Seifert Manifolds

K. B. LeeUniversity of Oklahomakblee@math.ou.edu

At Sogang UniversityJune 7–9, 2011

New Titles | FAQ | Keep Informed | Review Cart | Contact Us

Quick Search (Advanced Search ) Browse by Subject

Seifert FiberingsKyung Bai Lee, University of Oklahoma, Norman, OK, and Frank Raymond, University of Michigan, Ann Arbor, MI

Preface Preview Material Table of Contents Supplementary Material

SEARCH THIS BOOK:

Mathematical Surveys and Monographs

2010; 396 pp; hardcover

Volume: 166

ISBN-10: 0-8218-5231-0

ISBN-13: 978-0-8218-5231-6

List Price: US$99

Member Price: US$79.20Order Code: SURV/166

Suggest to a Colleague

See also:

Classifying Spaces of Sporadic Groups - David J

Benson and Stephen D Smith

Seifert fiberings extend the notion of fiber bundle mappings by allowing some of the fibers to be singular. Away

from the singular fibers, the fibering is an ordinary bundle with fiber a fixed homogeneous space. The singular fibersare quotients of this homogeneous space by distinguished groups of homeomorphisms. These fiberings areubiquitous and important in mathematics. This book describes in a unified way their structure, how they arise, andhow they are classified and used in applications. Manifolds possessing such fiber structures are discussed andrange from the classical three-dimensional Seifert manifolds to higher dimensional analogues encompassing, forexample, flat manifolds, infra-nil-manifolds, space forms, and their moduli spaces. The necessary tools not coveredin basic graduate courses are treated in considerable detail. These include transformation groups, cohomology ofgroups, and needed Lie theory. Inclusion of the Bieberbach theorems, existence, uniqueness, and rigidity of Seifertfiberings, aspherical manifolds, symmetric spaces, toral rank of spherical space forms, equivariant cohomology,polynomial structures on solv-manifolds, fixed point theory, and other examples, exercises and applications attestto the breadth of these fiberings. This is the first time the scattered literature on singular fiberings is broughttogether in a unified approach. The new methods and tools employed should be valuable to researchers andstudents interested in geometry and topology.

Readership

Graduate students and research mathematicians interested in topology (transformation groups, manifolds, singularfiberings, and differential geometry).

AMS Home | Comments: webmaster@ams.org

© Copyright 2011, American Mathematical Society

Privacy Statement

Matches for: http://www.ams.org/bookstore-getitem/item=surv-166

1 of 1 05/04/2011 10:37 AM

Figure 1: Mathematical Surveys and Monographs, AMS

1

In this series of 3 lectures entitled Seifert Fiberings, I will try to explainthe subject matter in the simplest form.

Lecture 1 “Classical 3-dimensional Seifert manifolds”This will be devoted to the classical 3-dimensional Seifert manifolds. For

each geometry, we shall discuss the isometry groups. The the local charac-teristics of the singular fiber will be explained.

Lecture 2 “General Seifert Fibering and Generalization of Bieberbach Theo-rems” We define a general Seifert fibering. On a principal bundle, the groupof weakly equivariant maps is used as the uniformizing group. The exis-tence and uniqueness theorem for the construction is stated. This lecturealso contains Generalizations of Bieberbach Theorems, which is a buildingblock for the singular fibers.

Lecture 3 “Applications” has three parts: Homologically injective torus op-erations, Fixed point theory, and Polynomial structures for solvmanifolds.

Lecture 1: Classical 3-dimensional Seifert manifoldsLecture 2: General Seifert Fibering and Generalization of Bieberbach TheoremsLecture 3: Applications

(a) Homologically injective torus operations(b) Applications to fixed-point theory(c) Polynomial structures for solvmanifolds

2

1. Classical 3-dimensional Seifert manifolds

There are 8 Seifert geometries in dimension 3.

χ > 0 χ = 0 χ < 0e = 0 R × S2 R3 R × H2

e 6= 0 S3 Nil SL(2, R)

plus two more, Sol = R2 ⋊ R and H3.Thurston’s geometrization theorem says that these are all in dimension

3.These model spaces have Riemannian metrics.Except for H3, all spaces have fibering structure:

G −−−−→ P −−−−→ B Isom0(P )

R −−−−→ R × S2 −−−−→ S2 R × SO(3)

S1 −−−−→ S3 −−−−→ S2 SO(4)

R −−−−→ R3 −−−−→ R2 E(3)0 = R3 ⋊ SO(3)

R −−−−→ Nil −−−−→ R2 Nil ⋊ SO(2)

R −−−−→ R × H2 −−−−→ H2 PSL(2, R) × R

R −−−−→ PSL(2, R) −−−−→ H2 (PSL(2, R) × R)/Z

R2 −−−−→ Sol −−−−→ R Sol

H3 PSL(2, C)

• Notation A⋊Q: Suppose Q acts on A. That is, there is a homomorphismϕ : Q → Aut(A). Then the semi-direct product A ⋊ Q is, as a set A × Q,but the group operation is

(a, α) · (b, β) = (a · ϕ(α)(b), αβ).

• Nil (=Heisenberg group) is the group of (3 × 3) matrices

Nil =

1 x z0 1 y0 0 1

: x, y, z ∈ R

It is diffeomorphic to R3. As a group, it is non-abelian, but 2-step nilpotentwith the center R (with x = y = 0).• SL(2, R) is the real (2× 2) matrices of determinant 1. It has the center Z2

consisting of {±I2}.• PSL(2, R) = SL(2, R)/{±I2} ∼= SO0(2, 1), the group of Mobius maps (lin-ear fractional transformations). We give the best Riemannian metric (com-ing from the Cartan-Killing form). It is diffeomorphic to S1 × H2, and the

universal covering group PSL(2, R) is diffeomorphic to R1 × H2. Of course,metrically they are different.

3

• Sol = R2 ⋊ R where t ∈ R acting on R2 by etI2. It is a solvable Lie groupwith nilradical R2.• PSL(2, C) ∼= SO0(3, 1).

Base=H2:

Look at two model spaces R × H2 and PSL(2, R). Both are principalR-bundles over H2.

R −−−−→ R × H2 −−−−→ H2 (trivial bundle)

R −−−−→ PSL(2, R) −−−−→ H2 (twisted bundle)

Compare the isometry groups Isom0(R × H2) and Isom0(PSL(2, R)).

1 −−−−→ R −−−−→ PSL(2, R) × R −−−−→ PSL(2, R) −−−−→ 1

1 −−−−→ R −−−−→ (PSL(2, R) × R×)/Z −−−−→ PSL(2, R) −−−−→ 1

The first one is a splitting extension, but the second is not.

Theorem 1.1 (Raymond-Kulkarni). Let

1 → Z → π → Q → 1

be a central extension, where Q is a Fuschian group. Then

(1) π embeds into Isom0(PSL(2, R)) if and only if [π] ∈ H2(Q; Z) hasan infinite order;

(2) π embeds into Isom0(R × H2) if and only if [π] ∈ H2(Q; Z) has afinite order.

• A Fuschian group is a discrete cocompact subgroup of PSL(2, R) (so thatit acts on H2 properly discontinuously with compact quotient). Therefore,a Fuschian group is a finite extension of a surface group.

• The action is free if and only if π is torsion-free (true if the space is con-

tractible manifold). In this case, the orbit space π\(R×H2) or π\PSL(2, R)is a manifold modeled on each geometry.

Base=R2:A similar statement holds for R3 and Nil. Both are principal R-bundles

over R2.R −−−−→ R3 −−−−→ R2 (trivial)

R −−−−→ Nil −−−−→ R2 (twisted)

The isometry groups Isom0(R3) = R3 ⋊ SO(3) and Isom0(Nil) are:

1 −−−−→ R3 −−−−→ R3 × SO(3) −−−−→ SO(3) −−−−→ 1

Isomf (R × R2)

1 −−−−→ R −−−−→ R × (R2 ⋊ SO(2)) −−−−→ R2 ⋊ SO(2) −−−−→ 1

1 −−−−→ R −−−−→ Nil ⋊ SO(2) −−−−→ R2 ⋊ SO(2) −−−−→ 1

4

Theorem 1.2. Let

1 → Z → π → Q → 1

be a central extension, where Q is a 2-dimensional crystallographic group.Then

(1) π embeds into Isom0(Nil) if and only if [π] ∈ H2(Q; Z) has an infiniteorder;

(2) π embeds into Isom0(R3) if and only if [π] ∈ H2(Q; Z) has a finiteorder.

• A 2-dimensional crystallographic group is a discrete cocompact subgroupof E(2) = R2 × O(2) (so that it acts on R2 properly discontinuously withcompact quotient).

Base=S2:Now compare R × S2 and S3. Both are principal bundles over S2.

R −−−−→ R × S2 −−−−→ S2 (trivial)

S1 −−−−→ S3 −−−−→ S2 (Hopf fibration, twisted)

The isometry groups Isom0(S3) = SO(4) = S3 × S3 and Isom0(R × S2)

are:1 −−−−→ R −−−−→ R × SO(3) −−−−→ SO(3) −−−−→ 1

1 −−−−→ S3 −−−−→ SO(4) −−−−→ SO(3) −−−−→ 1

In fact, SO(4) = (S3 × S3)/{±I4}.

Theorem 1.3. (1) A group Π is the fundamental group of a S3-geometryif and only if it is a finite subgroup of SO(4) acting freely (all clas-sified).

(2) A group Π is the fundamental group of a R × S2-geometry if andonly if it is virtually Z.

For q = x1 + x2i + x3j + x4k,

x1 −x2 −x3 −x4

x2 x1 −x4 x3

x3 x4 x1 −x2

x4 −x3 x2 x1

∈ S3

ℓ ,

x1 −x2 −x3 −x4

x2 x1 x4 −x3

x3 −x4 x1 x2

x4 x3 −x2 x1

∈ S3

r

Example 1.4. E(3) = R3 ⋊ O(3) is the group of isometries of R3 (3-dimensional Euclidean space). As a set, it is the Cartesian product R3×O(3),where O(3) is the orthogonal group. The group operation is given by

(a, A)(b, B) = (a + Ab, AB).

This acts on R3 by

(a, A) · x = a + Ax

5

for x ∈ R3. The matrix A is called the rotational part, and a is called thetranslational part of (a, A). Consider the subgroup Π ⊂ E(3) generated by

Π = 〈t1 = (e1, I), t2 = (e2, I), t3 = (e3, I), α = (a, A)〉,

where

e1 =

100

, e2 =

010

, e3 =

001

, a =

1

2e1, A =

1 0 00 −1 00 0 −1

.

Since ti · x = x + ei, each ti is a translation by the ith unit vector. On theother hand,

α · x =

1/200

,

1 0 00 −1 00 0 −1

·

x1

x2

x3

=

x1 + 1/2−x2

−x3

shows that α rotates the x2x3-plane by 180o, while it advances the x1 direc-tion by a half unit. Moreover,

α2 = t1.

Since

αt1α−1 = t1, αt2α

−1 = t−12 , αt3α

−1 = t−13 ,

the center of Π is generated by t1, and is isomorphic to Z. The quotient is

Q = Π/Z = 〈t2, t3, α〉 ∼= Z2 ⋊ Z2.

Clearly, Π ∩ R3 = Z3 (the translation part), and Π/Z3 = Z2 generated byα.

1 → Z3 = 〈t1, t2, t3〉 → Π → Z2 = 〈α〉 → 1

Thus our space M = Π\R3 is

M = Z2\(Z3\R3) = Z2\T

3.

We study this Z2-action in more detail. On R2,

α =

([00

],

[−1 00 −1

]),

and it has 4 fixed point when considered as an action on T 2,

(0, 0), (0, 12), (1

2 , 0), (12 , 1

2).

Look at a small disk-like neighborhood D around one of these points, say(0, 0). Our α acts on the preimage of the disk in the fibering S1 → S1 ×D2.Our α rotates D2 by 180 degrees while it advances by half a unit on thefiber S1. The result is: Start with a solid cylinder, twist 180 degree, andidentify the top and bottom. The core is identified right away, but all theother “fibers” will have length twice the core. Consequently this is not afibration. The core is called a singular fiber.

6

Definition 1.5. A Seifert manifold M is a 3-dimensional manifold withan effective S1-action; the orbit space B = S1\M is the base orbifold.S1 → M → B is called a Seifert fibering.

Theorem 1.6. All Seifert manifold are modeled on one of the 6 geometries(except for Sol and H3). Such a manifold has one and only one geometry(which is determined by the fundamental group).

The flat Riemannian manifold (or Euclidean space-form) of Example 1.4has an S1-action (rotation in the fiber 〈t1〉\R(e1)).

The subgroup R = {(se1, I) : s ∈ R} ⊂ E(3) normalizes (in facts, central-izes) our group Π. Therefore, it induces an action on the quotient. Sincet1 was in Π already, it is not effective. But S1 = R/〈t1〉 is effective onM = Π\R3. Thus, M is a Seifert manifold.

S1 → M → 〈α〉\T 2.

The orbit space of M by this S1-action is a 〈α〉\T 2, which is an orbifold(topologically, it is a manifold S2) with 4 singular points, of Seifert invariant(1, 2), (1, 2), (1, 2), (1, 2).

Example 1.7. Let Q1

Q1 = 〈a1, b1, . . . , a6, b6 | Π[ai, bi] = 1〉

be the fundamental group of a surface Σ group of genus 6. Let α be a maprotating the surface by 120 degree with two fixed points. Let Z3 = 〈α〉.Then Q = Q1 ⋊ Z3 is a Fuschian group. Form the product S1 × Σ, and liftthe action of α to this product in such a way that while it acts on the baseas before, it advances by 1

3 of a unit in the S1-direction. Then, this action

of new Z3 on S1 ×Σ is free. We have created an action of a group Π whichis an extension of Z

1 → Z → Π → Q = Q1 ⋊ Z3 → 1.

M = (S1 × Σ)/Z3 is a non-trivial Seifert manifold, with Seifert invariants(1, 3), (1, 3).

7

2. General Seifert Fibering

Let G be a Lie group, W a space. [G will replace S1 or R1 (fiber) and Wwill replace R2, S2 or H2 (base) of the classical Seifert fiberings.]

Then G acts on the space P = G × W as left translations, called ℓ(G).[This P is, in general, a principal G-bundle over W ]. Recall that a homeo-morphism

f : G × W → G × W

is weakly G-equivariant if and only if there exists a continuous automorphismαf of G so that

f(a · x, w) = αf (a)f(x, w)

for all a ∈ G and (x, w) ∈ G × W . The group of all weakly G-equivariantself-maps of G × W is denoted as TOPG(P ).

TOPG(P ) = weakly G-equivariant self-maps of G × W

= Normalizer of ℓ(G) in TOP(G × W )

= M(W, G) ⋊ (Aut(G) × TOP(W ))

• TOP(W ) =the group of all self homeomorphisms of W• M(W, G) = the group of all continuous maps from W to G.

Let (λ, α, h) ∈ TOPG(P ) and (x, w) ∈ G × W . ThenAction of TOPG(P ) on G × W is

(λ, α, h) · (x, w) = (α(x) · (λ(h(w)))−1, h(w)).

Group operation of TOPG(P ) is

(λ, α, h) · (λ′, α′, h′) = (λ · (α◦λ′◦h−1), αα′, hh′).

Any discrete subgroup Π of TOPG(P ) will act on P = G×W nicely. Sincethe action is weakly G-equivariant, the action respects the fiber structure ofthe total space G × W ,

G → G × W → W

Then we would get

Γ\G → Π\(G × W ) → Q\W

where Γ = Π ∩ ℓ(G), and Q = Π/Γ.This is an action of Π on P = G × W , where

1 → Γ → Π → Q → 1.

8

Definition 2.1. Let G be a Lie group, Γ a lattice of G, and W a space.Given

(1) a proper action ρ : Q → TOP(W ),

(2) a group extension 1 −→ Γ −→ Πp

−→ Q −→ 1,

A Seifert Construction for Π with the uniformizing group U ⊂ TOPG(P )(closed subgroup) is a homomorphism

θ : Π −−−−→ U

such that θ|Γ = ℓ(Γ) and the induced action of θ on the base is ρ.

Problem: Does there exist θ? If so, how many θ’s are there?

The smaller U is, the more restrictive the fiber space structure will be.Certainly, we took Isom(P ) as our U for all the 3-dimensional Seifert fiber-ings.

Consequences:If θ exists, the group Π acts on G×W properly, we get a space Π\(G×W ).

Further,Γ\G −−−−→ Π\(G × W ) −−−−→ Q\W

is a Seifert fibering.

Γ\G : typical fiber

Q\W : base orbifold

All singular fibers are finite quotients of Γ\G.

In the Seifert fibering in Example 1.4

S1 → M → 〈α〉\T 2,

regular fiber is S1, and the base orbifold is 〈α〉\T 2, topologically S2. It has4 singular fibers.

A Lie algebra g is said to be of type (R) if for each x ∈ g the eigenvaluesof ad(x) are real. A Lie group G is of type (R) if its Lie algebra is of type(R). This is also called completely solvable. A Lie group G is of type (E) ifexp : g → G is surjective, where g is the Lie algebra of G. Clearly,

Abelian =⇒ nilpotent =⇒ type (R) =⇒ type (E).

Example The solvable Lie group E(2)0 = R2 ⋊ SO(2) = R2 ⋊R, where t ∈ R

acts on R2 as

[cos t sin t− sin t cos t

], is not type (E). (Note that (x, 2π) has no

square root). However, Sol = R2 × R, where t ∈ R acts on R2 as etI2 is oftype (R).

9

If ρ1, ρ2 : Q → TOP(W ) are rigidly related (i.e., there exists h ∈ TOP(W )for which ρ2 = µ(h) ◦ ρ1)

Theorem 2.2 (Conner, Kamishima, Lee, Raymond, Wigner). For simplyconnected G = Rk, nilpotent, completely solvable or some semi-simple∗,

(1) θ : Π → U ⊂ TOPG(P ) always exists.(2) θ is unique up to conjugation by M(W, G).(3) If (Q, W ) is rigid, then θ is unique up to conjugation by TOPG(P ),

even if ℓ’s and ρ’s are different.

(2) means that: if θ1 and θ2 are two constructions, then there existsλ ∈ M(W, G) so that

θ2() = λ ◦ θ1() ◦ λ−1.

Then,

G × Wλ

−−−−→ G × Wy

yG×Wθ1(π)

λ−−−−→ G×W

θ2(π)

Γ\G Γ\Gy

yG×Wθ1(π)

λ−−−−→ G×W

θ2(π)yy

Q\W=

−−−−→ Q\W

Moreover, λ moves only along the fibers.

Special cases:If W is a point, then Q is a finite group, and

TOPG(G × {w}) = G ⋊ Aut(G).

G =Rk : flat manifold

G =Nilpotent : infra-nilmanifold

G =Solvable : infra-solvmanifold

In particular, (2) implies that Two infra-nilmanifolds with isomorphicπ1’s are diffeomorphic via an affine diffeomorphism.

Also, this characterizes all singular fibers; i.e., all singular fibers are infra-homogeneous spaces finitely covered by the typical fiber Γ\G.

Definition 2.3. Let Γ be a subgroup of a Lie group G. We say (G, Γ) has aUnique Automorphism Extension Property (UAEP) if every automorphismof Γ extends to an automorphism of G uniquely. For example, if G is aconnected, simply connected nilpotent Lie group, and Γ is any lattice, then(G, Γ) has UAEP.

10

Simply connected G = Rk, nilpotent, completely solvable or some semi-simple∗ with their lattice have this UAEP. For example, let Zk ⊂ Rk. Thenany automorphism of Zk extends uniquely to an automorphism of Rk, sinceGL(k, Z) ⊂ GL(k, R).

The existence and uniqueness/rigidity yield the following applications:

(1) Existence of closed K(Π, 1)-manifolds(2) Rigidity for Seifert fiberings(3) Lifting problem for homotopy classes of self-homotopy equivalences(4) Polynomial structures for solvmanifolds(5) Applications to fixed-point theory(6) Homologically injective torus operations(7) Maximal torus actions on solvmanifolds and double coset spaces(8) Toral rank of spherical space forms(9) Bounding problems

(10) and many more...

11

3. Generalization of Bieberbach’s Theorems

In the general Seifert fibering theory, our Lie group G will be usuallyRk or a simply connected nilpotent Lie group, or even a simply connectedsolvable Lie group. Let Γ be a lattice of G. Then the typical fiber Γ\G willbe a torus, nilmanifold, or solvmanifold; singular fibers are flat Riemannianmanifold, infra-nilmanifold, or infra-solvmanifold.

For G = Rn, Aut(Rn) = GL(n, R) and Aff(Rn) = Rn ⋊ Aut(Rn) =Rn ⋊ GL(n, R); and it contains the Euclidean group E(n) = Rn ⋊ O(n), thegroup of isometries of Rn.crystallographic group = discrete cocompact subgroup of E(n).

Bieberbach group=torsion-free crystallographic group.

Theorem 3.1 (Bieberbach).(A) (structure) Let Π ⊂ Rn ⋊ O(n) be a crystallographic group. Then

Γ = Π ∩ Rn is a lattice of Rn, and Γ has finite index in Π.(B) (Rigidity) Let Π, Π ′ ⊂ Rn ⋊ O(n) be crystallographic groups. Then

every isomorphism θ : Π → Π ′ is a conjugation by an element ofRn ⋊ GL(n, R).

(C) (Finiteness) In each dimension n, there are only finitely many Bieber-bach groups up to isomorphism.

Theorem 3.2. M is a flat Riemannian manifold (i.e., the sectional cur-vature is constant 0) if and only if M is of the form M = Π\Rn for someBieberbach group Π ⊂ E(n). Therefore, π1(Flat manifold)=Bieberbach group.

Corollary 3.3. (A) Every flat Riemannian manifold is finitely coveredby a flat torus.

(B) Homotopy equivalent flat manifolds are affinely diffeomorphic.(C) In each dimension n, there are only finitely many flat manifolds up

to affine diffeomorphism.

3.4. The Bieberbach theorems have been generalized to nilpotent Lie groups,or even to some solvable Lie groups (Auslander, Lee, Raymond, Dekimpe,Kamishima).

Let G be a connected, simply connected nilpotent Lie group, and let Cbe a compact subgroup of Aut(G). almost crystallographic group = discretecocompact subgroup of G ⋊ C.

almost Bieberbach group=torsion-free crystallographic group.Note: If G = Rn, then G ⋊ C = Rn ⋊ O(n) = E(n).

Theorem 3.5 (Genralized Bieberbach theorems).(A) (Auslander) Let Π ⊂ G ⋊ C be an almost crystallographic group.

Then Γ = Π ∩ G is a lattice of G, and Γ has finite index in Π.(B) (Lee-Raymond) Let Π, Π ′ ⊂ G⋊C be almost crystallographic groups.

Then every isomorphism θ : Π → Π ′ is a conjugation by an elementof G ⋊ Aut(G).

12

(C) (Lee-Raymond) Under each nilmanifold Γ\G, there are only finitelymany infra-nilmanifolds essentially covered by Γ\G, up to isomor-phism.

The proof uses group cohomology arguments.

Corollary 3.6. (A) Every infra-nilmanifold is finitely covered by a nil-manifold.

(B) Homotopy equivalent infra-nilmanifolds are affinely diffeomorphic.

A smooth compact manifold M is called almost flat if for any ε > 0 thereis a Riemannian metric gε on M such that diam(M, gε) ≤ 1 and |Kgǫ

| < ε.

Theorem 3.7 (Gromov-Ruh). M is almost flat if and only if it is infranil.(Compare with Theorem 3.2.)

A covering Γ\GΦ\−→ Finite group\(Γ\G) is essential if no element of

covering transformations is homotopic to the identity.

Example 3.8. Consider the 3-dimensional Heisenberg group and its lattices

G =

1 x z0 1 y0 0 1

: x, y, z ∈ R

, Γp =

1 ℓ np

0 1 m0 0 1

: ℓ, m, n ∈ Z

For each p > 0, Γp is a lattice containing Γ as a subgroup of index p. There-

fore, for each p > 0, there is a nilmanifold Γp\G. Note that Γ1\GZp

−−−−→ Γp\Gis not essential even though the spaces are distinct homologically, because

H1(Γp\G; Z) = Zp ⊕ Z2.

Similarly to Example 1.4, there is a holonomy Z2 infra-nilmanifold coveredby Γ1\G. Let

A : G → G

be an automorphism of G mapping

A :

1 x z0 1 y0 0 1

7−→

1 −x z0 1 −y0 0 1

.

Then A2 = id. Now let α = (a, A) ∈ G ⋊ Aut(G), where a =

1 0 12

0 1 00 0 1

.

Then α2 is the pure translation in Γ1 (with (ℓ = m = 0 and n = 1). LetΠ = 〈Γ1, α〉. Then

1 → Γ1 → Π → Z2 → 1

13

is exact. The group Π is torsion-free, and it acts on G freely, properlydiscontinuously, giving rise to an infra-nilmanifold Π\G. This is doublycovered by the nilmanifold Γ1\G.

The center Z of of Π is generated by α2, and

1 → Z → Π → Q → 1

is exact, where Q is again (as in the crystallographic case), Z2 ⋊ Z2.

“Homotopy equivalent infra-nilmanifolds are affinely diffeomorphic” is re-markable.

isometric =⇒ affinely diffeomorphic =⇒ diffeomorphic =⇒ homeomorphic =⇒ h.e.

Since infra-nilmanifolds are K(Π, 1)’s, if their π1’s are the same, then theyare h.e., and hence they are affinely diffeomorphic. [Think of the PoincareConjecture.]

14

4. Homologically injective torus operations

Theorem 4.1 (Conner-Raymond). Let G, a compact connected Lie group,act effectively on a closed aspherical manifold M . Then G is a k-torus andacts injectively; i.e., evx

∗ : π1(G, e) → π1(M, x) is injective (into the center).

There have been many efforts trying to split a manifold as a product oftwo manifolds.

Theorem 4.2 (Calabi, Lawson-Yau, Eberlein). Let M be a closed manifoldsof nonpositive sectional curvature. If π1(M) has nontrivial center Zk, thenM splits as M = T k ×Φ N , where N is a closed manifold of nonpositivesectional curvature and Φ is a finite Abelian group acting diagonally andfreely on the T k-factor as translations.

Theorem 4.3 (conner-Raymond). If a topological space X with H1(X; Z)finitely generated admits a homologically injective (topological) torus action(T k, X), then X splits as T k ×Φ N for some N , where Φ is a finite Abeliangroup acting diagonally and acting freely on the T k-factor as translations.

Definition 4.4. A normal subgroup A of C is said to be homologicallyinjective in C if the inclusion induces an injective homomorphism on thefirst homology, H1(A; Z) → H1(C; Z), or equivalently, A ∩ [C, C] = {1}.

Lemma 4.5. Let Γ be a group whose center Z(Γ) is a free Abelian groupof finite rank. Let 1 → Γ → Π → Q → 1 be an extension whose abstractkernel has finite image. Then the following are equivalent:

(1) [Π] has finite order in H2(Q;Z(Γ));(2) Π contains a normal subgroup Γ × Q′ such that Φ = Π/(Γ × Q′) is

a finite group;(3) Z(Γ) homologically injects into CΠ(Γ).

The Splitting Theorem 4.3 can now be generalized to Seifert fiberingswith typical fiber a compact solvmanifold of type (R).

Theorem 4.6. Let X be an Seifert fibering with typical fiber a compactsolvmanifold Γ\G of type (R). Assume that Π, with finitely generated center,

acts freely on X. Then the following are equivalent:

(1) The abstract kernel, Q → Out(Γ), of the associated exact sequence1 → Γ → Π → Q → 1, Π = π1(X), has finite image in Out(Γ) andthe center of Γ, Z(Γ), homologically injects into CΠ(Γ);

(2) X = (Γ\G) ×Φ X ′, where Φ is a finite group which acts diagonally,as affine maps on the first factor.

Example 4.7. Recall the two examples

1 → Z → Π → Q → 1

where Q =

([00

],

[−1 00 −1

]).

15

(1) Π1 ⊂ R3 ⋊ O(3) = E(3) was defined by

Π1 = 〈(e1, I3), (e2, I3), (e3, I3), α = (a, A)〉

where

e1 =

100

, e2 =

010

, e3 =

001

, a =

1

2e1, A =

1 0 00 −1 00 0 −1

.

Then α2 = (e1, I3).(2) Π2 ⊂ G ⋊ Aut(G) (G=Heisenberg group) was defined by

Π1 = 〈(e1, id), (e2, id), (e3, id), α = (a, A)〉

where

e1 =

1 1 00 1 00 0 1

, e2 =

1 0 00 1 10 0 1

, e3 =

1 0 10 1 00 0 1

, a =

1 0 12

0 1 00 0 1

,

and A : G → G is an automorphism

A :

1 x z0 1 y0 0 1

7−→

1 −x z0 1 −y0 0 1

.

Then α2 = (e3, id).Now both are central extensions of Z by Q = Z2 ⋊ Z2:

1 −−−−→ Z −−−−→ Π1 −−−−→ Z2 ⋊ Z2 −−−−→ 1, Π1 ⊂ Isom(R3)

1 −−−−→ Z −−−−→ Π2 −−−−→ Z2 ⋊ Z2 −−−−→ 1, Π2 ⊂ Isom(G)

However, [Π1] ∈ H2(Z2 ⋊ Z2; Z) has finite order, while [Π2] has infiniteorder. This means that the flat Riemannian manifold Π1\R3 “splits” as

Π1\R3 = S1 ×Z2

T 2

where Z2 = 〈α〉 acts diagonally, as a translation on S1 and as

[−1 00 −1

]on

T 2. However, the infra-nilmanifold Π2\G does not split.

16

5. Applications to fixed-point theory

Let M be a closed manifold, and let f : M → M be a continuous map.The Lefschetz number L(f) of f is defined by

L(f) :=∑

k

(−1)kTrace{(f∗)k : Hk(M ; Q) → Hk(M ; Q)}

To define the Nielsen number N(f) of f , we define an equivalence relationon Fix(f) as follows: For x0, x1 ∈ Fix(f), x0 ∼ x1 if and only if there existsa path c from x0 to x1 such that c is homotopic to f ◦ c relative to theend points. An equivalence class of this relation is called a fixed-point class(FPC) of f . To each FPC F , one can assign an integer ind(f, F ). An FPCF is called essential if ind(f, F ) 6= 0. Now,

N(f) := the number of essential fixed-point classes.

These two numbers give information on the existence of fixed-point sets.If L(f) 6= 0, every self-map of M homotopic to f has a nonempty fixed-pointset. The Nielsen number is a lower bound for the number of componentsof the fixed-point set of all maps homotopic to f . Even though N(f) givesmore information than L(f) does, it is harder to calculate.

Example 5.1. On S1, there is a self-map f which is homotopic to theidentity and fixed-point free.

What happens to S2? Let f : S2 → S2 be a self-map, homotopic to theidentity. (For example, think of the rotation around the axis connecting theNorth-South poles). Since f ≃ id, they induce the same homomorphisms onhomotopy and homology.

H0(S2; Z) = Z, H1(S

2; Z) = 0, H2(S2; Z) = Z,

and f∗ = id∗ is the identity map on each homology group. Thus,

L(f) = 1 − 0 + 1 = 2.

There is always fixed points (Lefschetz Fixed point theorem).

Theorem 5.2 (Kwasik-Lee, McCord). Let f : M → M be a homotopicallyperiodic map on an infra-solvmanifold. Then N(f) = L(f).

Let M = Π\G, G simply connected nilpotent Lie group. f being homo-topically periodic implies f has a homotopy representative (d, D) ∈ G ⋊Aut(G). Then the proof uses

L(f) =1

[Π : Λ]

∑L(αf) =

1

|Φ|

∑

A∈Φ

det(A∗ − D∗)

det A∗,

N(f) =1

[Π : Λ]

∑|N(αf)| =

1

|Φ|

∑

A∈Φ

|det(A∗ − D∗)|,

where the sum ranges over all α ∈ Π/Λ = Φ. Since f is homotopicallyperiodic, D and Φ still forms a finite group. Furthermore,

17

Lemma 5.3. Let B ∈ GL(n, R) with a finite order. Then det(I − B) ≥ 0.

Proof. Since B has finite order, it can be conjugated into the orthogonalgroup O(n). Since all eigenvalues are roots of unity, there exists P ∈GL(n, R) such that PBP−1 is a block diagonal matrix, with each blockbeing a (1 × 1)- or a (2 × 2)-matrix. All (1 × 1)-blocks must be D = [±1],and hence det(I − D) = 0 or 2. For a (2 × 2)-block, it is of the form[

cos t sin t− sin t cos t

]. Consequently, each (2 × 2)-block D has the property that

det(I − D) = (1 − cos t)2 + sin2 t = 2(1 − cos t) ≥ 0. ¤

Since |detA∗| = 1 and det(A∗ − D∗)/ det A∗ = det(I − D∗A−1∗ ) ≥ 0 by

the above Lemma (note: DA−1 has a finite order), we have

det(A∗ − D∗)

det A∗=

∣∣∣∣det(A∗ − D∗)

detA∗

∣∣∣∣ =|det(A∗ − D∗)|

|detA∗|= |det(A∗ − D∗)|.

This proves the theorem.

Example 5.4. Let Π be same as in Example 1.4. Let g = (d, D) be givenby

d =

000

, D =

3 0 00 1 10 1 2

.

Let f : M → M be the map induced from g. There are six conjugacyclasses of g; namely, g and αg, αt1g, αt21g, αt31g, and αt41g. Each class hasexactly one fixed point. Clearly, det(I − D) = +2 and det(I − AD) = −10.Therefore, the first fixed point has index +1 and the rest have index −1.Consequently, L(f) = −4, while N(f) = 6.

For the “Averaging Formula”, one needs even more general version ofBieberbach’s second theorem (Proof by group cohomology), which is niceby itself.

Theorem 5.5. Let S be a connected and simply connected solvable Lie groupof type type(R). Let π, π′ ⊂ Aff(S) be finite extensions of lattices of S. Thenany homomorphism θ : π → π′ is semi-conjugate by an “affine map”. Thatis, for any homomorphism θ : π → π′, there exist d ∈ S and a homomor-phism D : S → S such that θ(α) ◦ (d, D) = (d, D) ◦ α, or the followingdiagram is commutative

S(d,D)−−−−→ S

yα

yθ(α)

S(d,D)−−−−→ S

for all α ∈ π.

Theorem 5.6 (JB-KB Lee). (Averaging Formula) Let M be an infra-solvmanifoldof type type(R) and f : M → M be any self map. Suppose that N is a regular

18

covering of M which is a solvmanifold with fundamental group K. Assumethat f∗(K) ⊂ K. Then

N(f) =1

[π : K]

∑N(f),

where the sum ranges over all the liftings f of f onto N . In particular,N(f) ≥ |L(f)|.

19

6. Polynomial structures for solvmanifolds

John Milnor asked if every torsion-free polycyclic-by-finite group Γ occursas the fundamental group of a compact, complete affinely flat manifold. Thisis equivalent to asking if Γ can act on RK properly as affine motions withΓ\RK compact.

Example 6.1. Let

Γ =

1 ℓ n0 1 m0 0 1

: ℓ, m, n ∈ Z

Then it embeds into Aff(3) as

Γ =

1 −12m 1

2ℓ0 1 00 0 1

,

nℓm

∈ GL(3, R) ⋉ R3

so that it acts on R3 transitively.Note that, this affine map maps

zxy

7−→

z − 12mx + 1

2ℓy + nx + ℓy + m

,

and the entries are polynomials of degree 1. Under this embedding, R3/Γ iscompact.• All affine maps are polynomial maps of degree ≤ 1.

However, Benoist constructed an example of a 10-step nilpotent group Γof Hirsch length 11 which does not admit an affine structure. This examplewas generalized to a family of examples by Burde and Grunewald. Burdeconstructs counterexamples of nilpotency class 9 and Hirsch length 10.

A polynomial diffeomorphism f of Rn is a bijective polynomial transfor-mation of Rn for which the inverse mapping is again polynomial.

P(Rn) = the group of all polynomial diffeomorphisms.Affine diffeomorphisms clearly are polynomial diffeomorphisms of degree

less than or equal to 1; smooth actions could be considered as being poly-nomial of infinite degree.

Example 6.2. Let p, q : R2 → R2 be such that

p(x, y) = (y + 1, x + y2) and q(x, y) = (y − x2 + 2x − 1, x − 1).

Clearly, they are inverse to each other in P(R2).

Definition 6.3. A representation θ : Γ → Aff(RK) which yields a properaction with θ(Γ)\RK compact is called an affine structure on Γ. It is alsocommon to call θ(Γ) an affine crystallographic group (ACG).

Analogously to the affine structure, a representation θ : Γ → P(RK)which yields a proper action with θ(Γ)\RK compact is called a polynomialstructure on Γ; θ(Γ) is called a polynomial crystallographic group.

20

Theorem 6.4 (Dekimpe-Igodt-Lee, Dekimpe-Igodt). Every polycyclic-by-finite group Γ admits a polynomial structure of bounded degree. That is,Γ can act on RK properly as polynomial diffeomorphisms so that Γ\RK iscompact. Moreover, all polynomials involved consist entirely of a boundeddegree.

The construction of this polynomial structure is a special case of an iter-ated Seifert fiber space construction, which can be achieved here because ofa very strong cohomology vanishing theorem.

The vector space P(RK , Rk) has GL(Rk) × P(RK)-module structure, via

∀(g, h) ∈ GL(Rk) × P(RK), ∀p ∈ P(RK , Rk) : (g,h)p = g ◦ p ◦ h−1.

The resulting semidirect product P(RK , Rk) ⋊ (GL(Rk) × P(RK)) embedsinto P(Rk+K) as follows: ∀p ∈ P(RK , Rk), ∀g ∈ GL(Rk), ∀h ∈ P(RK) :

∀x ∈ Rk, ∀y ∈ RK : (p, g, h)(x, y) = (g(x) − p(h(y)), h(y)).

The crux of the construction is the iteration of the following procedure.Let

1 → Zk → Π → Q → 1

be an exact sequence with abstract kernel ϕ : Q → GL(k, R). Let

ρ : Q → P(RK)

be a representation which yields a proper action of Q on RK with Q\RK

compact. We try to find a homomorphism

θ : Π → P(RK , Rk) ⋊ (GL(k, R) × P(RK))

so that the diagram

1 // Zk //

i

²²

Π //

θ

²²

Q //

ϕ×ρ

²²

1

1 // P(RK , Rk) // P(RK , Rk) ⋊ (GL(k, R) × P(RK)) // GL(k, R) × P(RK) // 1,

where i : Zk → Rk ⊂ P(RK , Rk) is the standard translations, is commuta-tive.Note that

P(RK , Rk) ⊂ M(RK , Rk) and P(RK) ⊂ TOP(RK),

and therefore,

P(RK , Rk) ⋊ (GL(k, R) × P(RK))⊂

−−−−→ P(RK+k)

∩

y ∩

y

M(RK , Rk) ⋊ (GL(k, R) × TOP(RK))⊂

−−−−→ TOP(RK+k);

21

Consequently, the problem reduces to showing (this is the crux of the argu-ment)

H2(Q; P(RK , Rk)) = 0. ¤