #coker( f1∗ − f2∗), but (if m > 1)MC( f1, f2)=∞.

Ulrich Koschorke 5

An important special case of our invariants are the degrees

deg#( f ) := ω#( f ,∗), deg( f ) := ω( f ,∗), deg( f ) := ω( f ,∗) (1.14)

of a given map f :M→N (here ∗ denotes a constant map).

Example 1.7 (homotopy groups). LetM be the sphere Sm; in view of the previous examplewe may also assume that n≥ 2.

Then, given [ fi]∈ πm(N ,∗i), i= 1,2, ∗1 �=∗2, we can identify Ω#( f1, f2),Ωm−n(E( f1,f2); ϕ) and Ωm−n(M;ϕ) with the corresponding groups in the top line of the diagram

πm(Sn∧Ω(N)+

) stabilizeΩfrm−n(ΩN) Ωfr

m−n

πm(N)deg#

degdeg

(1.15)

(This is possible since the loop space ΩN occurs as a typical fiber of the natural projectionp : E( f1, f2)→ Sm, cf. [12, Section 7], and [13].)

Furthermore, after deforming the maps f1 and f2 until they are constant on oppositehalf spheres in Sn, we see that

ω(f1, f2

)= ω( f1,∗2)

+ ω(∗1, f2

), (1.16)

and similarly for ω# and ω.Thus it suffices to study the degree maps in diagram (1.15). They turn out to be group

homomorphisms which commute with the indicated natural forgetful homomorphisms.It can be shown (cf. [13]) that deg#( f ) is (a strong version of) the Hopf-Ganea invari-

ant of [ f ]∈ πm(N) (w.r. to the attaching map of a top cell in N , compare [5, 6.7]), while

deg( f ) is closely related to (weaker) stabilized Hopf-James invariants ([12, 1.14]).

Special case: M = Sm, N = Sn, n≥ 2. Here deg# is injective and we see that

N( f ,∗)≤N#( f ,∗)=MCC( f ,∗)=⎧⎨⎩

0 if f is null homotopic,

1 otherwise,(1.17)

for all maps f : Sm → Sn. There are many dimension combinations (m,n), where the

equality N( f ,∗) = N#( f ,∗) is also valid for all f or, equivalently, where deg is injec-tive (compare, e.g., our Remark 4.2 below or [12, 1.16]). However, if n�= 1,3,7 is odd andm= 2n− 1, or if, for example, (m,n)= (8,4),(9,4),(9,3),(10,4),(16,8),(17,8),(10 +n,n)for 3 ≤ n ≤ 11, or (24,6), then there exists a map f : Sm → Sn such that 0 = N( f ,∗) <N#( f ,∗)= 1 (compare [12, 1.17]).

Very special case: M = S3,N = S2. Here

deg : π3(S2)∼= Z−→Ωfr

1

(ΩS2)∼= Z2⊕Z (1.18)


captures the Freudenthal suspension and the classical Hopf invariant of a homotopy class

[ f ]; therefore deg is injective (and so is deg# a fortiori).On the other hand the invariant deg( f ) ∈Ωfr

1∼= Z2 (which does not involve the path

space E( f ,∗)) retains only the suspension of f . The corresponding homological invariantμ(deg( f ))∈H1(S3;Z) vanishes altogether.

Finally let us point out that our approach can also be applied fruitfully to study linkingphenomena. Consider, for example, a link map

f = f1� f2 :M1�M2 −→N ×R (1.19)

(i.e., the closed manifolds M1 and M2 have disjoint images). Just as in the case of twodisjoint closed curves in R3 the degree of linking can be measured to some extend by thegeometry of the overcrossing locus: it consists of that part of the coincidence locus of theprojections to N , where f1 is bigger than f2 (w.r. to the R-coordinate). Here the normalbordism/path space approach yields strong unlinking obstructions which, in addition,turn out to distinguish a great number of different link homotopy classes. Moreover itleads to a natural notion of Nielsen numbers for link maps (cf. [10]).

2. The path space E(f1, f2)

A crucial feature of our approach to Nielsen theory is the central role played by thespace E( f1, f2). It yields the Nielsen decomposition of coincidence sets in a very natu-ral geometric fashion. In the defining (1.6) NI denotes the space of all continuous pathsθ : I := [0,1]→N with the compact—open topology. The starting point/endpoint fibra-tion NI →N ×N pulls back, via the map

(f1, f2

):M −→N ×N , (2.1)

to yield the Hurewicz fibration

p : E(f1, f2

)−→M (2.2)

defined by p(x,θ) = x. Given a coincidence point x0 ∈M, the fiber p−1({x0}) is just theloop space Ω(N , y0) of paths in N starting and ending at y0 = f1(x0)= f2(x0); let θ0 de-note the constant path at y0.

Proposition 2.1. The sequence of group homomorphisms

··· −→ πk+1(M,x0

) f1∗− f2∗−−−−−→ πk+1(N , y0

) incl∗−−−→ πk(E(f1, f2

),(x0,θ0

))

p∗−−→ πk(M,x0

)−→ ··· −→ π1(M,x0

) (2.3)

is exact. Moreover, the fiber inclusion incl : Ω(N , y0)→ E( f1, f2) induces a bijection of thesets

R(f1, f2

)= π1(N , y0

)/Reidemeister equivalence−→ π0

(E(f1, f2

)), (2.4)

where two classes [θ],[θ′]∈ π1(N , y0)= π0(Ω(N , y0)) are called Reidemeister equivalent if[θ′]= f1∗(τ)−1 · [θ] · f2∗(τ) for some τ ∈ π1(M,x0).

Ulrich Koschorke 7

The proof is fairly evident. In fact, we are dealing here essentially with the long exacthomotopy sequence of the fibration p.

3. Normal bordism

In this section we recall some standard facts about a geometric language which seems wellsuited to describe relevant coincidence phenomena in arbitrary codimensions.

Let X be a topological space and let ϕ be a virtual real vector bundle over X , that is, anordered pair (ϕ+,ϕ−) of vector bundles written ϕ= ϕ+−ϕ−.

A singular ϕ-manifold in X of dimension q is a triple (C,g ,g), where(i) C is a closed smooth q-dimensional manifold;

(ii) g : C→ X is a continuous map;(iii) g : TC ⊕ g∗(ϕ+) → g∗(ϕ−) is a stable vector bundle isomorphism (i.e., we can

first add trivial vector bundles of suitable dimensions on both sides).Two such triples (Ci,gi,gi), i= 0,1, are bordant if there exists a compact singular (q+

1)-dimensional ϕ-manifold (B,b,b) in X with boundary ∂B = C0�C1 such that b and b,when restricted to ∂B, coincide with the corresponding data gi and gi at Ci, i = 0,1 (viavector fields pointing into B along C0 and out of B along C1). The resulting set of bordismclasses, with the sum operation given by disjoint unions, is the qth normal bordism groupΩq(X ;ϕ) of X with coefficients in ϕ.

Example 3.1. Let G denote the trivial group or the (special) orthogonal group (S)O(q′),q′ > q + 1. For any topological space Y let ϕ+ be the classifying bundle over BG, pulledback to X = Y ×BG, while ϕ− is trivial. Then Ωq(X ;ϕ) is the standard (stably) framed,oriented or unoriented qth bordism group of Y (cf., e.g., [4, I.4 and 8]).

For every virtual vector bundle ϕ over a topological space X there are well knownHurewicz homomorphisms

μ : Ωq(X ;ϕ)−→Hq(X ; Zϕ

), q ∈ Z, (3.1)

into singular homology with local integer coefficients Zϕ (which are twisted like the ori-entation line bundle ξϕ = ξϕ+ ⊗ ξϕ− of ϕ); they map a normal bordism class [C,g ,g] to the

image of the fundamental class [C]∈Hq(C; ZTC) by the induced homomorphism g∗.In most cases μ leads to a big loss of information. However for q ≤ 4 this loss can often

be measured so that explicit calculations of (and in)Ωq(X ;ϕ) are possible (in particular sowhen ϕ is highly nontrivial), see [9, Theorem 9.3]. We obtain for example , the followinglemma.

Lemma 3.2. Assume X is path connected. Then the following hold.(i)

Ω0(X ;ϕ)μ

∼= H0(X ; Zϕ

) =⎧⎨⎩Z if w1(ϕ)= 0,

Z2 else.(3.2)


(ii) The following sequence is exact:

Ω2(X ;ϕ)μ−→H2

(X ; Zϕ

) w2(ϕ)−−−−→ Z2δ1−−→Ω1(X ;ϕ)

μ−→H1(X ; Zϕ

)−→ 0. (3.3)

Here δ1(1) is represented by the invariantly parallelized unit circle, together with a con-stant map, and

w1(ϕ)=w1(ϕ+)+w1

(ϕ−),

w2(ϕ)=w2(ϕ+)+w1

(ϕ+)w1

(ϕ−)

+w2(ϕ−)

+w1(ϕ−)2

(3.4)

denote Stiefel-Whitney classes of ϕ.

The setting of (normal) bordism groups provides also a first rate illustration of thefact that the geometric and differential topology of manifolds on one hand, and homo-topy theory on the other hand, are often but two sides of the same coin. Indeed, if ϕ−

allows a complementary vector bundle ϕ−⊥ (such that ϕ− ⊕ϕ−⊥ is trivial), then the well-known Pontryagin-Thom construction allows us to interpret Ωq(X ;ϕ), q ∈ Z, as a (sta-ble) homotopy group of the Thom space of ϕ+ ⊕ ϕ−⊥ which consists of the total spaceof ϕ+ ⊕ ϕ−⊥ with one point “added at infinity” (compare, e.g., [4, I, 11 and 12]). Thusthe methods of algebraic topology offer another (and often very powerful) approach tocomputing normal bordism groups (cf., e.g., [4, Chapter II]).

Example 3.3. The Thom space of the vector bundle ϕ = Rk over a one-point space isthe sphere Sk = Rk ∪ {∞}. Hence the framed bordism group Ωfr

q := Ωq({point};ϕ) is

canonically isomorphic to the stable homotopy group πSq := limk→∞πq+k(Sk) of spheres.It is computed and listed, for example, in Toda’s tables (in [14, Chapter XIV]) wheneverq ≤ 19.

For further details and references concerning normal bordism see, for example, [6] or[9].

4. The invariants

In this section we discuss the invariants ω( f1, f2) andN( f1, f2) based on normal bordism,as well as their sharper (nonstabilized) versions ω#( f1, f2) and N#( f1, f2). We refer to [12]for some of the details and proofs (see also [13]).

In the special case when the map ( f1, f2) :M→N ×N is smooth and transverse to thediagonal

Δ= {(y, y)∈N ×N | y ∈N}, (4.1)

the coincidence set

C = C( f1, f2)= ( f1, f2

)−1(Δ)= {x ∈M | f1(x)= f2(x)

}(4.2)

Ulrich Koschorke 9

C

M ( f1, f2)

N

N

N ×N

Δ

N ×N

Figure 4.1. A generic coincidence manifold and its normal bundle.

is a smooth submanifold of M. It comes with the maps

E(f1, f2

)

p

C

g

gM

(4.3)

defined by g(x)= x and g(x)= (x, constant path at f1(x)= f2(x)), x ∈ C.The normal bundle of C in M is described by the isomorphism

ν(C,M)∼= ( f1, f2)∗(

ν(Δ,N ×N))∼= f ∗1 (TN) | C (4.4)

(see Figure 4.1) which yields

g : TC⊕ f ∗1 (TN) | C ∼=−−→ TM | C. (4.5)

Define

ω(f1, f2

):= [C, g,g]∈Ωm−n

(E(f1, f2

); ϕ), (4.6)

ω(f1, f2

):= [C,g,g]= p∗

(ω(f1, f2

))∈Ωm−n(M;ϕ), (4.7)

where

ϕ := f ∗1 (TN)−TM, ϕ := p∗(ϕ). (4.8)

Invariants with precisely the same properties can be constructed in general. Indeed,apply the preceding procedure to a smooth map ( f ′1 , f ′2 ) which is transverse to Δ andapproximates ( f1, f2).

Also apply the isomorphism Ω∗(E( f ′1 , f ′2 ); ϕ′) ∼= Ω∗(E( f1, f2); ϕ) induced by a smallhomotopy (cf. [12, 3.3]) to ω( f ′1 , f ′2 ) in order to obtain ω( f1, f2) and similarly ω( f1, f2).


Now consider the decomposition

ω(f1, f2

)= {ωA(f1, f2

)}∈Ωm−n(E(f1, f2

); ϕ)=

⊕

A

Ωm−n(A; ϕ |A) (4.9)

according to the various path components A∈ π0(E( f1, f2)) of E( f1, f2).

Definition 4.1. A pathcomponent of E( f1, f2) is called essential if the corresponding di-rect summand of ω( f1, f2) is nontrivial. The Nielsen coincidence number N( f1, f2) is thenumber of essential path components A∈ π0(E( f1, f2)).

Since we assume M to be compact, N( f1, f2) is a finite integer. It vanishes if and onlyif ω( f1, f2) does.

Remark 4.2. In Figure 4.1 we have neglected an important geometric aspect: C is muchmore than just an (abstract) singular manifold with an description of its stable normalbundle. If we keep track (i) of the fact that C is a smooth submanifold in M, and (ii)of the nonstabilized isomorphism (4.4), we obtain the sharper invariants ω#( f1, f2) andN#( f1, f2). Note, however, that the bordism set Ω#( f1, f2) in which ω#( f1, f2) lies has pos-sibly no group structure—the union of submanifolds may no longer be a submanifold.Also N#( f1, f2)= 0 if ω#( f1, f2)= 0, but the converse may possibly not hold in general—nulbordisms of individual coincidence components may intersect in M× I .

However, in the stable range m ≤ 2n− 2, ω#( f1, f2) contains precisely as much infor-mation as ω( f1, f2) does, and N#( f1, f2)=N( f1, f2).

Let us summarize, we have the (successively weaker) invariants ω#( f1, f2), ω( f1, f2),ω( f1, f2) and μ(ω( f1, f2)) = Poincare dual of the cohomological primary obstruction todeforming f1 and f2 away from one another (cf. [8, 3.3]); they are related by the naturalforgetful maps

Ω#( f1, f2) stabilize−−−−−→Ωm−n

(E(f1, f2

); ϕ) p∗−−→Ωm−n(M;ϕ)

μ−→Hm−n(M; Zϕ

)(4.10)

(cf. Remark 4.2, (4.3), and (3.1)). Only ω#( f1, f2) and ω( f1, f2) involve the path spaceE( f1, f2), thus allowing the definition of the Nielsen numbers N#( f1, f2) and N( f1, f2).

Example 4.3 the classical dimension setting m= n. Here the coincidence set

C(f1, f2

)=∐

A∈π0(E( f1, f2))

g−1(A) (4.11)

consists generically of isolated points (in this very special situation the stabilizing mapand the Hurewicz homomorphism μ in (4.10) lead to no significant loss of information).

In our approach, each Nielsen class is expressed as an inverse image of some pathcomponent A of E( f1, f2) (compare Proposition 2.1). The corresponding index

ωA(f1, f2

)∈Ω0(A; ϕ |A)∼=⎧⎨⎩Z if ω1(ϕ | A)= 0,

Z2 else,(4.12)

Ulrich Koschorke 11

(cf. Lemma 3.2) lies in Z2 precisely if w1(ϕ | A) �= 0 or, equivalently, if for some (andhence all) x0 ∈ g−1(A) there exists a class α ∈ π1(M,x0) such that f1∗(α) = f2∗(α) butw1(M)(α)�= f ∗1 (w1(N))(α) (cf. [12, 5.2]; this agrees with the criterion quoted in [1, page53 lines 5–6]). If π1(N) is commutative, then either the indices of all Nielsen classes areintegers, or they all lie in Z2. However, it is easy to construct examples (e.g., involvingmaps from the Klein bottle to the punctured torus) where both types of path componentsA∈ π0(E( f1, f2)) occur.

In any case our approach makes it clear from the outset where the indices of Nielsenclasses must take their values.

In the setting of fixed point theory (where f2 is the identity map on M =N) the tran-sition from the ω- to the ω-invariant (cf. (4.6) and (4.7)) which forgets the path spaceE( f1, f2) parallels the transition from Nielsen to Lefschetz theory—with all the loss ofinformation which this entails.

5. Finiteness conditions for the minimum number MC(f1, f2)

Consider the following possible conditions concerning the invariants defined in (1.2),(4.6), and (4.7):

(C1) MC( f1, f2)≤ 1 ;(C2) MC( f1, f2) is finite ;(C3) ω( f1, f2) lies in the image of the homomorphism

iE∗ :=⊕

A

iA∗ :⊕

A

Ωfrm−n −→

⊕

A

Ωm−n(A; ϕ |)=Ωm−n(E(f1, f2

); ϕ), (5.1)

where direct summation is taken over all A∈ π0(E( f1, f2)) and iA∗ is induced bythe inclusion of a point zA into the path componentA (and by a local orientationof ϕ at zA);

(C4) ω( f1, f2) lies in the image of a similarly defined homomorphism

i∗ : Ωfrm−n −→Ωm−n(M;ϕ). (5.2)

Proposition 5.1. Each of the first three conditions implies the next one.

Proof. Assume that the coincidence set C( f1, f2) is finite. If a generic pair ( f ′1 , f ′2 ) approx-imates ( f1, f2) closely enough then each component of C( f ′1 , f ′2 ) lies in a ball neighbour-hood of some x ∈ C( f1, f2); moreover the corresponding paths which occur in the con-struction of ω( f1, f2) lie entirely in a ball neighbourhood of y = f1(x)= f2(x) and hencecan be contracted into the constant path at y. Thus (C2) implies (C3). The propositionfollows. �

Our coincidence invariants ω( f1, f2) and ω( f1, f2) project to the obstructions

[ω(f1, f2

)]∈ coker(iE∗),

[ω(f1, f2

)]∈ coker i∗,(5.3)

which must vanish if MC( f1, f2) is to be finite.


For m−n= 0 these cokernels are trivial, MC( f1, f2) is actually finite and each integerd ≥ 0 can occur as the value of this minimum number for a suitable pair of maps (e.g.,for self maps of degrees d and 0 on S1).

If m−n= 1 the cokernels in (5.3) are isomorphic—via the Hurewicz homomorphismμ (cf. (3.1))—to H1(E( f1, f2); Zϕ) and H1(M; Zϕ), respectively, (compare Lemma 3.2). Infact μ vanishes on the image of iE∗ and of i∗, respectively, whenever m− n ≥ 1, but ingeneral the resulting homomorphisms on the cokernels will not be injective whenm−n >1 (cf. [12, 9.3]).

Remark 5.2. The finiteness criterion in Proposition 5.1 can be sharpened to yield a non-stabilized version involving ω#( f1, f2).

For self-coincidences there is a partial converse of Proposition 5.1.

Theorem 5.3. If m< 2n− 2 and f1 is homotopic to f2, then the four conditions (C1)–(C4)are equivalent.

Proof. These conditions are compatible with homotopies of f1 and f2 (cf. [12, 3.3] andthe discussion following (4.4)). Hence we may assume that f1 = f2 =: f .

Recall that the self-coincidence invariant ω( f , f ) is just the singularity obstructionω(R

˜, f ∗(TN)) to sectioning the vector bundle f ∗(TN) over M without zeroes (cf. [11,

Theorem 2.2]).Now assume that m< 2n− 2 and ω( f , f )= i∗(ω0) for some ω0 ∈Ωfr

m−n ∼= πSm−n. Thenthere exists a map u∂ : Sm−1 → Sn−1 whose (stable) Freudenthal suspension correspondsto ω0. Now consider the trivial bundle f ∗(TN) | Bm over some compact ball Bm in Mand interpret u∂ as a nowhere zero section over ∂Bm = Sm−1.

We will extend u∂ to a section u of f ∗(TN) over all of M which vanishes only in thecentre point of Bm. Over the ball Bm we use the obvious “concentric” extension. Note,however, that generically the zero set of any extension of u∂ over Bm is a framed manifold

which represents ω0. Thus a generic extension of u∂ to the complement M −◦Bm must

have a nulbordant manifold of zeroes (representing ω( f , f )− i∗(ω0)= 0). According to[9, Theorem 3.7] these zeroes can be removed altogether.

The resulting section u of f ∗(TN) allows us to construct a “small” homotopy of f :for every x ∈M just deform f (x) somewhat in the direction of the tangent vector u(x)∈Tf (x)(N). We obtain a map which has only one coincidence point with f . �

6. The examples of the introduction

In view of the self-coincidence theorem in [11] and of our Theorem 5.3 the first exampleis a special case of the following proposition.

Proposition 6.1. Let ξ be an oriented real plane bundle over a closed smooth connectedmanifold N and let f :M := S(ξ)→N denote the projection of the corresponding unit circlebundle. Then, the following exist:

(i) the self-coincidence invariant ω( f , f ) (cf. (4.7)) vanishes if and only if the Eulernumber χ(N) of N lies in e(ξ)(H2(N ;Z)) ⊂ Z and χ(N) is even; here e(ξ) denotesthe Euler class of ξ,

Ulrich Koschorke 13

(ii) the finiteness obstruction μ(ω( f , f )) � [ω( f , f )] (cf. (5.3) and the subsequent dis-cussion in Section 5) vanishes if and only if χ(N)∈ e(ξ)(H2(N ;Z)).

Proof. We will extend the arguments of [11, Section 4]. Consider the commuting diagram

χ(N)

∈

Z2

ω( f , f )

∈ incl∗

Ω2(N ;−ξ)�

μ

Ωfr0 (N)

∂Ω1(M;− f ∗(ξ)

)

μ

H2(N ;Z)e(ξ)

w2(ξ)

Z H1(M;Z)

Z2 0

(6.1)

Here the vertical exact sequences are as described in Lemma 3.2. The horizontal linesare exact Gysin sequences of ξ in normal bordism and in homology (or, equivalently,oriented bordism), compare [9, 9.20 and 9.4]. As was shown in [11, Section 4], we haveω( f , f ) = χ(N) · ∂(1). Since the Stiefel-Whitney class w2(ξ) is the mod2 reduction ofe(ξ), the proposition follows. �

Next let us examine Example 1.5. If r ≥ 2k ≥ 2 then according to the theorem in the in-troduction of [11] only the “highest order part” 1.11 of the complete obstruction ω( f , f )(to deforming f away from itself) survives. Thus the finiteness obstruction [ω( f , f )] (cf.(5.3)) vanishes. If also k ≥ 2 then it follows from Theorem 5.3 that MC( f , f ) (and hencealso MCC( f , f )) equals 0 or 1 according to whether ω( f , f ) vanishes or not. It requiresusing deep results of homotopy theory and of other branches of algebraic topology to de-cide which of the two values occur actually, but it can be done at least for k ≤ 10 (cf. [11,Section 3]). However, the case k = 1 (where we deal with the standard projection fromSr−1 to real projective space) is elementary.

Next we turn to Example 1.6 where N = (S1)n. Use the Lie group structure to replace( f1, f2) by the pair ( f ,∗) which consists of the quotient f = f1 · f −1

2 and of the constantmap taking values at the unit element of (S1)n. This does not change the coincidence setsand data significantly.

Since each torus (S1)k is a K(Zk,1)-space the homotopy class of f is determined by f∗ :H1(M;Z)→H1((S1)n;Z). Moreover, if the image of f∗ has rank k < n, then f factors upto homotopy through the lower-dimensional torus (S1)k and hence through (S1)n−{∗}.On the other hand, if the image of f∗ has rank n (or, equivalently, the Reidemeister setR( f ,∗)∼= coker f∗ is finite) then—according to Remark 1.3—all Reidemeister classes areessential and hence N( f ,∗)= #R( f ,∗), provided ω( f ,∗)�= 0. This holds, in particular,


if the invariant

ω(coll◦ f ,∗)∈Ωm−n(M;−TM) (6.2)

which corresponds to the bordism class of the stably coframed manifold C( f ,∗) =f −1({∗})⊂M, is nontrivial. Here the map

coll :N = (S1)n −→N/(N −

◦B)∼= Sn (6.3)

collapses the complement of an open ball to a point. The induced cohomology homomor-phisms coll∗, and ( f ◦ coll)∗, respectively, map a generator of Hn(Sn;Z) to the cup prod-uct ι1 ··· ιn ∈Hn((S1)n;Z), and to the Poincare dual of μ(ω( f ,∗)), respectively, (compare(4.10) and [8, 3.3]). Our claims concerning Example 1.6 in the introduction follow nowfrom Section 5.

Finally note that the facts described in Example 1.7 follow mainly from the discussionin [12] (see 1.14–1.17 as well as Sections 7 and 8); the calculation of Ωfr

1 (ΩS2) can beunderstood easily with the help of our Lemma 3.2.

Let us put the role of the path space E( f1, f2) and its influence on the relative strengthof our invariants into perspective (compare diagram (4.10)).

In the self coincidence situation f1 = f2 := f the fibration p : E( f , f )→M allows aglobal section s by constant paths; therefore ω( f , f ) = s∗(ω( f , f )) is precisely as strongas the (usually much weaker) invariant ω( f , f ) which does not involve any path spacedata. As Examples 1.4 and 1.5 illustrate, ω( f , f ) may nevertheless capture decisive andvery delicate information (which is also registered to some extend by the Nielsen numberN( f , f ) in spite of the fact that it can take only the values 0 and 1).

In Example 1.6 our path space approach serves to decompose coincidence sets intoNielsen classes. However, it does not seem to enrich the higher-dimensional homotopytheoretical aspects of their data very much (as the torus is aspherical; compare Proposition2.1). Still, all natural numbers can occur here as Nielsen numbers of suitable maps f1 andf2.

In contrast, in Example 1.7 the higher-dimensional topology of E( f1, f2) turns out tobe potentially very rich (e.g., when N = Sn, n ≥ 2) and able to capture much more thanjust the decomposition into Nielsen classes.

Acknowledgment

This work was supported in part by the Deutsche Forschungsgemeinschaft and AARMS(Canada).

References

[1] S. A. Bogatyı, D. L. Goncalves, and H. Zieschang, Coincidence theory: the minimization problem,Proceedings of the Steklov Institute of Mathematics 225 (1999), no. 2, 45–77.

[2] R. B. S. Brooks, On removing coincidences of two maps when only one, rather than both, of themmay be deformed by a homotopy, Pacific Journal of Mathematics 40 (1972), no. 1, 45–52.

Ulrich Koschorke 15

[3] R. F. Brown, Wecken properties for manifolds, Nielsen Theory and Dynamical Systems (Mass,1992), Contemp. Math., vol. 152, American Mathematical Society, Rhode Island, 1993, pp. 9–21.

[4] P. E. Conner and E. E. Floyd, Differentiable Periodic Maps, Ergebnisse der Mathematik und ihrerGrenzgebiete, N. F., vol. 33, Academic Press, New York; Springer, Berlin, 1964.

[5] O. Cornea, G. Lupton, J. Oprea, and D. Tanre, Lusternik-Schnirelmann Category, MathematicalSurveys and Monographs, vol. 103, American Mathematical Society, Rhode Island, 2003.

[6] J.-P. Dax, Etude homotopique des espaces de plongements, Annales Scientifiques de l’Ecole Nor-male Superieure. Quatrieme Serie (4) 5 (1972), 303–377 (French).

[7] A. Dold and D. L. Goncalves, Self-coincidence of fibre maps, Osaka Journal of Mathematics 42(2005), no. 2, 291–307.

[8] D. L. Goncalves, J. Jezierski, and P. Wong, Obstruction theory and coincidences in positive codi-mension, Bates College, preprint, 2002.

[9] U. Koschorke, Vector Fields and Other Vector Bundle Morphisms—a Singularity Approach, LectureNotes in Mathematics, vol. 847, Springer, Berlin, 1981.

[10] , Linking and coincidence invariants, Fundamenta Mathematicae 184 (2004), 187–203.[11] , Selfcoincidences in higher codimensions, Journal fur die Reine und Angewandte Mathe-

matik 576 (2004), 1–10.[12] , Nielsen coincidence theory in arbitrary codimensions, to appear in to appear in Jour-

nal f’ur die Reine und Angewandte Mathematik, 2003, http:/www.math.uni-siegen.de/topology/publications.html.

[13] , Nonstabilized Nielsen coincidence invariants and Hopf-Ganea homomorphisms, preprint,2005, http://www.math.uni-siegen.de/topology/publications.html.

[14] H. Toda, Composition Methods in Homotopy Groups of Spheres, Annals of Mathematics Studies,no. 49, Princeton University Press, New Jersey, 1962.

Ulrich Koschorke: Universitat Siegen, Emmy Noether Campus, Walter-Flex Street 3,D-57068 Siegen, GermanyE-mail address: [email protected]

FIXED POINT SETS OF MAPS HOMOTOPIC TO A GIVEN MAP

CHRISTINA L. SODERLUND

Received 3 December 2004; Revised 20 April 2005; Accepted 24 July 2005

Let f : X → X be a self-map of a compact, connected polyhedron and Φ⊆ X a closed sub-set. We examine necessary and sufficient conditions for realizing Φ as the fixed point setof a map homotopic to f . For the case where Φ is a subpolyhedron, two necessary condi-tions were presented by Schirmer in 1990 and were proven sufficient under appropriateadditional hypotheses. We will show that the same conditions remain sufficient when Φ isonly assumed to be a locally contractible subset of X . The relative form of the realizationproblem has also been solved for Φ a subpolyhedron of X . We also extend these results tothe case where Φ is a locally contractible subset.

Copyright © 2006 Christina L. Soderlund. This is an open access article distributed un-der the Creative Commons Attribution License, which permits unrestricted use, distri-bution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

Let f : X → X be a self-map of a compact, connected polyhedron. For any map g , denotethe fixed point set of g as Fixg = {x ∈ X | g(x) = x}. In this paper, we are concernedwith the realization of an arbitrary closed subset Φ⊆ X as the fixed point set of a map ghomotopic to f .

Several necessary conditions for this problem are well known. If Φ = Fixg for somemap g homotopic to f , it is clear that Φ must be closed. Further, by the definition ofa fixed point class (cf. [1, page 86], [7, page 5]), all points in a given component of Φmust lie in the same fixed point class. Thus, as the Nielsen number (cf. [1, page 87], [7,page 17]) of any map cannot exceed the number of fixed point classes and as the Nielsennumber is also a homotopy invariant, the set Φ must have at least N( f ) components.In particular, if N( f ) > 0 then Φ must be nonempty. It is also necessary that f |Φ, therestriction of f to the set Φ, must be homotopic to the inclusion map i : Φ↩X .

In [12], Strantzalos claimed that the above conditions are sufficient if X is a compact,connected topological manifold with dimension�= 2, 4, or 5 and if Φ is a closed nonemptysubset lying in the interior of X with π1(X ,X −Φ)= 0. However, Schirmer disproved thisclaim in [10] with a counterexample and presented her own conditions, (C1) and (C2).


2 Fixed point sets of maps homotopic to a given map

Definition 1.1 [10, page 155]. Let f : X → X be a self-map of a compact, connected poly-hedron. The map f satisfies conditions (C1) and (C2) for a subset Φ⊆ X if the followingare satisfied (the symbol � denotes homotopy of paths with endpoints fixed and ∗ thepath product):

(C1) there exists a homotopy HΦ : Φ× I → X from f |Φ to the inclusion i : Φ↩X ,(C2) for every essential fixed point class F of f : X → X there exists a path α : I → X

with α(0)∈ F, α(1)∈Φ, and

{α(t)

}�{f ◦α(t)

}∗ {HΦ(α(1), t

)}. (1.1)

The latter condition, (C2), reflects Strantzalos’ error. He apparently overlooked theH-relation of essential fixed point classes of two homotopic maps (cf. [1, pages 87–92],[7, pages 9, 19]).

Schirmer showed that (C1) and (C2) are both necessary conditions for realizing Φ asthe fixed point set of any map g homotopic to f ([10, Theorem 2.1]). She then invokedthe notion of by-passing ([9, Definition 5.1]) to prove the following sufficiency theorem.A local cutpoint is any point x ∈ X that has a connected neighborhood U so that U −{x}is not connected.

Theorem 1.2 [10]. Let f : X → X be a self-map of a compact, connected polyhedron withouta local cutpoint and let Φ be a closed subset of X . Assume that there exists a subpolyhedronK of X such that Φ⊂ K , every component of K intersects Φ, X −K is not a 2-manifold, andK can be by-passed. If (C1) and (C2) hold for K , then there exists a map g homotopic to fwith Fixg =Φ.

Observe that Schirmer’s theorem permits Φ to be any type of subset, provided it lieswithin an appropriate polyhedron K . However, all the required conditions are placed onthe polyhedron K . If we wish to prove that Φ can be the fixed point set, then we shouldrequire that our conditions be on Φ itself. We can remedy this problem with a statementequivalent to that of Theorem 1.2.

Theorem 1.3. Let f : X → X be a self-map of a compact connected polyhedron without alocal cutpoint and let Φ be a closed subpolyhedron of X satisfying

(1) X −Φ is not a 2-manifold,(2) (C1) and (C2) hold for Φ,(3) Φ can be by-passed.

Then for every closed subset Γ of Φ that has nonempty intersection with every component ofΦ, there exists a map g homotopic to f with Fixg = Γ. In particular, if Φ is connected, thenevery closed subset of Φ (including Φ itself) is the fixed point set of a map homotopic to f .

Although Theorem 1.3 requires Φ to be a subpolyhedron, the subset Γ⊆Φ is subjectto few restrictions, thus preserving the broad scope of Schirmer’s original theorem.

In Section 3 we extend Theorem 1.3 to the case where Φ is a closed, locally contractiblesubset of X , but not necessarily a polyhedron. The result is given in Theorem 3.5. Sincethe class of closed, locally contractible spaces contains the class of compact, connected

Christina L. Soderlund 3

polyhedra, this extension broadens the scope of the sufficiency theorem. Moreover, poly-hedral structure is a global property, whereas local contractibility is a local property andthus presumably easier to verify.

We examine a similar question for maps of pairs in Section 4. For any map f : (X ,A)→(X ,A) of a polyhedral pair, Ng ([8]) presented necessary and sufficient conditions forrealizing a subpolyhedron Φ ⊆ X as the fixed point set of a map homotopic to f via ahomotopy of pairs. Ng’s results solved a problem raised by Schirmer in [11]. Since Ng’stheory was never published, we include a sketch of his work for the convenience of thereader. We conclude by extending Ng’s results to the case where Φ is a closed, locallycontractible subset of X (Theorem 5.3).

It is assumed that the reader is familiar with the general definitions and techniques ofNielsen theory, as in [1, 7].

2. Neighborhood by-passing

LetX be a compact, connected polyhedron andΦ a subset ofX . We sayΦ can be by-passedin X if every path in X with endpoints in X −Φ is homotopic relative to the endpoints toa path in X −Φ.

The notion of by-passing plays a key role in relative Nielsen theory and in realizingfixed point sets. Currently, we wish to extend Theorem 1.3 to the case where Φ is a locallycontractible subset, but not necessarily a polyhedron (Theorem 3.5). To do so, we requirea property that is closely related to by-passing. This property is the subject of the nextdefinition.

Definition 2.1. A subset Φ of a topological space X can be neighborhood by-passed if thereexists an open set V in X , containing Φ, such that V can be by-passed.

If Φ is chosen to be by-passed, the next theorem suggests that adding the requirementthat Φ also be neighborhood by-passed does not affect our choice of Φ.

Theorem 2.2. If X is a compact, connected polyhedron, Φ ⊆ X is a closed, locally con-tractible subset, and if Φ can be by-passed, then Φ can be neighborhood by-passed.

Proof [3]. We prove this theorem in two steps. First we show that for any open neighbor-hoodU ofΦ, there exists a closed neighborhoodN ⊂U ofΦ, withX −N path connected.We then show that this neighborhood N can be chosen to be by-passed in X .Step 1. Given an open neighborhood U of Φ, there exists N ⊂ U , a closed neighborhoodof Φ, with X −N path connected. Let U ⊂ X be any open neighborhood of Φ. Choosea closed neighborhood M of Φ, contained in U . Then X −U can be covered by finitelymany components of X −M. (This follows from compactness since X −U is closed in Xand therefore compact.)

Since Φ can be by-passed in X , we can connect each pair of these components by apath in X −Φ. In particular, for each pair of components M′

i and M′j of X −M, choose

points xi ∈M′i and xj ∈M′

j and choose a path

pi j : I −→ (X −Φ) (2.1)


with

pi j(0)= xi, pi j(1)= xj (2.2)

(xi and xj can lie either in U or its complement).Next we find a closed neighborhood K of Φ, contained in M, such that K misses all

the paths pi j . This is possible since

(X − Int(M)

)∪ ({pi j})

(2.3)

is compact (where Int(M) denotes the interior of M).We will prove that there is exactly one path component of the complement of such K

which contains X −U .First, observe that each component M′

i of X −M must lie in a single component ofX −K . If this was false, then for each component K ′j of X −K which intersects M′

i , wecould write M′

i as a disjoint union of clopen sets,

M′i =

⊔

j

(M′

i ∩K ′j), (2.4)

contrary to the connectedness of M′i .

Now suppose there exist two different components M′i and M′

j of X −M, lying indifferent components of X −K . Then the path pi j , as defined above, lies entirely withinX −K (by definition ofK). But pi j must also intersect the two components ofX −K , thuscontradicting the connectedness of paths. Therefore, M′

i and M′j (and hence all compo-

nents of X −M) lie in a single component of X −K . This component therefore containsX −U .

Finally, let W be the path component of X −K containing X −U . We have

X −U ⊂W ⊂ X −K , (2.5)

and hence

Φ⊂ K ⊂ X −W ⊂U. (2.6)

Define N = X −W . Then N ⊂ U is a closed neighborhood of Φ with path connectedcomplement.Step 2. We can choose the closed neighborhood N from Step 1 to be a subset that can beby-passed in X : since X is a compact, connected polyhedron, it has a finitely generatedfundamental group at any basepoint. Choose a basepoint a∈ (X −Φ) and finitely manygenerators (loops)

ρ1, . . . ,ρn : I −→ X (2.7)

of π1(X ,a). As Φ can be by-passed, these loops may be homotoped off Φ. Thus without aloss of generality, we can rename these generators

ρ1, . . . ,ρn : I −→ (X −Φ). (2.8)


Let

P =n⋃

i=1

Im(ρi)

(2.9)

be a compact subset of X −Φ, where Im(ρi) denotes the image of the path ρi. Let U be anopen neighborhood of Φ with U ∩P =∅.

Then any loop α in X with basepoint a∈ X −Φ can be expressed as a word consistingof a finite number of loops in X −U . Thus, α is homotopic to a loop in X −U .

Now as in Step 1, choose N in U having path connected complement. Then by [9,Theorem 5.2], N may be by-passed. Choosing V = Int(N) completes the proof. �

3. Realizing subsets of ANRs as fixed point sets

Our present goal is to show that if the subset Φ in Theorem 1.3 is chosen to be locallycontractible, but not necessarily polyhedral, the results of this theorem still hold. In par-ticular, every closed subset of Φ that intersects every component of Φ can be realized asthe fixed point set of a map homotopic to f . We will prove this by constructing a sub-polyhedron of X that contains such Φ and also satisfies the hypotheses of Theorem 1.3.

Lemma 3.1. If Φ is a closed subset of a compact, connected polyhedron X and X −Φ isnot a 2-manifold, then there exists a closed neighborhood N of Φ such that X −N is not a2-manifold.

Proof. Since X −Φ is not a 2-manifold, there exists an element x ∈ X −Φ with the prop-erty that no neighborhood of x is homeomorphic to the 2-disk.

Let d denote distance in X and suppose d(x,Φ) = δ > 0. Then the closed δ/2-neighborhood N of Φ satisfies the property that X −N is not a 2-manifold. �

Definition 3.2. Let Y be a metric space with distance d and choose a real-valued constantε > 0. Given any topological space X , two maps f ,g : X → Y are ε-near if d( f (x),g(x)) < εfor every x ∈ X . A homotopy H : X × I → Y is called an ε-homotopy if for any x ∈ X ,diam (H(x× I)) < ε.

Here we assume the usual definition of diameter: given a subsetA⊆ X and the distanced on X , diam(A)= sup{d(x, y) | x, y ∈ A}. Thus,

diam(H(x× I))= sup

{d(H(x, t),H(x, t′)

) | t, t′ ∈ I}. (3.1)

Theorem 3.3 [4, Proposition 3.4, page 121]. If X is a metric ANR and Φ is a closed ANRsubspace of X , then for every ε > 0, there exists an ε-homotopy ht : X → X satisfying

(1) h0 = idX ,(2) ht(x)= x for all x ∈Φ, t ∈ I ,(3) there exists an open neighborhood U of Φ in X such that h1(U)=Φ.


The map ht is called a strong deformation retraction of the space U onto the subspaceΦ. We also say U strong deformation retracts onto Φ.

Lemma 3.4. Let f : X → X be a self-map of a compact, connected polyhedron and let Φ bea closed subset of X . Assume that there exists a subset B of X such that Φ⊆ B and B strongdeformation retracts onto Φ. If f satisfies (C1) and (C2) for Φ, then f satisfies (C1) and(C2) for B.

Proof. To verify (C1) for B, let R : B× I → B denote the strong deformation retractionfrom B onto Φ, and denote R(b, t) = rt(b) for any b ∈ B, t ∈ I . So r0(b) = b, r1(b) ∈Φ,and rt|Φ = idΦ. We will construct a homotopy HB : B× I → X from f | B to the inclusioni : B↩X .

Let H : B× I → X be the composition

H(b, t)=⎧⎨⎩f ◦ r2t(b) 0≤ t ≤ 1/2,

HΦ(r1(b),2t− 1

)1/2≤ t ≤ 1,

(3.2)

where HΦ is the homotopy given by (C1) on Φ. Then f is homotopic to r1 via H .Next we can construct a homotopy HB : B× I → X as follows:

HB(b, t)=⎧⎨⎩H(b,2t) 0≤ t ≤ 1/2,

R(b,2− 2t) 1/2≤ t ≤ 1.(3.3)

Observe that f |B is homotopic to the identity via HB. Thus, HB gives the desired homo-topy satisfying (C1) for B.

To prove (C2), choose any essential fixed point class F of f : X → X . As f satisfies (C2)for Φ, there exists a path α : I → X with α(0)∈ F and α(1)∈Φ⊆ B, whence α(1)∈ B.

We show that the homotopy HB : B× I → X constructed above can be viewed as anextension of HΦ : Φ× I → X . To see this, note that since R : B× I → B is a strong defor-mation retraction, for any x ∈Φ,

HB(x, t)=⎧⎨⎩H(x,2t) 0≤ t ≤ 1/2,

x 1/2≤ t ≤ 1,

H(x, t)=⎧⎨⎩f ◦ r2t(x)= f (x) 0≤ t ≤ 1/2,

HΦ(x,2t− 1) 1/2≤ t ≤ 1.

(3.4)

Thus for any x ∈Φ,

HB(x, t)=

⎧⎪⎪⎪⎨⎪⎪⎪⎩

f (x) 0≤ t ≤ 1/4,

HΦ(x,4t− 1) 1/4≤ t ≤ 1/2,

x 1/2≤ t ≤ 1,

(3.5)


and we sayHB|Φ is a reparametrization ofHΦ. Then by defining a continuous map φ : I →I by

φ(s)=

⎧⎪⎪⎪⎨⎪⎪⎪⎩

0 0≤ s≤ 1/4,

4s− 1 1/4≤ s≤ 1/2,

1 1/2≤ s≤ 1,

(3.6)

it is clear that HB|Φ =HΦ ◦ (id×φ), where id denotes the identity map on Φ and

(id×φ)(x,s)= (x,φ(s)), (3.7)

for any x ∈Φ, s∈ I . Therefore, HΦ is homotopic to HB|Φ via the homotopyH : (X × I)×I → X , defined by

H(x, t,s)=HΦ(x,φt(s)

), (3.8)

where

φt(s)= (1− t)φ(s) + ts. (3.9)

Finally, since f satisfies (C2) for Φ, we know that for any essential fixed point class Fof f , there exists a path α in X with α(0)∈ F, α(1)∈Φ, and

{α(t)

}�{f ◦α(t)

}∗ {HΦ(α(1), t

)}. (3.10)

From the above argument, {HΦ(α(1), t)}� {HB(α(1), t)}. Therefore,{α(t)

}�{f ◦α(t)

}∗ {HB(α(1), t

)}(3.11)

and f satisfies (C2) for B. �

As a consequence of the above results, we are now able to extend Theorem 1.3 to thecase where Φ is locally contractible.

Theorem 3.5. Let f : X → X be a self-map of a compact connected polyhedron without alocal cutpoint. Let Φ be a closed, locally contractible subspace of X satisfying

(1) X −Φ is not a 2-manifold,(2) f satisfies (C1) and (C2) for Φ,(3) Φ can be by-passed.

Then for every closed subset Γ of Φ that has nonempty intersection with every component ofΦ, there exists a map g homotopic to f with Fixg = Γ. In particular, if Φ is connected, thenevery closed subset of Φ (including Φ itself) is the fixed point set of a map homotopic to f .

The proof of this theorem requires a polyhedral construction known as the star coverof a subset. Let K be a triangulation of X . We write X = |K|. Then for any vertex v of K ,define the star of v, denoted StK (v), to be the union of all closed simplices of which v is avertex. Then for any subspace Φ⊆ X , the star cover of Φ is

StK (Φ)=⋃

v∈ΦStK (v). (3.12)


(4, 0) (8, 0) (11, 0)

Figure 3.1. A locally contractible fixed point set.

Proof of Theorem 3.5. We can assume Φ�=∅ as, otherwise, this theorem reduces to [10,Lemma 3.1]. Since X is a polyhedron, let K be a triangulation of X = |K|. By [2, Propo-sition 8.12, page 83], Φ is a finite-dimensional ANR. Thus, Theorem 3.3 gives an openneighborhoodU of Φ that strong deformation retracts onto Φ. Since Φ can be by-passed,Theorem 2.2 implies that there exists another open neighborhoodV of Φ such thatV canbe by-passed. The set V may be chosen to lie inside U . Choose a star cover StK ′(Φ) of Φwith respect to a sufficiently small subdivision K ′ of K such that StK ′(Φ)⊂V . Then (C1)and (C2) hold for StK ′(Φ) (Lemma 3.4). Further, the subdivision K ′ can be chosen smallenough so that X − StK ′(Φ) is not a 2-manifold (Lemma 3.1).

By the construction of star covers, each component of StK ′(Φ) contains a componentof Φ. If every component of Φ, in turn, intersects a given closed subset Γ⊂Φ, then eachcomponent of the star cover intersects Γ. As star covers are themselves polyhedra, theresult follows from Theorem 1.3. �

We close this section with an example of a self-map f on a compact, connected poly-hedron X , with a locally contractible subset Φ that is not a finite polyhedron, for whichthere exists g homotopic to f with Fixg =Φ.

Example 3.6. Consider the space

X = {(x, y)∈R2 | 4≤ (x− 4)2 + y2 ≤ 49}

, (3.13)

the annulus in R2 centered at the point (4,0), with outer radius 7 and inner radius 2 (seeFigure 3.1). Let f : X → X be the map flippingX over the x-axis. That is, f (x, y)= (x,−y).

Clearly Fix( f ) lies on the x-axis and f has exactly two fixed point classes,

F1 ={

(x,0) | −3≤ x ≤ 2}

, F2 ={

(x,0) | 6≤ x ≤ 11}. (3.14)


We define Φ=D∪Z∪{(8,0)} where

D = {(x, y) | (x+ 1)2 + y2 ≤ 1}

,

Z =∞⋃

k=1

([0,zk

]∪ [0,z−k]).

(3.15)

For each positive integer k, [0,zk] denotes the line segment in R2 from the point (0,0) tothe point (1/k,1/k2), and [0,z−k] is the line segment from (0,0) to (1/k,−1/k2).

First we show that Φ is locally contractible. At the origin, a sufficiently small neigh-borhood contracts via straight lines. Also for each k, given any point on the line segment[0,zk], we can find a neighborhood that does not contain any other segment of Φ, andhence contracts along the segment [0,zk]. Lastly, it is clear that D is itself locally con-tractible.

The subset Φ is also clearly closed and can be by-passed in X . Thus, it remains to beshown that f satisfies (C1) and (C2) for Φ.

To verify (C1), observe that Φ is homotopy equivalent to F1 ∪ F2. Let r : Φ→ F1 ∪F2 and s : F1 ∪ F2 → Φ, where s ◦ r � idΦ and r ◦ s � idF1∪F2 . We have the sequence ofhomotopies

f |Φ = f |Φ ◦ idΦ � f |Φ ◦ (s◦ r)= s◦ r � idΦ, (3.16)

where the second equality holds true because f is the identity map on F1∪F2.To prove (C2), we must find an appropriate path αi for each class Fi (i= 1,2). For F1,

we can choose α1 to be the constant path at the point (−1,0), and for F2 we can choose α2

to be the constant path at the point (8,0). The point at which we define αi is unimportant,provided that the point lies in the intersection of Φ with the fixed point class. It is clearthat αi(0)∈ Fi and αi(1)∈Φ for i= 1,2. Moreover, the required homotopy holds trivially,thus proving (C2).

Therefore by Theorem 3.5, Φ is the fixed point set of a map homotopic to f . It is clearthat Φ is not a finite polyhedron, thus showing that there exist interesting sets that satisfythe hypotheses of Theorem 3.5, but do not satisfy the hypotheses of Theorem 1.3.

4. Polyhedral fixed point sets of maps of pairs

Given a compact polyhedral pair (X ,A), let Z = cl(X −A) denote the closure of X −A.For any subset Φ ⊆ X , let ΦA = A∩Φ. We call (Φ,ΦA) a subset pair of (X ,A). For anymap f : (X ,A)→ (X ,A), denote the restriction f |A as fA : A→ A. We write f �A g if thereexists a homotopy of pairsH : (X ,A)× I → (X ,A) from f to g where (X ,A)× I denotes thepair (X × I ,A× I). If f �A g, it follows that fA � gA via the restriction of the homotopyto A.

In [8], Ng developed the following definition and theorems. As all the proofs can befound in [8], we provide only a sketch of each proof here. All references to (C1) and (C2)are to Schirmer’s conditions, as stated in Definition 1.1.

Definition 4.1. Let f : (X ,A)→ (X ,A) be a map of a compact polyhedral pair. The mapf satisfies conditions (C1′) and (C2′) for a subset Φ ⊆ X if the following are satisfied


(the symbol � denotes the usual homotopy of paths with endpoints fixed and ∗ the pathproduct):(C1′) there exists a homotopy H : (Φ,ΦA)× I → (X ,A) from f |Φ to the inclustion i :

Φ↩X and the map fA satisfies (C1) and (C2) for ΦA in A where HΦA = H|ΦA×I ,(C2′) for every essential fixed point class F of f intersecting Z, there exists a path α : I →

Z with α(0)∈ F∩Z, α(1)∈Φ, and

{α(t)

}�{f ◦α(t)

}∗ {H(α(1), t)}. (4.1)

Theorem 4.2. Let f : (X ,A)→ (X ,A) be a map of a compact polyhedral pair. If f satisfiesconditions (C1′) and (C2′) for a subset Φ⊆ X , then f satisfies (C1) and (C2) for Φ.

Sketch of proof. First observe that by choosing A to be the empty set, (C1′) implies (C1).To prove (C2), choose any essential fixed point class F of f . We can write

F= FA∪FZ , (4.2)

where

FA = F∩A, FZ = F− Int(A)= F∩Z. (4.3)

By [5, Theorem 1.1], there exists an integer-valued index indA( f ,FZ) such that

indA(f ,FZ

)= ind( f ,F)− ind(fA,FA

), (4.4)

where “ind” denotes the classical fixed point index.Suppose indA( f ,FZ)�= 0. Write

FZ = F1∪···∪Fk, (4.5)

where for each i between 1 and k, Fi denotes the intersection of F with a path componentof Z. It follows from [5] that indA( f ,FZ)=∑k

i=1 indA( f ,Fi). Then since indA( f ,FZ)�= 0,there exists at least one i for which indA( f ,Fi)�= 0. This Fi can be written as a finite unionof fixed point classes of f intersecting Z. At least one of these classes must be an essentialclass of f intersecting Z. Denote this class asG. Then by (C2′), there exists a path α : I → Zwith α(0)∈G⊆ F, α(1)∈Φ and

{α(t)

}�{f ◦α(t)

}∗ {H(α(1), t)}

, (4.6)

thus proving (C2) for this case.Next suppose that indA( f ,FZ) = 0. Then ind( fA,FA) �= 0, implying that FA is an es-

sential fixed point class of fA. From (C1′) there exists a path α : I → A with α(0) ∈ FA,α(1)∈ΦA ⊂Φ, and

{α(t)

}�{fA ◦α(t)

}∗ {HΦ(α(1), t

)}

= { f ◦α}∗ {H(α(1), t)}

,(4.7)

which proves (C2). �


Corollary 4.3. Let f : (X ,A)→ (X ,A) be a map of a compact polyhedral pair. If f satisfiesconditions (C1′) and (C2′) for a subset Φ ⊆ X , then Φ has at least N( f ) components andΦA has at least N( fA) components.

It is worthwhile to observe that in some cases, (C2′) is easy to check. In particular, fsatisfies (C2′) for Φ⊆ X if any of the following is satisfied ([8]):

(1) N( f |Z)= 0,(2) X is simply connected,(3) Φ⊆ Fix f ∩Z and Φ intersects every essential fixed point class of f on Z.

Theorem 4.4 (necessity). Let f : (X ,A)→ (X ,A) be a map of a compact polyhedral pairand let Φ be a subspace of X . If there exists a map g �A f with Fixg =Φ, then f satisfies(C1′) and (C2′) for Φ.

Sketch of proof. Let H denote the homotopy of pairs from f to g . It is clear that by lettingH =H|Φ×I and applying [10, Theorem 2.1], f satisfies (C1′).

To prove (C2′), choose any essential fixed point class F of f intersecting Z. It followsfrom [13, Theorem 2.7] that there exists an essential fixed point class G of g intersectingZ, to which F is H-related. Thus, there exists a path α : I → Z with α(0) ∈ F, α(1) ∈Φ,and

{α(t)

}�{H(α(t), t

)}

�{H(α(t),0

)}∗ {H(α(1), t)}

= { f ◦α(t)}∗ {H(α(1), t

)}.

(4.8)

Therefore f satisfies (C2′) for Φ. �

Theorem 4.5 (Ng’s finiteness theorem). Let f : (X ,A)→ (X ,A) be a map of a compactpolyhedral pair in whichX andA have no local cutpoints. Suppose (Φ,ΦA) is a subpolyhedralpair such that

(1) A−ΦA is not a 2-manifold,(2) ΦA can be by-passed in A,(3) f satisfies (C1′) for Φ.

Then there exists a map g �A f via a homotopyH : (X ,A)× I → (X ,A) that extends H suchthat Fixg = Φ∪ Zo, where Zo is a finite subset of X −A and each point of Zo lies in theinterior of a maximal simplex of X .

Sketch of proof. To construct the homotopy H , we will build three homotopies H1, H2,and H3, and take their composition.

From conditions (1)–(3), we can apply [10, Lemma 3.1] to show that there exists amap g1,A homotopic to fA with Fixg1,A =ΦA via a homotopy HA : A× I → A that is anextension of H|ΦA×I . Consider the homotopy H1,A : (A∪Φ,A)× I → (X ,A) defined by

H1,A(x, t)=⎧⎨⎩HA(x, t) (x, t)∈A× I ,H(x, t) (x, t)∈Φ× I. (4.9)


By the homotopy extension property, there is a homotopy H1 : (X ,A)× I → (X ,A) that isan extension ofH1,A; let f1(x)=H1(x,1). It is easy to check that Φ⊆ Fix f1, Fix f1|A =ΦA,and H1 extends H .

To construct H2, choose a strong deformation retraction R : St(A∪Φ)× I → St(A∪Φ) of a star cover of A∪Φ onto the set A∪Φ. We will abbreviate StA∪Φ for the star coverSt(A∪Φ). We can define H2 : (X ,A)× I → (X ,A) to be an extension of the compositionf1 ◦R : (StA∪Φ,A)× I → (X ,A). Setting f2(x)=H2(x, t), it is easy to check that Fix f2|A =ΦA and Fix f2 =A∪ (X − StA∪Φ).

By a careful application of the Hopf construction, we can find a map f3 : cl(X − StA∪Φ)→ X that is ε-homotopic to f2|cl(X−StA∪Φ), where f3 has only a finite number of fixed points,each lying in the interior of a maximal simplex of X . Let H3,cl : cl(X − StA∪Φ)× I → Xdenote the homotopy from f2|cl(X−StA∪Φ) to f3. We construct another homotopy H′

3 :(∂(StA∪Φ)∪A∪Φ,A)× I → (X ,A) as follows:

H′3(x, t)=

⎧⎨⎩H3,cl(x, t) (x, t)∈ ∂(StA∪Φ

),

f2(x) (x, t)∈ A∪Φ.(4.10)

Then H′3 can be extended to a homotopyH3,St : (StA∪Φ,A)× I → (X ,A). Finally, we define

H3 : (X ,A)× I → (X ,A) by

H3(x, t)=⎧⎨⎩H3,St(x, t) x ∈ StA∪Φ,

H3,cl(x, t) x ∈ cl(X − StA∪Φ

).

(4.11)

One can check that if we let H be the composition of the homotopies H1, H2, and H3

and define g(x)=H(x,1), we complete the proof. �

Theorem 4.6 (Ng’s sufficiency theorem #1). Let f : (X ,A)→ (X ,A) be a map of a compactpolyhedral pair in whichX andA have no local cutpoints. Suppose (Φ,ΦA) is a subpolyhedralpair such that

(1) A−ΦA and all components of X − (A∪Φ) are not 2-manifolds,(2) f satisfies (C1′) and (C2′) for Φ,(3) ΦA can be by-passed in A, Φ can be by-passed in X−A and ∂A can be by-passed in Z.

Then there exists a map g �A f with Fixg =Φ.

Sketch of proof. From Theorem 4.5, there exists a map g1 �A f via a homotopy H : (X ,A)× I → (X ,A) that extends H with Fixg1 =Φ∪Zo, where Zo is a finite subset of X −Aand each point of Zo lies in the interior of a maximal simplex of X . To construct thedesired map g, we use a sequence of homotopies relative to A∪Φ.

Using techniques from [6, 9], one can show that the following procedures are possiblein this scenario.

(1) Given any two points x, y ∈ Fixg1∩ (X −A) that lie in the same fixed point classof g1 intersecting Z, we can delete the point x from Fixg1 by an appropriate ho-motopy. This requires the assumption that every component of X − (A∪Φ) isnot a 2-manifold.

(2) If x ∈ Fixg1∩ (X −A) and y is any point in Z∩Φ that lies in the same fixed pointclass as x, we can delete the point x from Fixg1 by an appropriate homotopy. In


addition to the first assumption, this requires that ∂A can be by-passed in Z andΦ can be by-passed in X −A.

(3) Any point x ∈ Fixg1 ∩ (X −A) with ind( f ,x) = 0 can be removed in the usualway.

After a finite number of applications of the above procedures, we achieve a new mapg �A f . If g is fixed point free on X − (Φ∪A), we are done. If Fixg ∩ (X − (Φ∪A)) �=∅, then any point x ∈ Fixg ∩ (X − (Φ∪A)) forms an entire essential fixed point classof g. A slight modification of the proof of [10, Lemma 3.1] shows that this scenario isimpossible. �

In the original statement of Theorem 4.6, Ng required that no component of A−ΦA

be a 2-manifold. However, this assumption is not required for the proof and thereforeomitted.

Theorem 4.7 (Ng’s sufficiency theorem #2). Let f : (X ,A)→ (X ,A) be a map of a compactpolyhedral pair in whichX andA have no local cutpoints. Suppose (Φ,ΦA) is a subpolyhedralpair such that

(1) A−ΦA and all components of X − (A∪Φ) are not 2-manifolds,(2) f satisfies (C1′) and (C2′) for Φ,(3) ΦA can be by-passed inA, Φ can be by-passed in X−A, and ∂A can be by-passed in Z.

Then for every closed subset Γ of Φ that has nonempty intersection with every component ofΦA and every component of Φ∩Z, there exists a map g �A f with Fixg = Γ.

Sketch of proof. Let K be a triangulation of X = |K|. As in the proof of [10, Theorem 3.2],we can find a subpolyhedronN in a subdivision K ′ of K such that f |N is a proximity mapwith only a finite number of fixed points, all lying in Γ⊆ Int(N). Let α(x, y, t) be definedas in [1, Lemma 1, page 124]. Define a homotopy HN : (N ,N ∩A)× I → (X ,A) by

HN (x, t)= α(x, f (x), 1− t(1−d(x,Γ)))

, (4.12)

where d denotes the usual distance function. It is not difficult to check thatHN is a specialhomotopy (cf. [6, page 751]) on ∂N × I and that FixHN (x,1) = Γ. Next, we can extendHN to a new homotopy H : (X ,A)× I → (X ,A) that is special on cl(X −N). If we letg(x)=H(x,1), then g �A f and Fixg = Γ. �

5. Locally contractible fixed point sets of maps of pairs

We wish to extend Ng’s work (in particular, Theorem 4.7) to the case where Φ is locallycontractible, but not necessarily a polyhedron. To do so, we first prove a useful lemmaand theorem.

Lemma 5.1. Let f : (X ,A)→ (X ,A) be a map of a compact polyhedral pair and let (Φ,ΦA)be a subset pair in which Φ is closed in X . Assume that there exists a subset B of X suchthat Φ⊆ B and the pair (B,B∩A) strong deformation retracts onto (Φ,ΦA) via a retractionR : (B,B∩A)× I → (B,B∩A). If f satisfies (C1′) and (C2′) for Φ, then f satisfies (C1′)and (C2′) for B.


Proof. First observe that since f satisfies (C1′) for Φ, fA satisfies (C1) and (C2) for ΦA

in A. Then Lemma 3.4 shows that fA satisfies (C1) and (C2) for B∩A. This proves thesecond statement in (C1′). A simple modification of the proof of Lemma 3.4 proves thefirst statement in (C1′) and also proves that f satisfies (C2′) for B. �

Theorem 5.2. Let f : (X ,A) → (X ,A) be a map of a compact polyhedral pair. Suppose(Φ,ΦA) is a subset pair in which both Φ and ΦA are closed, locally contractible subsets of X .Then there exists a subset B of X such that Φ⊆ B and the pair (B,B∩A) strong deformationretracts onto (Φ,ΦA) via a retraction of pairs �t : (B,B∩A)→ (B,B∩A).

Proof. We will construct three homotopies and then take their composition to obtain anexplicit strong deformation retraction of pairs. First, we must establish some terminology.

Since X itself is an ANR embedded in Euclidean space, there exists a neighborhoodV ⊂Rn (n > 0) of X that strong deformation retracts onto X . Let

ρt :V −→ X (5.1)

denote this strong deformation retraction. The subset Φ is also a finite-dimensional ANR([2, Proposition 8.12, page 83]). Thus, there exists a neighborhood U ⊂ X that strongdeformation retracts onto Φ. Let

ϕt :U −→Φ (5.2)

denote this strong deformation retraction. Define

ε=min(d(X ,Vc

),d(Φ,Uc

)), (5.3)

where Vc and Uc denote the complements of V and U in Rn and X , respectively.Choose any three positive real numbers ε1, ε2, and ε3 so that

ε1 + ε2 + ε3 < ε. (5.4)

The subsets Φ and ΦA are both finite-dimensional ANR’s. Thus, there exist neighbor-hoods U1,U2 ⊂ X and strong deformation retractions Rt : U1 →Φ and rt : U2 →Φ thatare ε1- and ε2-homotopies, respectively (Theorem 3.3). In other words, for each x ∈U1,

d(Rt(x),Rt′(x)

)< ε1, for any t, t′ ∈ I (5.5)

and for each x ∈U2,

d(rt(x),rt′(x)

)< ε2, for any t, t′ ∈ I. (5.6)

Notice that although both Rt and ϕt are strong deformation retractions of neighborhoodsof Φ onto itself, we neither require that U1 =U nor that Rt|U1∩U = ϕt|U1∩U .

Next define δ1,δ2 > 0 by

δ1 = d(Φ,X −U1

), δ2 = d

(ΦA,X −U2

), (5.7)


and let Δ =min(δ1,δ2). Define B to be a Δ-neighborhood of Φ. Then Φ ⊂ B ⊆ U1 andΦA ⊂ B∩A⊆U2. By restricting Rt to B and rt to B∩A, we can view these maps as strongdeformation retractions of B onto Φ and of B∩A onto ΦA, respectively.

Next, since the subset A is a subpolyhedron of X , it must also be a finite-dimensionalANR. Thus, there exists a neighborhood W ⊂ X containing A and a strong deformationretraction ψt : W → A that is an ε3-homotopy. For ease of notation, we will let x′ = ψ1(x)for any x ∈W .

Let us choose Ω= d(cl(B∩A),X −W) and define a set

C = {x ∈ B−B∩A | d(x, cl(B∩A))≤Ω

}. (5.8)

For any x ∈ C, let

d(x, cl(B∩A)

)= s≤Ω, d(x,Φ)= q < Δ. (5.9)

Finally, we let β =max(Ω,Δ) > 0, and we define our first homotopy to be the map Ht :(B,B∩A)→ (B,B∩A) defined by

Ht(x)=

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

x x ∈ B−C,

ρ1−t

(⟨ψt(x),x

⟩1β

)x ∈ C, q = s,

ρ1−t

([⟨ψt(x),x

⟩qβ

+ (s− q)x]

1s

)x ∈ C, q < s,

ρ1−t

([⟨ψt(x),x

⟩ sβ

+ (q− s)ψt(x)]

1q

)x ∈ C, q > s,

(5.10)

where 〈ψt(x),x〉 = (β− s)ψt(x) + sx.We must check that Ht is defined on C. By the definition of C, if x ∈ C then x ∈

B∩W . Thus, ψt(x) is defined. Next, since s≤Ω≤ β, for each t ∈ I the expression 〈ψt(x),x〉(1/β)= [(β− s)ψt(x) + sx](1/β) represents a point lying on the straight path in Rn be-tween ψt(x) and x. Since ψt(x) is an ε3-homotopy, the length of this path must be lessthan ε3. Moreover,

ε3 < ε =min(d(X ,Vc

),d(Φ,Uc

))(5.11)

and ψt(x), x ∈ X . Thus all points on this straight path must lie in V , whence ρ1−t(〈ψt(x),x〉(1/β) exists.

To see that Ht(x) is defined for x ∈ C with q < s, observe that for each t ∈ I the expres-sion

[⟨ψt(x),x

⟩qβ

+ (s− q)x]

1s=[q[(β− s)ψt(x) + sx

]1β

+ (s− q)x]

1s

(5.12)

represents a point lying on the straight path in Rn between x and some point lying be-tween ψt(x) and x. Thus, the length of the path in (5.12) must be less than or equal tothe length of the straight path between ψt(x) and x. In short, all points in this expression


must lie in V . Thus, ρ1−t([〈ψt(x),x〉(q/β) + (s− q)x](1/s)) is defined. A similar argumentholds for x ∈ C with q > s. Therefore, Ht(x) is defined for all x ∈ C.

It is straightforward to check that Ht is continuous, H0 = idB, and Ht is a homotopy ofpairs.

We define the second homotopy of pairs in the composition to be Jt : (B,B ∩A) →(Φ,Φ∩A) such that

Jt(x)=

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

rt(x) x ∈ B∩A,

Rt(x) x ∈ (B−B∩A)−C,

ρt

(⟨rt(x′),Rt(x)

⟩1β

)x ∈ C, q = s,

ρt

([⟨rt(x′),Rt(x)

⟩qβ

+ (s− q)Rt(x)]

1s

)x ∈ C, q < s,

ρt

([⟨rt(x′),Rt(x)

⟩ sβ

+ (q− s)rt(x′)]

1q

)x ∈ C, q > s.

(5.13)

It is clear that Jt is defined and continuous outside C. However, we must again checkthat Jt is defined on C. For any x ∈ C, we have x ∈ B∩W and x′ ∈ B∩A. Thus, Rt(x)and rt(x′) are defined. Next, since s ≤Ω ≤ β, the expression 〈rt(x′),Rt(x)〉(1/β) = [(β−s)rt(x′) + sRt(x)](1/β) represents a point lying on the straight path in Rn between rt(x′)and Rt(x). Now for each t ∈ I the distance from Rt(x) to rt(x′) satisfies the followinginequality:

d(Rt(x),rt(x′)

)≤ d(Rt(x),x)

+d(x,x′) +d(x′,rt(x′)

)

= d(Rt(x),R0(x))

+d(ψ0(x),ψ1(x)

)+d(r0(x′),rt(x′)

)

< ε1 + ε3 + ε2 < ε.

(5.14)

Since ε =min(d(X ,Vc),d(Φ,Uc)) and Rt(x),rt(x′) ∈ X , all points on the straight pathbetween rt(x′) and Rt(x) must lie in V . Therefore, the expression

ρt

(⟨rt(x′),Rt(x)

⟩1β

)= ρt

([(β− s)rt(x′) + sRt(x)

]1β

)(5.15)

is defined for x ∈ C. Moreover, as a composition of continuous functions, the expressionis continuous. Therefore, Jt is defined and continuous for x ∈ C with q = s.

For x ∈ C with q < s or q > s, a similar argument to that of the proof above forHt showsthat Jt(x) is defined and continuous. It is straightforward to check that Jt is continuouson X and that Jt is a homotopy of pairs.


We denote the third and final homotopy in the construction by Kt : (B,B ∩ A) →(Φ,ΦA), where

Kt(x)=

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

r1(x) x ∈ B∩A,

R1(x) x ∈ (B−B∩A)−C,

ϕt

(⟨r1(x′),R1(x)

⟩1β

)x ∈ C, q = s,

ϕt

([⟨r1(x′),R1(x)

⟩qβ

+ (s− q)R1(x)]

1s

)x ∈ C, q < s,

ϕt

([⟨r1(x′),R1(x)

⟩ sβ

+ (q− s)r1(x′)]

1q

)x ∈ C, q > s.

(5.16)

We must again check that our homotopy is defined on C. For x ∈ C, all points in theexpression 〈r1(x′),R1(x)〉(1/β)= [(β− s)r1(x′) + sR1(x)](1/β) lie on the straight path be-tween r1(x′) and R1(x). From (5.14), the length of this path must be less than ε. Sinceε = min(d(X ,Vc),d(Φ,Uc)) and r1(x′),R1(x) ∈ Φ, all points on this path must lie inU . Therefore, ϕt(〈r1(x′),R1(x)〉(1/β))= ϕt([(β− s)r1(x′) + sR1(x)](1/β)) is defined. Theproof that Kt(x) is defined for x ∈ C with q < s or q > s is similar to that of Ht and Jt, andhence omitted. It is straightforward to check that Kt is continuous on X and that Kt is ahomotopy of pairs.

Finally, we define our strong deformation retraction of pairs as �t : (B,B ∩ A) →(Φ,ΦA), where

�t(x)=

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

H3t(x) 0≤ t ≤ 1/3,

J3t−1(x) 1/3≤ t ≤ 2/3,

K3t−2(x) 2/3≤ t ≤ 1.

(5.17)

One can check that for all x ∈ B, �1/3(x) = H1(x) = J0(x) and �2/3(x) = J1(x) = K0(x).It remains to show that �t is indeed a strong deformation retraction of pairs. In otherwords, we must check that �0 = idB, �1(B) ⊆ Φ, �1(B ∩A) ⊆ ΦA, and for all t ∈ I ,�t|Φ = idΦ and �t|ΦA = idΦA . By the construction of �t, it is clear that �0 =H0(x)= idB,�1(B∩A)⊆ΦA, and �t|ΦA = idΦA .

To see that �1(B) ⊆Φ, observe that �1(x) = J1(x) for all x. It is clear that J1(x) ∈Φfor all x ∈ B−C. For x ∈ C, the point J1(x) is obtained by evaluating ϕ1 at some point inU . Since ϕt :U →Φ is a strong deformation retraction, ϕ1(U)⊆Φ. Therefore, J1(x)∈Φfor all x ∈ C, and hence for all x ∈ B.

Finally, to see that �t|Φ = idΦ, choose any x ∈ Φ. If x ∈ Φ∩ B − C then �t(x) =Rt(x) = x. If x ∈ C∩Φ, then d(x,Φ) = 0 = q < s, whence Ht(x) = ρ1−t(x) = x, Kt(x) =ρt(Rt(x)) = x, and Jt(x) = ρt(R1(x)) = x. Therefore, �t|Φ = idΦ, which completes theproof. �

We now use Ng’s results with Lemma 5.1 and Theorem 5.2 to show that the hypothesesand results from Theorem 4.7 hold for all locally contractible closed subsets of X .


Theorem 5.3. Let f : (X ,A)→ (X ,A) be a map of a compact polyhedral pair in which Xand A have no local cutpoints. Suppose (Φ,ΦA) is a subset pair in which Φ, ΦA, and Φ∩Zare closed, locally contractible subsets of X such that

(1) A−ΦA and all components of X − (A∪Φ) are not 2-manifolds,(2) f satisfies (C1′) and (C2′) for Φ,(3) ΦA can be by-passed in A, Φ∩Z can be by-passed in Z, and ∂A can be by-passed in Z.

Then for every closed subset Γ of Φ that has nonempty intersection with every component ofΦA and every component of Φ∩Z, there exists a map g �A f with Fixg = Γ.

Proof. If Φ=∅, then this theorem reduces to a special case of [10, Lemma 3.1]. Thus, wemay assume Φ �=∅. Let K be a triangulation of X = |K|. By [2, Proposition 8.12, page83], both Φ and ΦA are finite-dimensional ANR’s.

From Theorem 5.2, there exists a subset B of X such that Φ⊆ B and the pair (B,B∩A)strong deformation retracts onto the pair (Φ,ΦA). Lemma 3.1 guarantees that we can finda star cover StK ′(Φ) of Φ with respect to a sufficiently small subdivision K ′ of K such thatStK ′(Φ) ⊆ B and the sets A− (StK ′(Φ)∩A) and X − (A∪ StK ′(Φ)) are not 2-manifolds.It follows from Lemma 5.1, by restricting the retraction, that f satisfies (C1′) and (C2′)for StK ′(Φ).

By assumption, ΦA can be by-passed inA. SinceA is a polyhedron, Theorem 2.2 showsthat ΦA can be neighborhood by-passed in X . Likewise, Φ∩Z can be neighborhood by-passed in Z. Therefore K ′ may be chosen with mesh small enough so that StK ′(Φ)∩Acan be by-passed in A and StK ′(Φ)∩Z can be by-passed in Z. Then StK ′(Φ)∩ (X −A)can also be by-passed in Z. Thus any path with endpoints in X −A is homotopic to a pathin Z = cl(X −A). But ∂A can also be by-passed in Z, implying that such a path must behomotopic to a path in X −A. Therefore StK ′(Φ)∩ (X −A) can be by-passed in X −A.

Now by the construction of star covers, each component of StK ′(Φ) contains a com-ponent of Φ and every component of Φ is contained in a component of StK ′(Φ). Chooseany closed subset Γ of Φ, having nonempty intersection with every component of ΦA

and every component of Φ∩Z. Then Γ also has nonempty intersection with every com-ponent of StK ′(Φ)∩A and with every component of StK ′(Φ)∩Z. Finally, since K ′ is atriangulation of both X and A, the set StK ′(Φ)∩A is a subpolyhedron of A and thus itselfa polyhedron. Therefore, the result follows from Theorem 4.7. �

Corollary 5.4. Let f : (X ,A)→ (X ,A) be a map of a compact polyhedral pair. Suppose(Φ,ΦA) is a subset pair in which both Φ and ΦA are closed, locally contractible subsets of Xsuch that

(1) A−ΦA and all components of X − (A∪Φ) are not 2-manifolds,(2) f satisfies (C1′) and (C2′) for Φ,(3) ΦA can be by-passed in A, Φ∩Z can be by-passed in Z, and ∂A can be by-passed in Z.

Then there exists a map g �A f with Fixg =Φ.

Notice that in the statement of Theorem 5.3, we require both Φ and ΦA to be locallycontractible. If Φ is locally contractible, it does not necessarily follow that ΦA is locallycontractible. For instance, the intersection of the subset Φ in Example 3.6 with the curvey = 1/x2 (x ≥ 0) is an infinite sequence of discrete points converging to (0,0) and there-fore not locally contractible at the origin.


We conclude this paper with an example of a map f : (X ,A)→ (X ,A) of a polyhedralpair, having a locally contractible subset pair (Φ,ΦA), for which there exists g �A f withFixg =Φ.

Example 5.5. Consider the subset Φ in Example 3.6. Let A=Φ∩R where R denotes theclosed rectangle

R= {(x, y)∈R2 | −2≤ x ≤ 0, −ε ≤ y ≤ ε}, (5.18)

for any positive real number ε < 1/2. Then ΦA = A, vacuously implying that A−ΦA isnot a 2-manifold and that ΦA can be by-passed in A. As we saw in Example 3.6, the onlycomponent ofX − (A∪Φ)= X −Φ is not a 2-manifold. It is also easy to check that Φ∩Zcan be by-passed in Z and that ∂A can be by-passed in Z.

Let f : (X ,A)→ (X ,A) be the map flippingX over the x-axis. That is, f (x, y)= (x,−y),as in Example 3.6. It remains to show that f satisfies (C1′) and (C2′) for Φ.

To see that f satisfies (C1′), recall from Example 3.6 that Φ is homotopy equivalent toF1∪F2, the union of the two fixed point classes of f . Likewise,ΦA is homotopy equivalentto F1. As f (ΦA)⊆ΦA, the homotopyHΦ from (C1) in Example 3.6 also maps ΦA to itself.In other words, we can write HΦ : Φ× I → X as the homotopy of pairs H : (Φ,ΦA)× I →(X ,A).

Next we must show that fA satisfies (C1) and (C2) for ΦA. The restriction HΦA =H|ΦA×I provides the necessary homotopy from fA to the inclusion i|A, proving (C1). Tosee (C2), observe that fA has only one essential fixed point class F= F1∩A. By choosingthe path α : I → Z to be a constant path at any point in F, we see that fA satisfies (C2) forΦA.

Finally, to check that f satisfies (C2′), observe that both essential classes F1 and F2

intersect Z. For F1, choose the path α : I → Z to be the constant path at the origin. As theorigin lies in both Z and Φ, the path α fulfills the requirements of (C2′). Similarly for F2,we can choose α : I → Z to be the constant path at the point (8,0).

Thus f satisfies all the hypotheses of Theorem 5.3, implying that for every closed sub-set Γ of Φ that has nonempty intersection with ΦA and with both components of Φ∩Z,there exists a map g �A f with Fixg = Γ. In particular, there exists a map homotopic to fvia a homotopy of pairs whose fixed point set is Φ itself.

Acknowledgment

I would like to thank Jerzy Dydak of the University of Tennessee, who provided the proofof Theorem 2.2.

References

[1] R. F. Brown, The Lefschetz Fixed Point Theorem, Scott, Foresman, Illinois, 1971.[2] A. Dold, Lectures on Algebraic Topology, Die Grundlehren der mathematischen Wissenschaften,

vol. 200, Springer, New York, 1972.[3] J. Dydak, private communication, 2003.[4] S.-T. Hu, Theory of Retracts, Wayne State University Press, Michigan, 1965.


[5] J. Jezierski, A modification of the relative Nielsen number of H. Schirmer, Topology and Its Appli-cations 62 (1995), no. 1, 45–63.

[6] B. J. Jiang, On the least number of fixed points, American Journal of Mathematics 102 (1980),no. 4, 749–763.

[7] , Lectures on Nielsen Fixed Point Theory, Contemporary Mathematics, vol. 14, AmericanMathematical Society, Rhode Island, 1983.

[8] C. W. Ng, Fixed point sets of maps of pairs, Ph.D. thesis, University of California at Los Angeles,California, 1995.

[9] H. Schirmer, A relative Nielsen number, Pacific Journal of Mathematics 122 (1986), no. 2, 459–473.

[10] , Fixed point sets in a prescribed homotopy class, Topology and Its Applications 37 (1990),no. 2, 153–162.

[11] , A survey of relative Nielsen fixed point theory, Nielsen Theory and Dynamical Systems(South Hadley, MA, 1992), Contemp. Math., vol. 152, American Mathematical Society, RhodeIsland, 1993, pp. 291–309.

[12] P. Strantzalos, Eine charakterisierung der fixpunktmengen bei selbstabbildungen kompakter man-nigfaltigkeiten aus einer homotopieklasse, Bulletin de l’Academie Polonaise des Sciences. Serie desSciences Mathematiques, Astronomiques et Physiques 25 (1977), no. 8, 787–793.

[13] X. Z. Zhao, Estimation of the number of fixed points on the complement, Topology and Its Appli-cations 37 (1990), no. 3, 257–265.

Christina L. Soderlund: Department of Mathematics, California Lutheran University,60 West Olsen Road 3750, Thousand Oaks, CA 91360-2700, USAE-mail address: [email protected]

COINCIDENCE CLASSES IN NONORIENTABLE MANIFOLDS

DANIEL VENDRUSCOLO

Received 15 September 2004; Revised 20 April 2005; Accepted 21 July 2005

We study Nielsen coincidence theory for maps between manifolds of same dimensionregardless of orientation. We use the definition of semi-index of a class, review the defi-nition of defective classes, and study the occurrence of defective root classes. We prove asemi-index product formula for lifting maps and give conditions for the defective coinci-dence classes to be the only essential classes.

Copyright © 2006 Daniel Vendruscolo. This is an open access article distributed underthe Creative Commons Attribution License, which permits unrestricted use, distribution,and reproduction in any medium, provided the original work is properly cited.

1. Introduction

In [2, 6] the Nielsen coincidence theory was extended to maps between nonorientabletopological manifolds. The main idea to do this is the notion of semi-index (a nonnegativeinteger) for a coincidence set.

Let f ,g :M→N be maps between closed n-manifolds without boundary. If we defineh = ( f ,g) : M → N ×N as usual, then we may assume that h is in a transverse position,that is, the coincidence set Coin( f ,g)= {x ∈M | f (x)= g(x)} is finite and for each coin-cidence point x there is a chart Rn×Rn =U ⊂N ×N such that (U , ( f ,g)(M)∩U ,ΔN ∩U) corresponds to (Rn×Rn,Rn× 0,0×Rn) (see [6] for details).

We say that two coincidence points x, y ∈ Coin( f ,g) are Nielsen related if there is apath γ : [0,1]→M with γ(0) = x, γ(1) = y such that f γ is homotopic to gγ relative tothe endpoints. In fact, this is an equivalence relation whose equivalence classes are calledcoincidence classes of the pair ( f ,g).

Let x, y ∈ Coin( f ,g) belong to the same coincidence class and let γ be a path estab-lishing the Nielsen relation between them. We choose a local orientation μ0 of M in x anddenote by μt the translation of μ0 along γ(t).

Definition 1.1 [6, Definition 1.2]. We will say that two points x, y ∈ Coin( f ,g) are R-related (xRy) if and only if there is a path γ establishing the Nielsen relation between them


2 Coincidence classes in nonorientable manifolds

such that the translation of the orientation h∗μ0 along a path in the diagonal Δ(N) ⊂N ×N homotopic to hγ in N ×N is opposite to h∗μ1. In this case the path γ is calledgraph-orientation-reversing.

Since ( f ,g) is transverse, Coin( f ,g) is finite. Let A⊂ Coin( f ,g), then A can be repre-sented as A= {a1,a2, . . . ,as; b1,c1, . . . ,bk,ck} where biRci for any i and aiRaj for no i�= j.The elements {ai}i of this decomposition are called free.

Definition 1.2. In the above setup the semi-index of the pair ( f ,g) in A = {a1, . . . ,as;b1,c1, . . . ,bk,ck} is the number of free elements s denoted by |ind|( f ,g ;A) of A.

This definition makes sense, since it does not depend on a decomposition (c.f. [2,6]). Moreover the semi-index is homotopy invariant, it is well defined for all continuousmaps, and ifU ⊂M is an open subset such that Coin( f ,g)∩U is compact, we can extendthis definition to that of the semi-index of a pair on the subset U , which is denoted by| ind|( f ,g;U).

Definition 1.3. A coincidence class C of a transverse pair ( f ,g) is called essential if| ind|( f ,g;C)�= 0.

In [5] Jezierski investigates whether a coincidence point x ∈ Coin( f ,g) satisfies xRx.Such points can occur only when M or N are nonorientable, in which case they are calledself-reducing points. This is a new situation (see [5, Example 2.4]) that cannot occur nei-ther in the orientable case nor in the fixed point context.

Definition 1.4 [5, Definition 2.1]. Let x ∈ Coin( f ,g) and let H ⊂ π1(M), H′ ⊂ π1(N)denote the subgroups of orientation-preserving elements. We define

Coin( f#,g#)x ={α∈ π1(M,x) | f#(α)= g#(α)

},

Coin+( f#,g#)x = Coin( f#,g#)x ∩H.(1.1)

Lemma 1.5 [5, Lemma 2.2]. Let f ,g :M→N be transverse and x ∈ Coin( f ,g). Then xRx ifand only if Coin+( f#,g#)x �= Coin( f#,g#)x ∩ f −1

# (H′) (in other words, if there exists a loop αbased at x such that f α∼ gα and exactly one of the loops α or f α is orientation-preserving).

Definition 1.6. A coincidence class C is called defective if C contains a self-reducing point.

Lemma 1.7 [5, Lemma 2.3]. If a Nielsen class C contains a self-reducing point (i.e., C isdefective), then any two points in this class are R-related, and thus

| ind|( f ,g;C)=⎧⎪⎨⎪⎩

0 if #C is even;

1 if #C is odd.(1.2)

Daniel Vendruscolo 3

2. The root case

In [1] we can find a different approach to extend the Nielsen root theory to the nonori-entable case. They use the concept of orientation-true map to classify maps between man-ifolds of the same dimension in three types (see also [7, 8]).

Definition 2.1. A map f is orientation-true if for each loop α∈ π1(M), f α is orientation-preserving if and only if α is orientation-preserving.

Definition 2.2 [1, Definition 2.1]. Let f :M→N be a map of manifolds. Then three typesof maps are defined as follows.

(1) Type I: f is orientation-true.(2) Type II: f is not orientation-true but does not map an orientation-reversing loop

in M to a contractible loop in N .(3) Type III: f maps an orientation-reversing loop in M to a contractible loop in N .

Further, a map f is defined to be orientable if it is of Type I or II, and nonorientableotherwise.

For orientable maps they describe an Orientation Procedure [1, 2.6] for root classes.This procedure uses local degree with coefficients in Z. For maps of Type III the sameprocedure is possible only with coefficients in Z2. Then they define the multiplicity of aroot class, that is an integer for orientable maps and an element of Z2 for maps of TypeIII.

Now if we consider the root classes of a map f as the coincidence classes of the pair( f ,c) where c is the constant map, we have.

Theorem 2.3. Let f : M → N be a map between closed manifolds of the same dimension,without boundary.

(i) If f is orientable, then no root class of f is defective.(ii) If f is of Type III, then all root classes of f are defective.

Proof. If f is orientable and α is a loop in M, f α ∼ 1 implies that α is orientation-preserving. On the other hand by Lemma 1.5, a coincidence class C of the pair ( f ,c)is defective if and only if there exists a point x ∈ C and a loop α at x such that f α∼ 1 andα is orientation-reversing.

Now if f is a Type III map, then there exists a loop α ∈ π1(M,x0) such that α isorientation-reversing and f α ∼ 1. Let x ∈ Coin( f ,g) be a root. We fix a path β from xto x0. Then γ = βαβ−1 is a loop based at x, orientation-reversing and f γ ∼ 1. Thus x is aself-reducing root. �

In fact [1, Lemma 4.1] shows the equality between the multiplicity of a root class andits semi-index.

Theorem 2.4. Let M and N be closed manifolds of the same dimension, without boundarysuch that M is nonorientable and N is orientable. If f : M → N is a map, then all essentialroot classes of f are defective.


Proof. There is no orientation-true maps from a nonorientable to an orientable manifold.If f is a Type II map then by [1, Lemma 3.10] deg( f ) = 0 and f has no essential rootclasses. The result follows by Theorem 2.3. �

We use the ideas of Theorem 2.3 to state.

Lemma 2.5. Let f ,g : M → N be two maps between manifolds of the same dimension. Ifthere exist a coincidence point x0 and a graph-orientation-reverse loop α based in x0 suchthat f α is in the center of π1(N , f (x0)), then all coincidence points of the pair ( f ,g) areself-reducing points.

Proof. Let x1 ∈ Coin( f ,g). We fix a path β from x0 to x1 and we will show that for theloop γ = β−1αβ, the loops f γ and gγ are homotopic and γ is orientation-reverse. In factf γ ∼ gγ means f β−1 · f α · f β ∼ gβ−1 · gα · gβ hence f α · ( f β · gβ−1)∼ ( f β · gβ−1) · gα.The last holds, since the homotopy class of f α∼ gα belongs to the centre of π1(N , f (x0)).On the other hand γ = β−1 ·α ·β is orientation-reverse, since so is α.

Corollary 2.6. Let f ,g : M → N be two maps between manifolds of the same dimensionsuch that f#(π1(M)) is contained in the center of π1(N). If ( f ,g) has a defective class, thenall classes of ( f ,g) are defective. �

In particular this is true for π1(N) commutative.

3. Covering maps

Let M and N be compact, closed manifolds of the same dimension, let f ,g : M → N betwo maps such that Coin( f ,g) is finite, and let p : M→M and q : N →N be finite regular

coverings such that there exist lifts f , g : M→ N of the pair f ,g :

Mf

g

p

N

q

Mf

g N

(3.1)

Under such hypotheses there is a bijection between the set of Deck transformations,D(M), of the covering space M and the group (π1(M))/(p#(π1(M))). We fix a point x0 ∈M and for each Deck transformation α we choose a path γ in M, from x0 to α(x0). Then,if α is the projection of γ, the formula

D(M) α−→ [α]∈ π1(M, p

(x0))

p#(π1(M, x0

)) (3.2)

gives such bijection. It is easy to see that such bijection is an isomorphism of groups.


The above isomorphism and a fixed lift f determine the homomorphism from thegroup D(M) to D(N) for which the diagram

D(M)f∗,x0

D(N)

π1(M, p(x0))

p#(π1(M, x0))

f# π1(N ,q( f (x0)))

q#(π1(N , f (x0)))

(3.3)

commutes. This homomorphism is given by the equality

f∗,x0 (α)(f (x)

)= f α(x), ∀α∈D(M), ∀x ∈ M. (3.4)

The same construction can be done for map g and we have the following.

Lemma 3.1. Let x0 ∈ Coin( f , g) and α ∈D(M). Then α(x0) ∈ Coin( f , g) if and only if

f∗,x0 (α)= g∗,x0 (α) where x0 = p(x0).

Corollary 3.2. Let x0 ∈ Coin( f , g) and x0 = p(x0). Then p−1(x0)∩Coin( f , g) have ex-

actly #Coin( f∗,x0 , g∗,x0 ) elements.

Lemma 3.3. Let x0 and x′0 be two coincidences of the pair ( f , g) such that p(x0)= p(x′0)=x0, and let γ be the unique element of D(M) such that γ(x0)= x′0. The points x0 and x′0 are

in the same coincidence class of ( f , g) if and only if there exists γ ∈ π1(M,x0) such that(i) [γ]∈ (π1(M,x0))/(p#(π1(M, x0))) corresponds to γ;

(ii) f#(γ)= g#(γ).

Proof. (⇒) If x0 and x′0 are in the same coincidence class of ( f , g), there exists a path β

from x0 to x′0 establishing the Nielsen relation, (i.e., f β ∼ gβ).

Take γ = pβ ∈ π1(M,x0). We can see that [γ] = γ and f γ = q f β ∼ qgβ = gγ, thismeans that f#(γ)= g#(γ).

(⇐) The lift γ of γ starting at x0 is a path from x0 to x′0 establishing the Nielsen relation,

(i.e., f γ ∼ g γ). �

If γ is a loop in a manifold, we say that sign(γ)= 1 or −1 if γ is orientation-preservingor orientation-reversing, respectively.

Corollary 3.4. In Lemma 3.3, if the points x0 and x′0 are in the same coincidence class

of ( f , g), then x0Rx′0 if and only if sign( f∗,x0 (γ)) · sign(γ) = −1. In this case, x0 is a self-

reducing coincidence point.

Proof. First we note that since f#(γ) = g#(γ), f∗,x0 (γ) = g∗,x0 (γ) and we have that

sign( f∗,x0 (γ)) · sign(γ) = −1 if and only if the paths γ and γ in the proof of Lemma 3.3are both graph orientation-reversing. �

If we denote by jx0 the natural projection from π1(M,x0) to D(M) and by Coin( f#,g#)x0

the set {α∈ π1(M,x0) | f#(α)= g#(α)}, we have the following.


Corollary 3.5. If x0 is a coincidence of the pair ( f ,g), then the set p−1(x0)∩Coin( f , g)

can be partitioned in (#Coin( f∗,x0 , g∗,x0 ))/(# jx0 (Coin( f#,g#)x0 )) disjoint subsets, each ofthem with # jx0 (Coin( f#,g#)x0 ) elements all of them Nielsen related (therefore they are con-

tained in the same coincidence class of the pair ( f , g)). Moreover, no two points of differentsubsets are Nielsen related.

Lemma 3.6. Let x0, x1 be coincidence points in the same coincidence class of the pair ( f ,g), αbe a path from x0 to x1 establishing the Nielsen relation, x0, x′0 coincidence points of the pair

( f , g) such that p(x0)= p(x′0)= x0, and γ the unique element of D(M) such that γ(x0)= x′0.If α and α′ are the two liftings of α starting at x0 and x′0 respectively then:

(i) α(1) and α′(1) are coincidence points of the pair ( f , g);(ii) α(1) (α′(1)) is in the same coincidence class as x0 (x′0);

(iii) p(α(1))= p(α′(1))= x1;(iv) γ(α(1))= α′(1).(v) If α is a graph orientation-reversing-path (in this case x0Rx1), then α and α′ are graph

orientation-reverse-paths (in this case x0Rx1 and x′0Rx′1).

Proof. (i), (ii), and (iii) are known (we prove using covering space theory). To prove (iv)we notice that γ(α(0))= γ(x0)= x0

′ = α′(0) implies γ(α(1))= α′(1).To prove (v), we use [2, Lemma 2.1, page 77]. �

Theorem 3.7. Let M and N be compact, closed manifolds of the same dimension, let f ,g :M → N be two maps, and let p : M →M and q : N → N be finite coverings such that there

exist lifts f , g : M → N of the pair ( f ,g). If C is a coincidence class of the pair ( f , g), thenC = p(C) is a coincidence class of the pair ( f ,g) and

| ind|( f , g; C)=

⎧⎪⎨⎪⎩

s · k(mod2) if C is defective;

s · k otherwise,(3.5)

where s= | ind|( f ,g,C), k = # j(Coin( f#,g#)x0 ) and x0 ∈ C.

Proof. Since | ind| is homotopy invariant, we may assume that Coin( f ,g) is finite. Thefact that C = p(C) is a coincidence class of the pair ( f ,g) is known. We choose a pointx0 ∈ C. Since Coin( f ,g) is finite, we can suppose C = {x1, . . . ,xs; c1,c′1, . . . ,cn,c′n} whereeach xi is free, and for all pairs cj , c′j we have cjRc′j .

Now we choose paths {αi}i, 2≤ i≤ s; {βj} j and {γ j} j , 1≤ j ≤ n (see Figure 3.1) suchthat

(i) αi is a path in M from x1 to xi establishing the Nielsen relation;(ii) βj is a path in M from x1 to cj establishing the Nielsen relation;

(iii) γ j is a graph-orientation-reversing path in M from cj to c′j .

Assume that C is not defective. We notice that p−1({c1,c′1, . . . ,cn,c′n})∩ C splits into thepairs of points {γrj(0), γrj(1)}where γrj is the lift of γrj(0) starting from a point c ri ∈ p−1(ci).By Lemma 3.6 (v) the points γrj(0), γrj(1) are R-related. For the same reason no two points


x1 α2 x2· · ·

αs

xs

β1

c1 c′1

βn

γ1· · ·

γn

cn c′n

Figure 3.1. The class C and the chosen paths.

from p−1({x1, . . . ,xs}) are R-related. Thus

| ind|( f , g; C)= #p−1({x1, . . . ,xs}

)= | ind|( f ,g ,C) · k = s · k. (3.6)

Now we assume that C is defective. Then each point from C is self-reducing hence soalso is each point in C (Lemma 3.6 (v)). Now

| ind|( f , g; C)= #C(mod 2)

= k(s+ 2n)(mod2)

= k · s(mod2).

(3.7)

�

4. Twofold orientable covering

Let M and N be compact closed manifolds of same dimension such that M is nonori-entable and N is orientable; let f ,g : M → N be two maps, and let p : M →M be the

twofold orientable covering of M. We define f , g : M→N by f = f p and g = g p:

Mf

gp

Mf

g N

(4.1)

Lemma 4.1. Under the above conditions, if C is a coincidence class of the pair ( f ,g), then

p−1(C)⊂ Coin( f , g) is such that(1) p−1(C) can be divided in two disjoint sets C and C′, such that p(C)= p(C′)= C;

(2) if x1, x2 ∈ C (or C′), then x1 and x2 are in the same coincidence class of ( f , g);

(3) C and C′ are in the same coincidence class of the pair ( f , g) if and only ifC is defective.

Proof. We make q : N →N as the identity map in the Corollaries 3.2, 3.4 and Lemma 3.6.�

Corollary 4.2. Under the hypotheses of Lemma 4.1 we have

(1) if C is not defective, then C and C′ are two coincidence classes of the pair ( f , g) such

that ind( f , g, C)=− ind( f , g, C′) and | ind( f , g , C)| = | ind|( f ,g ,C);


(2) if C is defective, then C ∪ C′ is a unique coincidence class of the pair ( f , g) with

ind( f , g, C∪ C′)= 0.

Proof. It is useful to remember that the pair ( f , g) is a pair of maps between orientable

manifolds and that ind( f , g, C) are the indices of the coincidence class C. Since the indexand the semi index are homotopy invariants, we may assume that Coin( f ,g) is finite.

(1) SinceM is nonorientable, the antipodism ofA : M→ M, that is, the map exchang-ing the points in p−1(x) reverses the orientation of M. On the other hand A(C)=C′, hence ind( f , g; C′)= ind( f , g;A(C))= ind( f A−1, gA−1; C)=− ind( f , g; C).

(2) As above we deduce that for x, x′ ∈ p−1(x), ind( f , g ; x)= ind( f , g; x′), hence ind( f ,g; p−1(x))= 0. �

Corollary 4.3. Under de hypotheses of Lemma 4.1 we have

(1) L( f , g)= 0;

(2) N( f , g) is even;

(3) N( f ,g)≥ (N( f , g))/2;

(4) if N( f , g)= 0, then all coincidence classes with nonzero semi-index of the pair ( f ,g)are defective.

Proof. We have that p(Coin( f , g))= Coin( f ,g), and in the pair ( f , g) the pre-image, byp, of a defective class of the pair ( f ,g) has index zero. �

5. Applications

Theorem 5.1. Let f ,g :M→N be two maps between closed manifolds of the same dimen-sion such thatM is nonorientable andN is orientable. Suppose thatN is such that for all ori-entable manifolds M′ of the same dimension of N and all pairs of maps f ′,g′ : M′ →N wehave that L( f ′,g′)= 0 implies that N( f ′,g′)= 0. Then all coincidence classes with nonzerosemi-index of the pair ( f ,g) are defective.

Proof. The hypotheses on N are enough to show, using the notation of the proof of

Lemma 4.1, that N( f , g) = 0. So by Corollary 4.3, all coincidence classes with nonzerosemi-index of the pair ( f ,g) are defective. �

We notice that the hypotheses on the manifoldN in Theorem 5.1, in dimension greaterthan two, are equivalent to the converse of Lefschetz theorem. In dimension two thesehypotheses are not equivalent but necessary for the converse of Lefschetz theorem.

Remark 5.2. The following manifolds satisfy the hypotheses on the manifold N inTheorem 5.1:

(1) Jiang spaces [3, Corollary 1];(2) nilmanifolds [4, Theorem 5];(3) homogeneous spaces of a compact connected Lie group G by a finite subgroup K

[3, Theorem 4].


Acknowledgments

This work was made during a postdoctoral year of the author at Laboratoire Emile Picard,Universite Paul Sabatier (Toulouse, France). We would like to thank John Guaschi andClaude Hayat-Legrand for the invitation and hospitality, Peter N.-S. Wong for helpfulconversations, and the referee for his critical reading and a number of helpful suggestions.This work was supported by Capes-BEX0755/02-8 (International Cooperation Capes-Cofecub Project no. 364/01).

References

[1] R. F. Brown and H. Schirmer, Nielsen root theory and Hopf degree theory, Pacific Journal of Math-ematics 198 (2001), no. 1, 49–80.

[2] R. Dobrenko and J. Jezierski, The coincidence Nielsen number on nonorientable manifolds, TheRocky Mountain Journal of Mathematics 23 (1993), no. 1, 67–85.

[3] D. L. Goncalves and P. N.-S. Wong, Homogeneous spaces in coincidence theory, MatematicaContemporanea 13 (1997), 143–158, 10th Brazilian Topology Meeting (Sao Carlos, 1996), (P.Schweitzer, ed.), Sociedade Brasileira de Matematica.

[4] , Nilmanifolds are Jiang-type spaces for coincidences, Forum Mathematicum 13 (2001),no. 1, 133–141.

[5] J. Jezierski, The semi-index product formula, Polska Akademia Nauk. Fundamenta Mathematicae140 (1992), no. 2, 99–120.

[6] , The Nielsen coincidence theory on topological manifolds, Fundamenta Mathematicae 143(1993), no. 2, 167–178.

[7] P. Olum, Mappings of manifolds and the notion of degree, Annals of Mathematics. Second Series58 (1953), 458–480.

[8] R. Skora, The degree of a map between surfaces, Mathematische Annalen 276 (1987), no. 3, 415–423.

Daniel Vendruscolo: Departamento de Matematica, Universidade Federal de Sao Carlos,Rodovia Washington Luiz, Km 235, CP 676, 13565-905 Sao Carlos, SP, BrazilE-mail address: [email protected]

REDUCING THE NUMBER OF FIXED POINTS OF SOMEHOMEOMORPHISMS ON NONPRIME 3-MANIFOLDS

XUEZHI ZHAO

Received 5 September 2004; Revised 15 March 2005; Accepted 21 July 2005

We will consider the number of fixed points of homeomorphisms composed of finitelymany slide homeomorphisms on closed oriented nonprime 3-manifolds. By isotopingsuch homeomorphisms, we try to reduce their fixed point numbers. The numbers ob-tained are determined by the intersection information of sliding spheres and sliding pathsof the slide homeomorphisms involved.

Copyright © 2006 Xuezhi Zhao. This is an open access article distributed under the Cre-ative Commons Attribution License, which permits unrestricted use, distribution, andreproduction in any medium, provided the original work is properly cited.

1. Introduction

Nielsen fixed point theory (see [1, 4]) deals with the estimation of the number of fixedpoints of maps in the homotopy class of any given map f : X → X . The Nielsen numberN( f ) provides a lower bound. A classical result in Nielsen fixed point theory is: any mapf : X → X is homotopic to a map with exactly N( f ) fixed points if the compact polyhe-dron X has no local cut point and is not a 2-manifold. This includes all smooth manifoldswith dimension greater than 2.

It is also an interesting question whether the Nielsen number can be realized as thenumber of fixed points of a homeomorphism in the isotopy class of a given homeomor-phism. In fact, it is just what J. Nielsen expected when he introduced the invariant N( f ).Assume that X is a closed manifold. The answer to this question is obviously positivefor the unique closed 1-manifold. A positive answer was given by Jiang and Guo [5] for2-manifolds, and was given by Kelly [7] for manifolds of dimension at least 5.

In [6], Jiang, Wang and Wu proved that for any closed oriented 3-manifold X whichis either Haken or geometric, any orientation-preserving homeomorphism f : X → X isisotopic to a homeomorphism with N( f ) fixed points ([6, Theorem 9.1]). If Thurston’sgeometric conjecture is true, all nonprime 3-manifolds are of this type.

In this paper, we will consider a certain class of homeomorphisms of closed, oriented3-manifolds that have a connected sum decomposition into prime factors, namely irre-ducible manifolds and copies of S2 × S1, and at least two factors (nonprime manifolds).


2 Fixed points of slide homeomorphisms

It is known from work of Kneser and Milnor that in the oriented setting, the prime andirreducible factors of the decomposition are unique. We examine homeomorphisms thatcan be expressed as the composition of finitely many slide homeomorphisms. A so-calledslide homeomorphism is the identity away from a certain stratified open neighborhood,the sliding set, of a torus, and is defined by a family of rotation-like transformations onthis set. According to McCullough’s result (see [8]), an arbitrary homeomorphism of a re-ducible 3-manifold can be expressed as the composition of homeomorphisms that comesin four types, one of which is that of slide homeomorphisms.

In [9], the author considered the Nielsen numbers and fixed points of homeomor-phisms which are compositions of m slide homeomorphisms on nonprime 3-manifolds.The fixed point index of the complement of the union of the sliding sets was proved to bezero. When m = 2, we found presentations that are in some sense “standard,” for whichthe fixed point numbers, the fixed point class coordinates and the fixed point indices forall fixed points can be determined. Thus, we were able to give some estimating bounds onthe Nielsen numbers of such kinds of homeomorphisms. The present paper is a continu-ation of [9]. We will generalize the results for m= 2 there to the case where m can be anarbitrary positive integer. We will focus on a geometrical method to reduce the number offixed points in any given isotopy class of such a homeomorphism. The lower bound prop-erty of Nielsen number implies that our number of fixed points yields an upper boundfor Nielsen number.

The remaining sections are organized as follows. In Section 2, we will fix notationwhich will be used throughout this paper, and recall the definition of slide homeomor-phism. In Section 3, we will show (Lemma 3.4) that away from the sliding set, f can beisotoped to a fixed point free homeomorphism by an arbitrary small isotopy. Althougheach component of this set has zero fixed point index ([9, Theorem 3.2]), the resulthere is not very obvious because we are considering fixed points up to isotopy ratherthan homotopy. In Section 4, fixed points over the sliding set are considered. It is ar-gued that f is isotopic to a homeomorphism with finitely many fixed points, and thatthe size of this fixed point set is expressible in terms self-intersection data for the slid-ing set (Proposition 4.6). Reducing the number of fixed points for homeomorphisms inthe isotopy class of f then involves controlling in some sense the number of self inter-sections; our main result (Theorem 4.11) gives a lower bound for this number. Finally, ashort Section 5 shows that in some cases, one may simplify and “optimize” the sliding setso that the bound in Section 4 can be further lowered, that is, the number of fixed pointscan be further reduced.

2. Conventions and notations

In this section, we will make necessary conventions in notation, which will be used inlater sections.

(1) The underlying manifoldM. In this paper, the manifoldM is assumed to be a closedoriented 3-manifold, which is nonprime. It is known thatM can be written as a connectedsum of finitely many prime 3-manifolds, that is, M =M1#M2#···#Mn′#···#Mn′+n′′ , inwhich Mi is irreducible for 1≤ i≤ n′ and Mi = S2× S1 for n′ + 1≤ i≤ n′ +n′′. The non-prime property implies that n′ +n′′ > 1.

Xuezhi Zhao 3

Take a 3-sphere and remove n′ + 2n′′ open discs to obtain a punctured 3-cell W withn′ + 2n′′ boundary components. We then have that M =W ∪ (∪n′+n′′

i=1 M′i ), where M′

i =Mi − Int(Di) for 1 ≤ i ≤ n′ and M′

i = S2 × I for n′ + 1 ≤ i ≤ n′ + n′′ (see [8]). Each M′i

admits the orientation coincident with that of M, and each ∂M′i inherits the orientation

of M′i .

(2) Slide homeomorphisms. Let S be an oriented essential 2-sphere in M, which isorientation-preservingly isotopic to a boundary component of ∂M′

j . Let α : I →M bea path without self intersection in M such that α∩M′

j = α∩ S = {α(0),α(1)}. Take tworegular neighborhoods N ′ and N ′′ (N ′ ⊂ Int(N ′′)) of α∪ S in M. Then Int(N ′′ −N ′) hastwo components which are homeomorphic to S2× (0,1) and T2× (0,1) respectively. Wewrite the latter as T(S,α).

Pick a coordinate function c : T(S,α)→ T2× (0,1), where the points in T2× (0,1) arelabeled by (θ,ϕ, t), such that the θ-line, c−1(θ,∗,∗), is parallel to the oriented path α andthe t-line c−1(∗,∗, t) moves radially away from the path αwhen the value of t is increased.

A slide homeomorphism s :M→M determined by α and S is defined by

s(x)=⎧⎨⎩c−1(θ + 2πt,ϕ, t) if x = c−1(θ,ϕ, t)∈ T(S,α),

x otherwise,(2.1)

denoted by s(S,α). The sets T(S,α), S and α are said to be respectively the sliding set,sliding sphere and sliding path of s(S,α).

(3) Orientations and isotopies. Since all manifolds under consideration are oriented,including sliding spheres and sliding paths, isotopies here are considered to be ambientand orientation-preserving. For example, ifM =M1#M2 is a connected sum of two primemanifolds, ∂M′

1 and ∂M′2 are not regarded as isotopic.

(4) Fundamental groups and path classes. Consider the constructionM=W∪(∪n′+n′′i=1 M′

i )of M. We choose a point x0 in W as its base point. To any path γ with ending points in Wthere corresponds uniquely an element 〈γ∗γγ−1∗∗〉 in π1(M,x0), where γ∗ and γ∗∗ are pathfrom x0 to γ(0) and γ(1) in W respectively. By abuse of notation, we write it simply as〈γ〉. Choose xj ∈ ∂M′

j as base point of M′j for j = 1,2, . . . ,n′ +n′′. Thus, each π1(M′

j ,xj) isembedded into π1(M,x0) in a natural way as above, and hence π1(M,x0) is the free prod-uct of π1(M′

j ,xj), j = 1,2, . . . ,n′ +n′′. We write simply as π1(M,x0)= π1(M′1)∗π1(M′

2)∗···∗π1(M′

n′+n′′), which is also equal to π1(M1)∗π1(M2)∗···∗π1(Mn′+n′′).(5) The homeomorphism f . From now on, f is assumed to be a homeomorphism com-

posed of finitely many slide homeomorphisms, that is, f = s(Sm,αm) ◦ s(Sm−1,αm−1) ◦··· ◦ s(S1,α1). The union ∪m

j=1T(Sj ,αj) of all sliding sets is said to the sliding set of f . Fora simplification in notation, we write sm′′···m′ for the composition s(Sm′′ ,αm′′) ◦ s(Sm′′−1,αm′′−1)◦ ··· ◦ s(Sm′ ,αm′) for any m′ and m′′ (1≤m′ <m′′ ≤m). In particular, s(Sj ,αj) issimply written as s j .

(6) General position. Consider the slide homeomorphisms whose composition is f . Wecan ensure the sliding paths α1,α2, . . . ,αm, and sliding spheres S1,S2, . . . ,Sm are in generalposition relative to the set ∪m

j=1{αj(0),αj(1)}. Thus, these sliding paths have no intersec-tion, and αi intersects with Sj transversally for i = j. Since each sliding sphere is isotopicto a component of −∂W , we can arrange these sliding spheres to be disjoint. In this sit-uation, if each sliding set T(Sj ,αj) is in a small neighborhood of αj ∪ Sj , the number


αi

q(i, j;1)

q(i, j;2)

Sj

· · · · · ·

Figure 2.1

of components of intersection of two sliding sets T(Sj′ ,αj′) and T(Sj′′ ,αj′′) is equal tothe number of points in (αj′ ∪ Sj′)∩ (αj′′ ∪ Sj′′) for all j′ and j′′ with j′ = j′′. In thissituation, we say that the sliding set ∪m

j=1T(Sj ,αj) of f is in general position.(7) Components B(∗,∗;∗) of the intersection of sliding sets. If the sliding set∪m

j=1T(Sj ,αj)is in general position, the points in αi ∩ Sj (i = j) are denoted by q(i, j;1),q(i, j;2), . . . ,q(i, j;|αi∩Sj |) (see Figure 2.1), where the last subscript indicates the order in αi ∩ Sj alongthe direction of αi, that is, α−1

i (q(i, j;k′)) < α−1i (q(i, j;k′′)) in I = [0,1] if and only if k′ <

k′′. The corresponding components of T(Si,αi)∩T(Sj ,αj) nearby are written as B(i, j;1),B(i, j;2), . . . ,B(i, j;|αi∩Sj |). Obviously, we have

Proposition 2.1. If the sliding set ∪mj=1T(Sj ,αj) is in general position, then each B(∗,∗;∗) is

homeomorphic to a solid torus, and T(Si,αi)∩T(Sj ,αj)= (�|αi∩Sj |k=1 B(i, j;k))� (�|αj∩Si|l=1 B( j,i;l))for any i and j, where i, j = 1,2, . . . ,m with i = j.

3. Removing fixed points on the complement of sliding set

Consider our homeomorphism f . Since the fixed point set of each slide homeomorphsimsi is just M−T(Si,αi), the points in the complement M−∪m

j=1T(Sj ,αj) of the sliding setof f are totally contained in the fixed point set of f . In [9], we proved that this isolatedfixed point set has zero fixed point index. In this section, we will show that this fixed pointset can be removed by arbitrary small isotopy.

The following definition is originally from [2].

Definition 3.1. Let Γ : N → TN be a vector field on a compact smooth n-manifold N .The manifold N is said to be a manifold with corners for the vector field Γ if Γ has nosingular points on ∂N and if ∂N can be considered a union of (n− 1)-manifolds (withboundaries) ∂oN , ∂+N and ∂−N with ∂+N ∩ ∂−N =∅ such that Γ(x) is tangent to ∂oNfor x ∈ ∂oN , points inward to N for x ∈ ∂−N and points outward from N for x ∈ ∂+N .

Clearly, for an n-manifold N with corners, both of ∂+N ∩ ∂oN and ∂−N ∩ ∂oN are(n− 2)-dimensional closed manifolds. A simple example is the following.

Xuezhi Zhao 5

Example 3.2. A constant vector field on R2 is given by Γ(x, y) = (1,0). Then the subsetN = [0,1]× [0,1] is a manifold with corners for such a vector field Γ, with ∂oN = [0,1]×{0,1}, ∂−N = {0}× [0,1] and ∂+N = {1}× [0,1].

The next lemma is a kind of generalization of the Poincare-Hopf vector field indextheorem. There are some similar statements in dynamical system theory, see for example[3, Lemma A.1.3].

Lemma 3.3. Let N be a 3-manifold with corners for a vector field Γ. If the boundary ∂N isa disjoint union of m-copies of a sphere such that ∂oN is a disjoint union of m-copies of anannulus, and either ∂+N or ∂−N is a disjoint union of m-copies of a disc, then we can changeΓ relative to a neighborhood of ∂N in N into a nonsingular vector field Γ′.

Proof. Through a coordinate function, each component of ∂N can be regarded as one ofthe following:

Ck ={

(x, y,z) : |x| ≤ 4, (y− 8k)2 + z2 = 4 or x =±4, (y− 8k)2 + z2 ≤ 4}

, (3.1)

where k = 1,2, . . . ,m. Since ∂+N ∩ ∂−N =∅, we may assume that

∂oN =∪mk=1

{(x, y,z) : |x| ≤ 4, (y− 8k)2 + z2 = 4

},

∂−N =∪mk=1

{(x, y,z) : x = 4, (y− 8k)2 + z2 ≤ 4

},

∂+N =∪mk=1

{(x, y,z) : x =−4, (y− 8k)2 + z2 ≤ 4

}.

(3.2)

Regard a neighborhood of ∂N as a subset outside of the cylinders:

Dk ={

(x, y,z) : |x| ≤ 4, (y− 8k)2 + z2 ≤ 4}

, k = 1,2, . . . ,m. (3.3)

Since N is a manifold with corners for the vector field Γ, Γ points inward for the cylinders(outward for N) at ∂+N and points outward for the cylinders (inward for N) at ∂−N . Wehave that Γ(p)∈ {(x, y,z) : x > 0} for p ∈ ∂+N ∪ ∂−N . It is not difficult to prove that therestriction of Γ at each component of ∂N , as a map from a sphere to R3 −{0}, has zerodegree. Hence, we can extend Γ to the union ∪m

k=1Dk such that there is no singular pointon ∪m

k=1Dk.Since N ∪ (∪m

k=1Dk) is a closed 3-manifold, its Euler characteristic number is zero.Using standard methods in differential topology, we can deform Γ into a nonsingular Γ′

relative to a neighborhood of ∪mk=1Dk. Then Γ′ |N is our desired vector field. �

Using this lemma, we can prove the following lemma.


Lemma 3.4. Assume that the sliding set of f is in general position. Given any positive numberε, there is an isotopy F :M× I →M from f to f ′ satisfying:

(i) d(F(x, t), f (x)) < ε for any x ∈M and any t ∈ I ,(ii) the support set {x ∈M : F(x, t) = f (x) for some t ∈ I} of F is contained in the ε-

neighborhood Nε(M − ∪mj=1T(Sj ,αj)) of the complement of the sliding set

∪mj=1T(Sj ,αj) in M,

(iii) Fix( f ′)= Fix( f )− (M−∪mj=1T(Sj ,αj)).

Proof. Clearly, we can regard a neighborhood N(∂W) of ∂W in M as a subset of R3 sothat ∂W =−∪n′+2n′′

j=1 Cj , where

Cj ={

(x, y,z)∈ R3 : (x, y,z) : |x| ≤ 4, (y− 8 j)2 + z2 = 4

or x =±4, (y− 8 j)2 + z2 ≤ 4}

,(3.4)

having the orientation induced from R3. Since ∂W = −∪n′+n′′j=1 ∂M′

j , we may arrange sothat ∂M′

j = Cj for 1≤ j ≤ n′; ∂M′n′+ j = Cn′+2 j−1∪Cn′+2 j . The set W is located outside of

these Cj ’s with respect to the given orientation of Cj ’s.Clearly, we can construct a vector field Γ0 :M→ TM on M so that Γ0(p)= {1,0,0} for

any p in the neighborhood N(∂W) of ∂W in M, where

N(∂W)=∪n′+2n′′k=1

{(x, y,z)∈ R3 : (x, y,z) : |x| ≤ 3, 1≤ (y− 8k)2 + z2 ≤ 9

or 3≤ |x| ≤ 5, (y− 8k)2 + z2 ≤ 9}.

(3.5)

Thus, W and all M′j ’s are manifolds with corners for Γ0. Apply Lemma 3.3 to W and all

M′j ’s, we will get a nonsingular vector field Γ :M → TM on M so that Γ(p)= {1,0,0} for

any p ∈N(∂W).By definition of slide homeomorphism, each sliding sphere Sk is isotopic to a Cj in M.

We then have a well-defined correspondence μ : {1,2, . . . ,m} → {1,2, . . . ,n′ + 2n′′} suchthat Sk is isotopic to Cμ(k) in M for any k = 1,2, . . . ,m.

We take Sk to be the sphere outside of Cμ(k) by a distance of νk (0 < νk < 1). Moreover,we can arrange these ν1,ν2, . . . ,νm to have distinct values. Each sliding path αk attaches thecorresponding sliding sphere Sk at “top” and “bottom” perpendicularly. More precisely,αk(u) = (νk,8μ(k),2 + νk + u) and αk(1− u) = (νk,8μ(k),−2− νk + 1− u) for small u ∈I . Each point q( j,k;∗) in αj ∩ Sk lies on (x(q( j,k;∗)),8μ(k) + 2 + νk,0), where all possiblex(q( j,k;∗)) are distinct numbers in (−1,1) (see Figure 3.1).

We can make such an arrangement because any two sliding spheres and any two slid-ing paths have no intersection by the general position assumption. We then arrange thesliding set to lie in a sufficiently small neighborhood of ∪m

j=1(αj ∪ Sj).Let ξ :M×R→M be the flow generated by Γ. We will show that ξ( f (p), t) = p for all

points p in a small η-neighborhood Nη of M −∪mj=1T(Sj ,αj) in M provided t is small

enough.Case 1. If p ∈M−∪m

i=1T(Si,αi), then f (p)= p. Since Γ has no zero, we have that ξ( f (p),t)= ξ(p, t) = p when t is small enough.

Xuezhi Zhao 7

z

x

yαk(1)

Sk

αj′

q( j′ ,k;∗)

q( j,k;∗) αj

αk(0)

Figure 3.1

Case 2. If p ∈∪mi=1T(Si,αi), then there is a unique smallest number j with p ∈ T(Sj ,αj).

There are two subcases.Subcase 2.1. If s j(p) ∈∪m

i= j+1T(Si,αi), then f (p)= s j(p). By general position, we can ar-range αj so that Γ(αj(u)) does not parallel to the tangent vector of αj(u) at u for all u∈ I .Thus, ξ(·, t) will not push along (or opposite) to the direction that s j does. It follows thatξ( f (p), t) = p when p is closed to the boundary ∂T(Sj ,αj) of T(Sj ,αj) (see Figure 3.2).Subcase 2.2. If s j(p) ∈ ∪m

i= j+1T(Si,αi), then there is a unique smallest number k withk > j such that s j(p) ∈ T(Sk,αk). Notice that p is close to ∂(∪m

i=1T(Si,αi)). We have thats j(p) is also close to ∂T(Sk,αk) because the difference between p and s j(p) is small, sosk ◦ s j(p) will not meet any sliding set other than T(Sk,αk) and T(Sj ,αj). It follows thatf (p)= sk ◦ s j(p).

The component of T(Sk,αk)∩T(Sj ,αj) around p and f (p) have two types: B(k, j;∗) andB( j,k;∗). In the first type, we explain the behavior of ξ( f (p), t) in two parts of Figure 3.3.The first two stages from p to sk ◦ s j(p) is shown on the left part. The last stage is illus-trated in the right part, where s j(p) is behind f (p) = sk ◦ s j(p). Let p = (xp, yp,zp), wehave

(xp, yp,zp

) s j (xp, y′p,z′p

) sk (xp, y′′p ,z′p

) ξ(·, t) (x′′′p , y′′p ,z′p

). (3.6)

This implies that in R3, p and ξ( f (p), t) will have different x-values when t is smallenough. It follows that ξ( f (p), t) = p. The proof for the type B( j,k;∗) is the same.

Define an isotopy Fδ,η :M× I →M by

Fδ,η(p, t)=⎧⎪⎨⎪⎩

ξ(p,δt) if p ∈M−∪mj=1T

(Sj ,αj

),

ξ(p,max

{η−d(p,∪m

j=1∂T(Sj ,αj

)),0}δt)

if p ∈∪mj=1T

(Sj ,αj

).

(3.7)


p sj(p) = f (p)

T(Sj , αj)ξ( f (p), t)

αj

ξ(·, t)

Figure 3.2

T(Sj , αj)

Sj p

T(Sk, αk) sk◦s j(p)s j(p)

z

y

αk

T(Sj , αj)

pξ( f (p), t)

sk◦s j(p)

T(Sk, αk)

z

x

Figure 3.3

Note that the arguments for ξ still work for Fδ,η, so we can prove that Fδ,η( f (p), t) = pfor all t ∈ I and p in the η-neighborhood Nη of M −∪m

j=1T(Sj ,αj) in M. Thus, when δand η are small enough, Fδ,η will be a desired isotopy. �

Corollary 3.5. Any slide homeomorphism is isotopic to a fixed point free map.

4. Fixed points on sliding sets

In this section, we try to reduce the fixed points of the homeomorphism f on its slid-ing set ∪m

j=1T(Sj ,αj). For an arbitrary fixed point x of f on its sliding set, we exam-ine its “trace” x,s1(x),s21(x), . . . ,sm···1(x) under the sliding homeomorphisms composingf . Lemma 4.1 will show that the sliding sets of individual slide homeomorphism meet-ing this trace is totally determined by x itself provided that each sliding set T(Sj ,αj) issmall enough. Hence, a fixed point x will determine a unique sequence consisting of the

Xuezhi Zhao 9

components of the intersection of sliding sets, which we call the accompanying sequence(Proposition 4.2). All the possible accompanying sequence will be given in Lemma 4.3.Next, we will isotope the given homeomorphism f so that different fixed points on slid-ing set of f have different accompanying sequences (Lemma 4.4). When the sliding setof f is in general position, there is a unique point (αi∪ Si)∩ (αj ∪ Sj) near an arbitrarycomponent of T(Si,αi)∩ T(Sj ,αj). Thus, in some sense, reducing the number of fixedpoints is equivalent to reducing the number of intersection points between the slidingpaths and sliding spheres. The minimal number MI({α1, . . . ,αm},{S1, . . . ,Sm}) of the in-tersection of sliding paths and sliding spheres gives a possible number of fixed points forhomeomorphisms in the isotopy class of f (Theorem 4.11). Since the Nielsen numberN( f ) is a lower bound of the number of fixed points for maps in the homotopy class off , the minimal number MI({α1, . . . ,αm},{S1, . . . ,Sm}) also provides an upper bound ofN( f ).

Lemma 4.1. If any three of these sliding sets T(Sj ,αj)’s have no common points, then to eachfixed point x of f there is associated a unique sub-sequence {i1, i2, . . . , ik} of {1,2, . . . ,m} withk ≥ 2 such that sik ◦ ··· ◦ si2 ◦ si1 (x)= x ∈ T(Si1 ,αi1 ), and such that si j−1 ◦ ··· ◦ si2 ◦ si1 (x)∈T(Sij ,αij ) for j = 2,3, . . . ,k.

Proof. Let x be a fixed point of f in ∪mi=1T(Si,αi). There is a unique minimal i such that

x ∈ T(Si,αi). We write this number as i1. A sequence {i1, i2, . . . , ik} will be defined induc-tively:

i j =min{n : n > ij−1, si j−1 ◦ ··· ◦ si1 (x)∈ T(Sj ,αj

)}. (4.1)

Since x ∈ T(Si1 ,αi1 ), we have si1 (x) = x. If there was no such a number i2, si1 (x) ∈ T(Si,αi)for all i > i1. Thus, f (x) = sm···1(x) = sm···i1 (x) = si1 (x). This would contradict the factthat x is a fixed point of f , so we always have that k ≥ 2.

By definition of i j , we have that sn···1(x) = si1 (x) if i1 ≤ n < i2, and that sn···1(x) =si2 ◦ si1 (x) if i2 ≤ n < i3. Inductively, we will get that sn···1(x) = sip−1 ◦ ··· ◦ si2 ◦ si1 (x) ifip−1 ≤ n < ip.

When our induction stops at a stage ip, we have that sip···1(x) = sip ◦ ··· ◦ si2 ◦ si1 (x)does not lie in any sliding set T(Sn,αn) with n > ip, so sm···ip ◦ sip−1 ◦ ··· ◦ si1 (x) = sip ◦sip−1 ◦ ··· ◦ si1 (x). It follows that f (x) = sm···1(x) = sip ◦ sip−1 ◦ ··· ◦ si1 (x). This point isjust x because x is a fixed point of f . Thus, this ip is the final number, say ik, in oursubsequence of {1,2, . . . ,m}.

Let us prove the uniqueness of such a subsequence. If there is another subsequence{ j1, j2, . . . , jl} satisfying the same conditions as {i1, i2, . . . , ik}, then we will get that x ∈T(Sj1 ,αj1 ). Since any three of the sliding sets have no common points, j1 is equal to eitheri1 or ik. If the last case happens, that is, j1 = ik, by the choice of ik, we have that s j(x)= xfor all j > ik. Thus, there would be no j2. It follows that i1 = j1.

Assume that jp = ip for p = 1,2, . . . ,n− 1. By the property of the subsequence {i1, i2, . . . ,ik}, we have sin−1 ◦ . . .◦ si1 (x)∈T(Sin ,αin); by the property of the subsequence { j1, j2, . . . , jl},we have s jn−1 ◦ . . . ◦ s j1 (x)∈ T(Sjn ,αjn). Our assumption implies that sin−1 ◦ . . . ◦ si1 (x) ands jn−1 ◦ . . . ◦ s j1 (x) are the same point. Since this point lies in the image of T(Sin−1 ,αin−1 ) =T(Sjn−1 ,αjn−1 ) under homeomorphism sin−1 = s jn−1 . It also lies in T(Sin−1 ,αin−1 ). Since any


three of the sliding sets have no common points, T(Sin ,αin), T(Sjn ,αjn) and T(Sin−1 ,αin−1 )are at most two different sets. Because in = in−1 and jn = jn−1 = in−1, the unique possibil-ity is that jn = in. Thus, we can prove by induction that jn = in for n= 1,2, . . . ,min{k, l}.

It remains to show that k = l. If k < l, then from the property of the subsequence{ j1, j2, . . . , jl}, we have that s jk ◦ ··· ◦ s j1 (x) ∈ T(Sjk+1 ,αjk+1 ). Since we have proved thatjn = in for n= 1,2, . . . ,k, s jk ◦ ··· ◦ s j1 (x)= sik ◦ ··· ◦ si1 (x)= x. Thus, x lies inT(Si1 ,αi1 )∩T(Sik ,αik )∩T(Sjk+1 ,αjk+1 ). Since ik > i1, jk+1 is equal to either i1 = j1 or ik = jk. This is acontradiction. Symmetrically, the case k > l cannot happen. �

For such a fixed point x, we write B1 for the component of T(Sik ,αik )∩T(Si1 ,αi1 ) con-taining x, and write Bj , j = 2,3, . . . ,k, for the component of T(Sij−1 ,αij−1 )∩ T(Sij ,αij )containing si j−1 ◦ ··· ◦ si2 ◦ si1 (x). The sequence {B1,B2, . . . ,Bk} is said to be the accom-panying sequence of x in the components of the intersection of sliding sets. Clearly, the set{B1,B2, . . . ,Bk} itself is just the set of all components of the intersection of sliding setscontaining sn···1(x) for some n. In other words, we have

Proposition 4.2. Let x be a fixed point of f and {i1, i2, . . . , ik} be its associated sub-sequenceof {1,2, . . . ,m}. Let {B1,B2, . . . ,Bk} be a set consisting of some components of the intersectionof sliding sets such that B1 is a component of T(Sik ,αik )∩T(Si1 ,αi1 ), and such that Bj , j =2,3, . . . ,k, is a component of T(Sij−1 ,αij−1 )∩T(Sij ,αij ). Assume that any three of these slidingsets have no common points. Then, {B1,B2, . . . ,Bk} is the accompanying sequence of the fixedpoint x of f if and only if x belongs to the following set:

sik ◦ sik−1 ◦ ··· ◦ si1 (B1)∩ sik ◦ sik−1 ◦ ··· ◦ si2 (B2)∩···∩ sik (Bk)∩B1. (4.2)

Lemma 4.3. Assume that the sliding set∪mj=1T(Sj ,αj) of f is in general position. If s j(B(i, j;k))

∩B(i′, j;k′) =∅ unless i= i′ and k = k′, then the accompanying sequence of each fixed pointof f in sliding sets has either one of the following forms:

{B(ik ,i1;∗),B(i1,i2;∗),B(i2,i3;∗), . . . ,B(ik−1,ik ;∗)

},

{B(i1,ik ;∗),B(i2,i1;∗),B(i3,i2;∗), . . . ,B(ik ,ik−1;∗)

},

(4.3)

where 1≤ i1 < i2 < ··· < ik ≤m (see Figure 4.1).

Proof. Let x be a fixed point of f in the sliding set with accompanying sequence {B1,B2, . . . ,Bk}. Then, by definition, there is a set {i1, i2, . . . , ik} with 1≤ i1 < i2 < ··· < ik ≤msuch that B1 is the component of T(Sik ,αik )∩T(Si1 ,αi1 ) containing x, and such that Bj ,j = 2,3, . . . ,k is the component of T(Sij−1 ,αij−1 )∩T(Sij ,αij ) containing si j−1 ◦ si2 ◦ si1 (x). Ifthere is a Bj is of the form B(∗,i j ;∗), that is, a component which is not near to αij , thenBj = B(ik ,i1;∗) for j = 1; Bj = B(i j−1,i j ;∗) for j = 1.

When j < k, we have that si j ◦ si j−1 ◦ ··· ◦ si1 (x)∈ si j (Bj)∈ Bj+1. Since si j (B(∗,i j ;∗)) doesnot meet any component of the form B(∗,i j ;∗) but itself, Bj+1 = B(i j ,∗;∗). Because Bj+1

lies in T(Sij+1 ,αij+1 ), we have Bj+1 = B(i j ,i j+1;∗). Similarly, we can prove that B1 = B(ik ,ii;∗) ifBk = B(ik−1,ik ;∗).

Notice that each Bj is only one of two types: either Bj = B(i j−1,i j ;∗) or B(i j ,i j+1;∗). Theabove arguments have shown that if one component Bj in an accompanying sequence isof the first type, the others are the same as it. Thus, we are done. �

Xuezhi Zhao 11

T(Sij−1 , αi j−1 )

Bj

B(i j−1 ,i j;∗)

si j (Bj)

B(l,i j;∗)

T(Sij , αi j )B(i j,∗;∗)

T(Sl, α∗)

Figure 4.1

This lemma in fact implies that the components in one accompanying sequence aredistinct. The next lemma will show that after some suitable isotopies on the slide home-omorphism, there is a one-to-one correspondence between the fixed point set on thesliding set and the set consisting of the above accompanying sequences.

Lemma 4.4. If the sliding set ∪mj=1T(Sj ,αj) is in general position, we can isotope the slide

homeomorphisms s(Si,αi), relative to a neighborhood of M − T(Si,αi), to s′i , where i = 1,2, . . . ,m, so that for each sequence � of components of the intersection of sliding sets of theform given in Lemma 4.3, there is unique fixed point of f ′ = s′m ◦ s′m−1 ◦ ··· ◦ s′1 with � asits accompanying sequence.

Moreover, the fixed point index of the fixed point of f ′ having

{B(ik ,i1; j1),B(i1,i2; j2),B(i2,i3; j3), . . . ,B(ik−1,ik ; jk)

}(4.4)

as its accompanying sequence is −I(ik ,i1; j1)I(i1,i2; j2) ··· I(ik−1,ik ; jk); the fixed point index of thefixed point of f ′ having

{B(i1,il ; j1),B(i2,i1; j2),B(i3,i2; j3), . . . ,B(il ,il−1; jl)

}(4.5)

as its accompanying sequence is (−1)lI(i1,il ; j1)I(i2,i1; j2) ··· I(il ,il−1; jl), where I(i, j;k) is the intersec-tion number of the oriented path αi and the oriented sphere Sj in M at the kth point q(i, j;k)

of αi∩ Sj .Remark 4.5. Although f ′ = s′m ◦ s′m−1 ◦ ··· ◦ s′1 is no longer a composition of standardslide homeomorphisms s′j , we still say that {B1,B2, . . . ,Bk} is the “accompanying sequence”of x in the following sense: B1 is the component of T(Sik ,αik )∩T(Si1 ,αi1 ) containing x,Bj , j = 2,3, . . . ,k is the component of T(Sij−1 ,αij−1 )∩T(Sij ,αij ) containing s′i j−1

◦ ··· ◦ s′i2 ◦s′i1 (x). The proof of Lemma 4.1 still works as long as s j and s′j have the same support setT(Sj ,αj) for each j.

Proof of Lemma 4.4. We will give the proof in three steps.Step 1. Isotope each s(Sj ,αj).


Consider an arbitrary component B(i, j;k) of the intersection of the sliding sets. Since itis a component near the kth point in αi∩ Sj , we can assume that

B(i, j;k) ⊂ c−1i

({(θ,ϕ, t) :

∣∣θ− θ(i, j;k;i)∣∣ < δ, 0≤ ϕ < 2π, 0 < t < 1

})

⊂ c−1j

({(θ,ϕ, t) :

∣∣θ− θ(i, j;k; j)∣∣ < δ,

∣∣ϕ− ϕ(i, j;k; j)∣∣ < δ, 0 < t < 1

}),

(4.6)

where δ > 0, and all θ(i, j;k;∗) and ϕ(i, j;k;∗) are constant. Note that the “length” of B(i, j;k) inT(Si,αi) and the “area” of B(i, j;k) in T(Sj ,αj) can be arbitrary small. The range of ci(B(i, j;k))in θ-coordinate, the range of cj(B(i, j;k)) in θ-coordinate and the range of cj(B(i, j;k)) in ϕ-coordinate can be arbitrarily small. All of B(∗,∗;∗)’s share the same δ.

By a small perturbation, we assume that the intervals [θ(i, j;k;∗) − δ, θ(i, j;k;∗) + δ] and[ϕ(i, j;k;∗)− δ, ϕ(i, j;k;∗) + δ] are disjoint for all possible i, j, k, ∗. Moreover, we can assumethat

[θ(i, j;k;i)− δ, θ(i, j;k;i) + δ

]⊂(π,

7π6

),

[θ(i, j;k; j)− δ, θ(i, j;k; j) + δ

]⊂(

0,π

6

). (4.7)

For i= 1,2, . . . ,m, we isotope s(Si,αi) to s′i so that

cis′i c−1i (θ,ϕ, t)=

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎩

(2πt+

π

6,ϕ,− θ

2π+

712

)if 0 < θ <

π

3,

512

< t <7

12,

(2πt− 5π

6,ϕ,− θ

2π+

1312

)if π < θ <

4π3

,5

12< t <

712

,

(θ + 2πt,ϕ, t) if 0 < t <16

or56< t < 1,

(4.8)

and so that s′i (x) = si(x) for x ∈ T(Si,αi). Thus s′i is isotopic to si relative to M −c−1i ({(θ,ϕ, t) : 1/6 < t < 5/6}). Thus, f ′ = s′m ◦ s′m−1 ◦ ··· ◦ s′1 is isotopic to f relative toM −∪m

i=1c−1i ({(θ,ϕ, t) : 1/6 < t < 5/6}) which is a neighborhood of M −∪m

i=1T(Si,αi) inM.

Since cj ◦ s′j c−1j preserves ϕ-levels, the condition that all possible intervals [ϕ(i, j;k;∗) −

δ, ϕ(i, j;k;∗) + δ] are disjoint implies that s′j(B(i, j;k))∩B(i′, j;k′) =∅ unless i = i′ and k = k′.Clearly, the sliding set here is in general position when δ is small enough. By Lemma 4.3,the accompanying sequence of each fixed point of f ′ is of one of two types listed there.Step 2. Fixed points having accompanying sequences of the first type.

Consider a sequence {B(ik ,i1; j1),B(i1,i2; j2),B(i2,i3; j3), . . . ,B(ik−1,ik ; jk)} of the components ofthe intersection of sliding sets.

Since B(ik ,i1; j1) ranges in t-direction from one component of ∂T(Si1 ,αi1 ) to the othercomponent of ∂T(Si1 ,αi1 ), its image under s′i1 will form a circle “parallel” to αi1 . Thus,s′i1 (B(ik ,i1; j1)) meets any component of the form B(i1,∗;∗). Note that B(i1,i2; j2) ∈ ci1 ({(θ,ϕ, t) :π < θ < 7π/6}). The behavior of s′i1 on B(ik ,i1; j1) (see (4.6), (4.7), and (4.8)) implies thats′i1 (B(ik ,i1; j1)) will be parallel to B(i1,i2; j2) in θ-direction of T(Si1 ,αi1 ). Thus, s′i1 (B(ik ,i1; j1))∩B(i1,i2; j2), which is a solid torus, also ranges in t-direction from one component of ∂T(Si2 ,αi2 ) to the other component of ∂T(Si2 ,αi2 ). Its image under si2 meets B(i2,i3; j3). We thenget a solid torus s′i2 (s′i1 (B(ik ,i1; j1))∩ B(i1,i2; j2))∩ B(i2,i3; j3) = s′i2 ◦ s′i1 (B(ik ,i1; j1))∩ s′i2 (B(i1,i2; j2))∩B(i2,i3; j3) in B(i2,i3; j3) (see Figure 4.2).

Xuezhi Zhao 13

B(ik ,i1; j1) B(i1 ,i2; j2)

s′i1 (B(ik ,i1; j1))

B(i2 ,i3; j3)

s′i2 (B(i1 ,i2; j2))

s′i2◦s′i1 (B(ik ,i1; j1))

Figure 4.2

Repeating the above argument, we will get a solid torus in B(ik ,i1; j1):

s′ik ◦ s′ik−1◦ ··· ◦ s′i1

(B(ik ,i1; j1)

)∩ s′ik ◦ s′ik−1◦ ··· ◦ s′i2

(B(i1,i2; j2)

)∩···∩s′ik(B(ik−1,ik ; jk)

)∩B(ik ,i1; j1).(4.9)

By Proposition 4.2 and the Remark 4.5 following the present lemma, a fixed point x of f ′

will be contained in this set if x has {B1,B2, . . . ,Bk} as its accompanying sequence. Notethat f ′ has unique fixed point on above set. Thus, the fixed point of f ′ with accompany-ing sequence {B1,B2, . . . ,Bk} is unique.

Let x∗ be the unique fixed point of f ′ in the set in (4.9). Then ci1 (x∗) is a fixed pointof ci1 ◦ f ′ ◦ c−1

i1 : U → ci1 (T(Si1 ,αi1 )), where U is the ci1 image of the set in (4.9). Usingthe coordinates of T2× I , the three eigenvalues λ1, λ2, λ3 of the derivative of ci1 ◦ f ′ ◦ c−1

i1

at ci1 (x∗) will satisfy the condition: one eigenvalue has absolute value greater than 1, theother two have absolute values less than 1. We assume that |λ1| > 1, |λ2| < 1 and |λ3| < 1.

From Figure 4.3, we know that the θ-direction of B(i, j;k) is mapped by s′j into the θ-direction of B( j,∗;∗) if I(i, j;k) > 0; the θ-direction of B(i, j;k′) is mapped by s′j into oppositionof the θ-direction of B( j,∗;∗) if I(i, j;k′) < 0. Thus, the eigenvalue λ1 > 1 if I(ik ,i1; j1)I(i1,i2; j2) ···I(ik−1,ik ; jk) = 1; and λ1 <−1 if I(ik ,i1; j1)I(i1,i2; j2) ··· I(ik−1,ik ; jk) =−1.

Note that the fixed point index of an isolated fixed point of a map is just (−1)κ, whereκ is the number of real eigenvalues which are greater than 1 of the derivative of this mapat this fixed point, provided that 1 is not an eigenvalue of this derivative (see [4, page12,3.2(2)]). We have that ind(ci1 ◦ f ′ ◦ c−1

i1 ,ci1 (x∗)) = −I(ik ,i1; j1)I(i1,i2; j2) ··· I(ik−1,ik ; jk), which isalso the fixed point index ind( f ′,x∗) by the commutativity of fixed point index.Step 3. Fixed points having accompanying sequences of the second type.


αi

αjS j

q(i, j;k′)q(i, j;k)

B( j,∗;∗)αj

s j(B(i, j;k))

B(i, j;k)

αi

Figure 4.3

Note that the inverse ( f ′)−1 = s′1 ◦ s′2 ◦ ··· ◦ s′m of f ′ is also a homeomorphism com-posed of finite isotoped slide homeomorphisms, where each s′j is isotopic to the slidehomeomorphism s(Sj ,−αj) determined by Sj and the inverse −αj of path αj . Clearly,the fixed point sets of f ′ and ( f ′)−1 are the same. Moreover, a fixed point of f ′ having{B(i1,il ;∗),B(i2,i1;∗),B(i3,i2;∗), . . . ,B(il ,il−1;∗)} as its accompanying sequence is also a fixed pointof ( f ′)−1 have an accompanying sequence of the first type discussed in last step.

Using the same argument as above, we can prove that there is a unique fixed point y∗of f ′ having {B(i1,il ;∗),B(i2,i1;∗),B(i3,i2;∗), . . . ,B(il ,il−1;∗)} as its accompanying sequence. Theonly difference is in the fixed point index. Because the three eigenvalues λ1, λ2, λ3 ofthe derivative of ci1 ◦ ( f ′)−1 ◦ c−1

i1 at ci1 (y∗) satisfy the conditions: |λ1| > 1, |λ2| < 1 and|λ3| < 1, the three eigenvalues μ1, μ2, μ3 of the derivative of ci1 ◦ f ′ ◦ c−1

i1 at ci1 (y∗) willsatisfy the conditions: |μ1| = |1/λ1| < 1, |μ2| = |1/λ2| > 1 and |μ3| = |1/λ3| > 1. Since bothof f ′ and f are orientation-preserving, we have that λ1λ2λ3 > 0 and μ1μ2μ3 > 0.

Note that at a point in αi ∩ Sj , the algebraic intersection number of αi with Sj is op-posite to the algebraic intersection number of −αi with Sj . If (−I(i1,il ; j1))(−I(i2,i1; j2))···(−I(il ,il−1; jl)) = (−1)lI(i1,il ; j1)I(i2,i1; j2) ··· I(il ,il−1; jl) > 0, by using the proof of the last step, wehave that λ1 > 1, and therefore 0 < μ1 = 1/λ1 < 1. Thus, μ2μ3 > 0. There are three possibil-ities: (1) μ2,μ3 > 1, (2) μ2,μ3 <−1 and (3) μ2 and μ3 are conjugate complex numbers. Ineach case, the number of real eigenvalues which are greater than 1 is even. We have thatind( f ′, y∗)= ind(ci1 ◦ f ′ ◦ c−1

i1 ,ci1 (y∗))= 1.If (−1)lI(i1,il ; j1)I(i2,i1; j2) ··· I(il ,il−1; jl) < 0, by using the proof of last step, we have that λ1 <

−1, and therefore −1 < μ1 = 1/λ1 < 0. Thus, μ2μ3 < 0. Hence, either μ2 < −1, μ3 > 1 orμ2 > 1, μ3 <−1. It follows that there is only one real eigenvalue which is greater than 1, soind( f ′, y∗)= ind(ci1 ◦ f ′ ◦ c−1

i1 ,ci1 (y∗))=−1.Combining these two cases, we have that ind( f ′, y∗)= (−1)lI(i1,il ; j1)I(i2,i1; j2) ··· I(il ,il−1; jl).

�

This lemma is a generalization of [9, Lemma 4.2]. The proof here is more descriptivethan the direct computation there. The fixed point class coordinates of these fixed pointscan be computed in the same way.

Proposition 4.6. Let f = s(Sm,αm) ◦ s(Sm−1,αm−1) ◦ ··· ◦ s(S1,α1) be a homeomorphismcomposed of finitely many slide homeomorphisms. Assume that the Sj ’s are pairwise disjoint,

Xuezhi Zhao 15

and that any αi and any Sj , i, j = 1,2, . . . ,m (i = j), intersect transversally. Then f is isotopicto a homeomorphism with

∑

1≤ j1<···<jk≤m

(∣∣αjk ∩ Sj1∣∣∣∣αj1 ∩ Sj2

∣∣···∣∣αjk−1 ∩ Sjk∣∣

+∣∣αj1 ∩ Sjk

∣∣∣∣αj2 ∩ Sj1∣∣···∣∣αjk ∩ Sjk−1

∣∣)(4.10)

fixed points.

Proof. The assumptions on αi’s and Sk’s imply that we can arrange the union of all slidingstes in general position provided T(Si,αi) is close to αi∪ Si for each i. Using above lemmaand Lemma 3.4, we get immediately our conclusion. �

By this proposition, the number |αi ∩ Sj| determines in some sense the number offixed points. In order to reduce the number of fixed points of such homeomorphisms,the intersection numbers (|αi∩ Sj|’s) should be reduced. In [9, page 184], we defined

MI(αi,Sj

)=: min{∣∣α∩ Sj

∣∣ : α� αj rel{0,1}, α has no self intersection}. (4.11)

From this definition, we have

Proposition 4.7. Let Sj be an oriented sphere isotopic to a component of the boundary∂M′

k( j) of a summand M′k( j), and let 〈αi〉 = a1b1a2b2 ···anbnan+1 where bl consists of words

in π1(M′k( j)), al does not contain any word in π1(M′

k( j)) and all al’s and bl’s are non-trivialexcept possibly for a1 and an+1. Then MI(αi,Sj) = 2n if M′

k( j) ∼= S2 × I ; MI(αi,Sj) = n ifM′

k( j)∼= S2× I . Here, the number n is just the number of word “groups” of 〈αi〉, consisting of

the words from π1(M′k( j)).

In particular, we have MI(αi,Sj)= 0 if Sj is isotopic to Si.

Proof. See [9, Proposition 4.4].It should be noticed that all MI ’s can not be minimized at same time if any two Si’s are

disjoint and if there are isotopic sliding spheres.

Example 4.8. Let M = T31 #T3

2 #T33 be the connected sum of three 3-dimensional tori. For

j = 1,2,3, we write gj1, gj2 and gj3 for the generators of the free abelian group π1(T3j ).

Let S1 and S3 be oriented spheres isotopic to the boundary of the summand T31 −

Int(D3), and S2 an oriented sphere isotopic to the boundary of the summand T32 −

Int(D3). Three paths are given by 〈α1〉 = g21, 〈α2〉 = g32g12g33, and 〈α3〉 = g31g22g33g23

(see Figure 4.4).

The numbers of |αi∩ Sj|’s in two cases are listed as follows:

α1 α2 α3

S1 — 2 2S2 2 — 4S3 0 2 —

α1 α2 α3

S1 — 2 0S2 2 — 4S3 2 2 —


S1

T33

T31

S3

α2

α3

α1

T32

S2

S1

T33

T31

S3

α2

α3

α1

T32

S2

Figure 4.4

Thus, the sum in Proposition 4.6 is

(∣∣α3∩ S1∣∣∣∣α1∩ S2

∣∣∣∣α2∩ S3∣∣+

∣∣α1∩ S3∣∣∣∣α2∩ S1

∣∣∣∣α3∩ S2∣∣)

+(∣∣α2∩ S1

∣∣∣∣α1∩ S2∣∣+

∣∣α1∩ S2∣∣∣∣α2∩ S1

∣∣)

+(∣∣α3∩ S1

∣∣∣∣α1∩ S3∣∣+

∣∣α1∩ S3∣∣∣∣α3∩ S1

∣∣)

+(∣∣α3∩ S2

∣∣∣∣α2∩ S3∣∣+

∣∣α2∩ S3∣∣∣∣α3∩ S2

∣∣).

(4.12)

In the case shown on the left, it is (8 + 0) + (4 + 4) + (0 + 0) + (8 + 8) = 32; in the othercase, it is (0 + 16) + (4 + 4) + (0 + 0) + (8 + 8)= 40.

Note that in both cases |αi∩ Sj| =MI(αi,Sj) except for (i, j)= (1,3) or (3,1). Since S3

and S1 are isotopic, we have thatMI(α1,S3)=MI(α3,S1)= 0. But, these two numbers cannot be realized simultaneously if the intersection of S1 and S3 is assumed to be empty. �

Thus, we need the following.

Definition 4.9. Given slide homeomorphisms s(S1,α1),s(S1,α1), . . . ,s(Sm,αm) whose com-position is f , we define MI({α1, . . . ,αm},{S1, . . . ,Sm}) to be:

minα′j ,S

′j

∑

1≤ j1<···<jk≤m

(∣∣α′jk ∩ S′j1∣∣∣∣α′j1 ∩ S′j2

∣∣···∣∣α′jk−1∩ S′jk

∣∣

+∣∣α′j1 ∩ S′jk

∣∣|α′j2 ∩ S′j1∣∣···∣∣α′jk ∩ S′jk−1

∣∣),(4.13)

where each α′j and S′j , j = 1,2, . . . ,m, range over all oriented paths and spheres such thatα′j and S′j are isotopic to αj and Sj , respectively, with α′j ∩ S′j = {α′j(0),α′j(1)}, and suchthat any two α′i ’s and any two S′j ’s have empty intersection.

In Example 4.8, we have MI({α1,α2,α3},{S1,S2,S3}) = 32. The relation between this“totally” minimal intersection number and the individual MI ’s is given by the followingproposition.

Xuezhi Zhao 17

Proposition 4.10. The number MI({α1, . . . ,αm},{S1, . . . ,Sm}) is greater or equal to thefollowing sum:

∑

1≤ j1<···<jk≤m

(MI

(αjk ,Sj1

)MI

(αj1 ,Sj2

)···MI(αjk−1 ,Sjk

)

+MI(αj1 ,Sjk

)MI

(αj2 ,Sj1

)···MI(αjk ,Sjk−1

)).

(4.14)

If any two sliding spheres are not isotopic, then the above two numbers are the same.

Now, we can state our main theorem.

Theorem 4.11. Let f = s(Sm,αm) ◦ s(Sm−1,αm−1) ◦ ··· ◦ s(S1,α1) be a homeomorphismwhich is composed of finitely many slide homeomorphisms. Then, f is isotopic to a homeo-morphism with MI({α1, . . . ,αm},{S1, . . . ,Sm}) fixed points.

Proof. By definition, MI({α1, . . . ,αm},{S1, . . . ,Sm}) can be realized as

∑

1≤ j1<···<jk≤m

(∣∣α′jk ∩ S′j1∣∣∣∣α′j1 ∩ S′j2

∣∣···∣∣α′jk−1∩ S′jk

∣∣

+∣∣α′j1 ∩ S′jk

∣∣∣∣α′j2 ∩ S′j1∣∣···∣∣α′jk ∩ S′jk−1

∣∣),(4.15)

where for each j = 1,2, . . . ,m, α′j and S′j are isotopic to αj and Sj , respectively with α′j ∩S′j = {α′j(0),α′j(1)}, and that any two α′i ’s and any two S′j ’s have no intersections. Thus,s(S′j ,α

′j) is isotopic to s(Sj ,αj). Applying Proposition 4.6 to the homeomorphism

s(S′m,α′m)◦ s(S′m−1,α′m−1)◦ ··· ◦ s(S′1,α′1), we will obtain our conclusion. �

By the lower bound property of Nielsen number, we immediately get the followingcorollary.

Corollary 4.12.

0≤N( f )≤MI({α1, . . . ,αm

},{S1, . . . ,Sm

}). (4.16)

5. Some remarks

In this final section, we will show that in some cases, the fixed point numbers can befurther reduced.

Consider our homeomorphism f . If some successive sliding spheres, say Sn,Sn+1, . . . ,Sn+p, are isotopic, we have

s(Sn+p,αn+p

)◦ ··· ◦ s(Sn+1,αn+1)◦ s(Sn,αn

)= s(Sn,βn), (5.1)

where 〈βn〉 = 〈αnαn+1 ···αn+p〉.Combine all possible slide homeomorphisms which are in succession and have the

same sliding spheres. We will get a shorter expression for f , denoted as follows:

f = s(Smp ,βmp

)◦ ··· ◦ s(Sm2 ,βm2

)◦ s(Sm1 ,βm1

), mp ≤m. (5.2)


Using the main theorem (Theorem 4.11), we can isotope f to a homeomorphism with

MI({βm1 ,βm2 , . . . ,βmp

},{Sm1 ,Sm2 , . . . ,Smp

})(5.3)

fixed points. This number is no more than MI({α1, . . . ,αm},{S1, . . . ,Sm}).In some cases, the two sliding spheres on two ends of the original expression of f are

isotopic, that is, Sm is isotopic to S1. This implies that Smp is isotopic to Sm1 . Consider thehomeomorphism

g = s(Sm1 ,βm1

)◦ s(Smp ,βmp

)◦ s(Smp−1 ,βmp−1

)◦ ··· ◦ s(Sm2 ,βm2

)

= s(Smp ,βmpβm1

)◦ s(Smp−1 ,βmp−1

)◦ ··· ◦ s(Sm2 ,βm2

).

(5.4)

Here, βmpβm1 can be considered as a path satisfying 〈βmpβm1〉 = 〈βmp〉〈βm1〉. Notice thatg = s(Sm1 ,βm1 ) ◦ f ◦ (s(Sm1 ,βm1 ))−1, that is, g is conjugate to f . The fixed point set of fand g are the same. Such a relation is preserved under isotopy. Thus, using the main the-orem (Theorem 4.11) again, we can isotope f to a homeomorphism with MI({βm2 , . . . ,βmp−1 ,βmpβm1},{Sm2 , . . . ,Smp}) fixed points.

Furthermore, if 〈βmpβm1〉 = 1 ∈ π1(M), we get that g = s(Smp−1 ,βmp−1 ) ◦ ··· ◦ s(Sm2 ,βm2 ), and therefore the resulting fixed point number is just MI({βm2 , . . . ,βmp−1 ,},{Sm2 , . . . ,Smp−1}), so we can repeat the above procedure if Smp−1 is isotopic to Sm2 .

Apply this method to Example 4.8, we will get a new homeomorphism

g = s(S3,α3α1)◦ s(S2,α2

)= s(S3,⟨g31g22g33g23g21

⟩)◦ s(S2,⟨g32g12g33

⟩). (5.5)

Thus, the homeomorphism is isotopic to one with MI({α2,α3α1},{S2,S3}) = 16 fixedpoints, and so is the homeomorphism s(S3,α3)◦ s(S2,α2)◦ s(S1,α1).

Acknowledgment

This paper was supported by Natural Science Foundation of Beijing. I would like to thankthe referee for many helpful comments and suggestions.

References

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Xuezhi Zhao 19

[8] D. McCullough, Mappings of reducible 3-manifolds, Geometric and Algebraic Topology, BanachCenter Publ., vol. 18, PWN—Polish Scientific, Warsaw, 1986, pp. 61–76.

[9] X. Zhao, On the Nielsen numbers of slide homeomorphisms on 3-manifolds, Topology and itsApplications 136 (2004), no. 1-3, 169–188.

Xuezhi Zhao: Department of Mathematics, Capital Normal University, Beijing 100037, ChinaE-mail address: [email protected]

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