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ISSN 1364-0380 (on line) 1465-3060 (printed) 779

Geometry & Topology GGGGGGGGG GG

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T TTTTTT

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Volume 8 (2004) 779–830

Published: 20 May 2004

ASD moduli spaces over four–manifolds

with tree-like ends

Tsuyoshi Kato

Department of Mathematics, Kyoto UniversityKyoto 606-8502, Japan

Email: [email protected]

Abstract

In this paper we construct Riemannian metrics and weight functions over Cas-son handles. We show that the corresponding Atiyah–Hitchin–Singer complexesare Fredholm for some class of Casson handles of bounded type. Using these,the Yang–Mills moduli spaces are constructed as finite dimensional smoothmanifolds over Casson handles in the class.

AMS Classification numbers Primary: 57M30, 57R57

Secondary: 14J80

Keywords: Yang–Mills theory, Casson handles

Proposed: Simon Donaldson Received: 16 October 2001

Seconded: Ronald Stern, Robion Kirby Revised: 29 March 2004

c© Geometry & Topology Publications

mailto:[email protected]

780 Tsuyoshi Kato

Introduction

0.A Review of previous works on exotic smooth structures on

open four–manifolds

Four–manifold theory has been deeply developed from two approaches. One isbased on geometry and functional analysis. In particular gauge theory, Yang–Mills theory or Seiberg–Witten theory, construct moduli spaces as the sets ofsolutions of nonlinear PDEs on four–manifolds. They have been discovered tocontain extremely rich information on smooth structure. In particular the con-struction of Donaldson’s invariant uses Yang–Mills moduli spaces. The invari-ant distinguishes many mutually-homeomorphic but non-diffeomorphic pairs ofsmooth four–manifolds. The other is based on topology, in particular Casson–Freedman’s theory. In high dimensional differential topology, Whitney’s trickto remove self-intersections of immersed discs has played one of the most impor-tant roles, but does not work in four dimensions. Casson’s idea was, instead ofremoving them, to increase self-intersections by attaching immersed two han-dles constructively so that all self-intersections were able to be pushed awayto infinity. Freedman verified that these infinite towers made from immersedtwo–handles were all homeomorphic to the standard open two–handle, and theyare called Casson handles. This allowed the complete classification of orientedsimply-connected topological four–manifolds by their intersection forms andKirby–Siebenmann classes [15].

Combination of Casson–Freedman’s theory with gauge theory provided verydeep results on smooth structures on four–manifolds. One of the most im-portant results on exotic R

4s was due to Taubes, who discovered uncountablymany exotic smooth structures on R

4 [27]. In the essential step, he constructedYang–Mills gauge theory on periodic open four–manifolds. Let X be an ori-ented topological four–manifold. X\pt can admit at least one smooth structure[16]. The idea was to verify that X\pt was not able to admit any end-periodicsmooth structure. In fact otherwise it would produce generic Yang–Mills mod-uli spaces over X\pt as smooth manifolds. On the other hand, detailed analysison the moduli spaces verified that it was impossible for such spaces to exist.Thus the smooth structure of the end was sufficiently complicated to obtainuncountably many exotic R

4s.

Geometry & Topology, Volume 8 (2004)

ASD moduli spaces over four–manifolds with tree-like ends 781

0.B Casson handles

Casson handles (CH) can be constructed inside smooth four–manifolds. Onedefines a smooth structure on CH by restriction. In another words, a towerconsists of smoothly-immersed two–handles, and so the tower itself admits asmooth structure. Each building block by an immersed two handle is called akinky handle. Even though any Casson handle is homeomorphic to the standardopen two–handle, many of them are not diffeomorphic to the standard one.In fact even for the simplest Casson handle, CH+ , consisting of a single andpositive kink at each stage, the following is known:

Lemma 0.1 [4, 5] CH+ is exotic in the sense that the attaching circle cannotbound a smoothly embedded disc inside CH+ .

For each signed infinite tree there corresponds a CH as a smooth four–manifold.In this paper we will define a subclass of signed infinite trees which are calledhomogeneous trees of bounded type. These are constructed iteratively by at-taching infinitely many half-periodic trees. A connected infinite subtree of ahomogeneous tree of bounded type is called a tree of bounded type. Noticethat a Casson handle of a tree of bounded type admits a smooth embeddingby another CH of a tree of homogeneously bounded type. In this paper wewill use the term Casson handle of bounded type to refer to any Casson handleconstructed from a signed tree of homogeneously bounded type.

Now any Casson handle of bounded type can be smoothly embedded in CH+

preserving the attaching circles. The above immediately implies the following:

Corollary 0.1 Any CH of bounded type is exotic.

Typically, smooth structures on Casson handles will have a deep effect on thesmooth types of four–manifolds which contain them. Thus Casson handles aresmooth open four–manifolds with the attaching region, and comprise a richclass among open four–manifolds. They will provide highly nontrivial examplesto study in open manifold theory. It is difficult to understand the importantnumerical relationship between the growth of signed trees and the complexity ofsmooth structures on the corresponding Casson handles. In [18] some relationbetween Stein structures and the number of kinks was found.


782 Tsuyoshi Kato

0.C Outline of the article

In this paper we study Yang–Mills gauge theory over Casson handles. Our finalaim will be to measure the complexity of smooth structures on Casson handlesby means of gauge theory. Roughly speaking in order to construct Yang–Millstheory on open four–manifolds, one needs to overcome two steps. The firstis the Fredholm theory of the linearized equation. The other is perturbationtheory, or transversality, where the setting of perturbation is different fromthe closed case (as we will explain below). A general procedure in Yang–Millstheory tells us that in a situation on four–manifolds where the Fredholm theoryis applicable, one can obtain Yang–Mills moduli spaces as generically smoothmanifolds of finite dimension. In order to induce information on smooth four–manifolds from these moduli spaces, they are required to be non-empty undergeneric perturbation, where transversality theory will enter effectively.

In this paper we show that for Casson handles of bounded type, one can imposecomplete Riemannian metrics and weight functions on them so that Fredholmtheory can be done. So we will obtain Yang–Mills moduli spaces on themas generically smooth manifolds of finite dimension. Perturbation theory onCasson handles is at present under development.

Let (S, g) be a Riemannian (possibly open) oriented four–manifold, and E → Sbe an SO(3) bundle. A connection A on E is called anti-self-dual (ASD)if its curvature form satisfies the equation F+

A ≡ FA + ∗FA = 0. Roughlyspeaking when S is closed, the set of ASD connections modulo the gauge groupis the Yang–Mills moduli space, which is generically a smooth manifold of finitedimension. In this case one does not have to take care so much on the underlyingfunction spaces. Any choice among various Sobolev spaces W k(S) (k large)gives the same moduli space.

When S is open, there are no standard choices of function spaces. Let w : S →[0,∞) be a smooth function. Then one obtains weighted Sobolev spaces W k

w(S).

Let us choose an ASD connection A0 , and consider two sets. One is M, theset of ASD connections such that their curvature forms are in L2 . The otheris M(A0) = {A : A is ASD, A − A0 ∈ W k

w}. In general these spaces arevery different and it seems difficult to study the geometry of the former space.In a periodic case, one can see that these give the same moduli space modulogauge transformation, which follows from the exponential decay estimate on thecurvature forms. A standard argument on transversality also works using thedecay estimate. For the case of Casson handles, it seems natural to regard thesespaces M and M(A0) as having mutually different natures. The exoticness of



smooth types of various Casson handles come from how the boundary solid toriare attached. These data are put in the cylindrical direction in our choice ofmetric. So it would be a delicate matter how these moduli spaces behave nearthe ends. The complexity of these moduli spaces will reflect that of the Cassonhandles.

In our class of Casson handles of bounded type, we will show that a Fredholmtheory can be constructed using M(A0). Our main concern here is analysis ofthe Atiyah–Hitchin–Singer complex:

0 −→ W k+1w ((Y, g))

d−→W k

w((Y, g); Λ1)d+

−→W k−1w ((Y, g); Λ2

+) −→ 0.

In this paper we will explicitly construct Riemannian metrics and weight func-tions over Casson handles, so that the above complex is Fredholm and its coho-mology groups are calculable when restricted to our class. This is the Fredholmpart which we mentioned above. One may generalize this to the case withcoefficients in the adjoint bundle Ad(g) of E . We denote by H∗

A′(AHS) thecohomology groups with coefficient at A′ ∈ M.

Suppose the above complex is Fredholm. Then roughly speaking the ASDmoduli space M(A0)/G has a local model H1

A′ = ker d+A′/ im dA′ at A′ . In

fact the moduli space has the structure of a finite-dimensional manifold at A′ ,when dA′ is injective and d+

A′ is surjective. In that case H1A′ is canonically

isomorphic to the tangent space of the moduli space at A′ . These propertiesare well known for closed four–manifolds. A parallel argument also works forthe open case. Once one fixes a ‘base’ L2 ASD connection A0 ∈ M, thentransversality also works for this case. Thus if one takes a generic metric g′

with respect to A0 , then the corresponding moduli space M(A0, g′, w)/G will

have the structure of a finite-dimensional manifold. Notice that these modulispaces are parameterized by A0 ∈ M, and g′ will depend on A0 . This causessome difficulty in perturbation theory.

0.D Main results

Here is the main theorem:

Theorem 0.1 Let S be a smooth oriented open four–manifold constructedby attaching to a zero handle Casson handles of homogeneous trees of boundedtype. Then there exists a complete Riemannian metric of bounded geometry gand an weight function w on S such that one can construct ASD moduli spacesover S as finite dimensional smooth manifolds with respect to (S, g,w).


784 Tsuyoshi Kato

This follows from the next two propositions. In this paper we will introduceadmissibility for a pair of a Riemannian metric and a weight function over anopen four–manifold (1.A). Then we will show the following:

Proposition 0.1 Let S be an open four–manifold. Suppose one can equip anadmissible pair (g,w) over S . Then one can construct ASD moduli spaces overS as finite-dimensional smooth manifolds.

In order to obtain admissible pairs over Casson handles of bounded type, we willuse an iterative method. The main construction in this paper is the following:

Proposition 0.2 Let S be in theorem 0.1. There exist a complete Rieman-nian metric and an weight function on S so that the corresponding Atiyah–Hitchin–Singer (AHS) complex over S becomes Fredholm. Moreover the metricand the weight function over S give an admissible pair.

Let us outline the construction of Riemannian metrics on Casson handles byrestricting to simple cases. Each building block of a Casson handle, namely akinky handle, is diffeomorphic to ♮(S1 × D3) with two attaching regions; oneis a tubular neighborhood of band sums of Whitehead links (this is connectedwith the previous block), and the other is a disjoint union of the standardopen subsets S1 ×D2 in ♯S1 × S2 = ∂(♮S1 ×D3) (this is connected with thenext block). The number of end-connected sums is exactly the number of self-intersections of the immersed two handle. The simplest Casson handles haveS1×D3 as their building blocks. We attach a Casson handle to the zero–handlealong the attaching circle and denote it by S = D4 ∪ CH .

Let us consider a simple Casson handle, say CH(R+), a periodic Casson handleby positive kinks. Unlike the Taubes construction, the building blocks here areopen. In order to make end-connected sums of building blocks isometrically, oneexplicitly equips the metrics on the building blocks. Then the building blockas an open manifold becomes an ‘open cylindrical’ manifold. As in the Taubesconstruction, one connects two attaching regions in a block. The result becomesa cylindrical manifold on which analysis is already well known. By equipping itwith a suitable weight function, one will apply the Fourier–Laplace transformbetween the cylindrical manifold and its periodic cover. By a kind of excision,one obtains a Fredholm AHS complex over S .

This method shows that once one obtains some suitable function spaces on anyopen manifolds, then the Fourier–Laplace transform works on their periodiccovers. We will use this observation iteratively. In general a Casson handle can



be expressed by an infinite tree with one end point and with a sign ± on eachedge. The next simplest Casson handle will be represented as follows. Let R+

be the half-line with the vertices {0, 1, 2, . . . }. We prepare another family ofhalf-lines {Ri

+}i=1,2,... assigned with indices. Then we obtain another infinitetree:

R(2) = R+ ∪i=1,2,... Ri+

where we connect i in R+ with 0 in Ri+ . For example one may assign − on

R+ and + on all {Ri+}i . Then one obtains the corresponding Casson handle

CH(R(2)). In this case the building blocks are diffeomorphic to Σ2 = (S1 ×D3)♮(S1 ×D3) along R+ . Σ2 has three attaching components. One is µ, thetubular neighborhood of the band sum of two Whitehead links as before. Wewill denote the others by µ′ and γ , where these represent a generator of π1(Σ2).In order to apply Fourier–Laplace transform, one takes end-connected sumstwice. Firstly one takes the end-connected sum between µ and µ′ as before. Theresult is an ‘open cylindrical’ manifold, since there still remains one attachingregion, γ . One takes the end-connected sum of this with CH(R+) along γ .In this manner, one obtains another open manifold, (Σ2/µ ∼ µ′)♮γCH(R+).Now we have already two kinds of analytic preparations. One is analysis forcylindrical manifolds, and the other for the half-periodic Casson handle as wehave explained. By a kind of excision argument, one can verify that the AHScomplex on the open manifold is Fredholm. Half of part of its periodic coveringis exactly CH(R(2)). Again by a Fourier–Laplace transform and excision, oneobtains the Fredholm AHS complex over S = D4 ∪ CH(R(2)).

These are simple examples, but the idea works for much more general cases ofCasson handles. One may iterate this construction inductively to more compli-cated Casson handles.

In the case of end-connected sums, the excision argument tells us that just thedifferentials have closed range. In order to see the finite-dimensionality of thecohomology, we will make explicit calculations. This is one point where weuse de Rham cohomology calculations. On this point a parallel argument bySeiberg–Witten theory seems to have some technical difficulty.

0.E Directions for further research

Finally let us indicate some possible developments arising from this kind ofanalysis, assuming perturbation theory. We would like to propose here someproblems on the study of smooth structures on Casson handles.


786 Tsuyoshi Kato

Let us consider algebraic surfaces, say the K3 surface. This decomposes topo-logically as 2| − E8|♯3(S

2 × S2), and contains six Casson handles. One mayguess that it would be impossible to do Yang–Mills gauge theory on Casson han-dles inside the K3 surface, or more generally inside many of algebraic surfaces.One can verify that at least one of Fredholm theory or perturbation theorybreaks down [21]. It seems reasonable to think that the smooth structure onthe Casson handles in K3 will be so complicated that one might not be able todo Fredholm theory. If perturbation theory could work on these, then one willbe able to tell that homogeneous Casson handles inside K3 should grow morethan exponentially. The argument is to construct Yang–Mills theory on Cassonhandles and it will lead to a contradiction by dimension-counting on the modulispaces.

Let us take two Casson handles, CH(T1) and CH(T2) where Ti are the corre-sponding signed trees. When T1 is embedded into T2 , then there is a smoothembedding, CH(T2) → CH(T1) preserving the attaching circles, but one can-not say about converse embeddings of CH(T1) into CH(T2) in general. Theabove argument suggests that any CH of bounded type will not be able toembed into Casson handles inside K3 preserving the attaching circles.

In our situation here, one treats Casson handles whose trees grow polynomially.In fact in our method, one might expect that as Casson handles grow nearexponentially, the continuous spectrum with respect to the AHS complex willapproach zero. It might be possible that even exponential growth is alreadytoo complicated to obtain Fredholm theory. On the other hand one does notknow concretely how the signed trees grow for the case of Casson handles in K3(for this direction, see [3]). In reality, any Casson handles of bounded type canappear in S2 ×S2\pt, and so it would be interesting to study smooth types onS2 × S2\pt arising from Casson handles of bounded type.

Next we will consider another problem. The re-imbedding theorem gives an-other Casson handle inside a six stage tower preserving the attaching circles.Now any Casson handle CH of bounded type can be embedded into one of thesimplest Casson handles CH± ≡ CH(R±), say into CH+ . Let CH+(n) be thenth stage of CH+ . One may consider a question whether CH can be smoothlyembedded into CH+(n) for some large n preserving the attaching circles. Forthis, we would like to outline a possible argument (see [5]). Let (Z, ∂0Z, ∂1Z) bea smoothly non-product h–cobordism with different Donaldson’s polynomialson two boundaries. By using Kirby calculus technique, one may find a decom-position Z = W ∪ U where W is smoothly product and CH+(n) appears inboth the ends, ∂0Z∩W and ∂1Z∩W . If CH could be smoothly embedded intoCH+(n), then one will obtain Yang–Mills moduli spaces over both of ∂iZ ∩W



whose ends are consisted by CH of bounded type. Since these are diffeomorphic,the Donaldson’s invariants over them will have the same numerical value, whichwould contradict the assumption. In the above situation, one would be ableto conclude that there are no smooth embeddings of CH into any finite-stageapproximations CH+(n) preserving the attaching circles.

The author would like to express his gratitude to the Max Planck Institut fortheir hospitality, during his visit. He also would like to express his appreciationof the referees for numerous comments and suggestions.

1 Yang–Mills moduli space

1.A Admissible four–manifolds

In this section, one will construct Yang–Mills moduli spaces over non compactfour–manifolds. We treat open four–manifolds which can admit Riemannianmetrics and weight functions such that they are able to construct a Fredholmcomplex.

Let Y be a non compact smooth four–manifold. Throughout this section, onealways assumes that Y is simply-connected, and simply-connected at infinity.

Let g be a complete Riemannian manifold of bounded geometry and w : Y →[0,∞) be a smooth function. Recall that a complete Riemannian metric is ofbounded geometry, if (1) the injectivity radius is more than a positive constantǫ > 0 at any point, and (2) for any l ≥ 0, supx∈Y |∇

lR| <∞, where ∇ and Rare with respect to g . Throughout this paper, we always assume that g is ofbounded geometry, and |∇l(w)|C0(Y ) <∞ for all l ≥ 1.

Let Λi(Y ) and Λ2+(Y ) be the exterior differentials on i–forms and self dual

2–forms with respect to g , i = 0, 1. Then we have the Atiyah–Hitchin–Singercomplex (without coefficient) as:

0 −→ C∞c (Y ; Λ0)

d−→ C∞

c (Y ; Λ1)d+

−→ C∞c (Y ; Λ2

+) −→ 0

where d+ = (1 + ∗) ◦ d. Using the L2 adjoint operator (here we do not use anyweight functions), we get the next elliptic operator:

P = d+ ⊕ d∗ : C∞c (Y ; Λ1) → C∞

c (Y ; Λ0 ⊕ Λ2+).

Let us introduce weighted Sobolev k–norms on Y by:

|u|W kw = (Σl≤k

∫exp(w)|∇lu|2vol)

1

2 .


788 Tsuyoshi Kato

We denote these Sobolev weighted spaces by W kw or L2

w (for k = 0). Then onehas a complex of bounded maps:

0 −→ W k+2w (Y ; Λ0)

d−→W k+1

w (Y ; Λ1)d+

−→W kw(Y ; Λ2

+) −→ 0

Now let us say that the pair (g,w) is admissible, if the following conditions aresatisfied:

(1) The above consists of a Fredholm complex, namely each differential hasclosed range, and its cohomology group is of finite dimension as a vector spaceover R. We denote their cohomology groups by H∗(AHS), ∗ = 0, 1, 2. Noticethat by the condition, one has L2

w adjoint operators d∗w and (d+)∗w .

(2) Y admits a partition Y = K ∪ Y1 ∪ . . . Yl such that (a) K is a compactsubset and each Yj is an open subset of Y , and (b) let us put D = d∗w ⊕ d+

or d ⊕ (d+)∗w . Then there is a positive constant Ck such that for any u ∈C∞

c (Yj ; Λ∗), ∗ = 1, 2, one has the bound |u|W k+1

w ≤ C|D(u)|W kw .

(3) For any f ∈ C1(Y ) with |df |L2w(Y ) < ∞, there is f ∈ R with f − f ∈

L2w(Y ).

(4) There is a compact subset K ⊂ Y and a homeomorphism Y \K ∼= S3 ×[0,∞).

In sections 2 to 5, we will construct admissible pairs for all Casson handlesconstructed from homogeneous signed trees of bounded type.

Proposition 1.1 Let (g,w) be an admissible pair on Y . Then H0(AHS) =H1(AHS) = 0 and H2(AHS) ≥ k = H2

+(Y ; R).

Proof Clearly H0(AHS) = H1(AHS) = 0 by the admissibility condition (3).We show dimH2(AHS) ≥ k . For this, one takes two steps.

Let H2cp(Y ; R)+ be a subspace of H2

cp(Y ; R) consisting of vectors with∫Yu∧u >

0. Then H2cp(Y ; R)+ is a linear subspace of H2(Y ; R) of dimension k , since

the natural map I : H2cp(Y ; R) → H2(Y ; R) gives an isomorphism.

Now by the above, one has dimH2cp(Y ; R)+ = k . For any element u ∈

C∞(Y ; Λ2), let us denote by u+ the projection to Λ2+ part. Then one defines:

p : H2cp(Y ; R)+ → H2(AHS)

by assigning [u] 7→ [u+]. This map is well defined. We show that p is aninjection.

Suppose [u+] = 0 ∈ H2(AHS). Then one has α ∈ W 1w(Y ; Λ1) such that

u+ = d+(α), or (u − d(α))+ = 0. Let us put v = u − d(α), and take the cup



product 〈v, v〉 =∫Yv∧ v . One can use the Stokes theorem to see

∫Yu∧d(α) =∫

Yd(u ∧ α) = 0 (u is a closed form). Thus one has the equality

∫Yv ∧ v =∫

Yu ∧ u ≥ 0. On the other hand since v does not have self-dual part, one has∫

Yv ∧ v ≤ 0. This shows u = d(α). This implies [u] = 0 ∈ H2

cp(Y ; R)+ since Igives an isomorphism. Thus one concludes dimH2 ≥ k .

Example 1.1 In general, dimH2cp(Y ; R)+ and dimH2(AHS) = ker(d+)∗w do

not coincide. One takes Y = R2×R

2 , where one equips the standard metric onR

2 , and the product one on Y . Let us choose a weight function w on Y . Weshow dimH2(AHS) is nonzero. Let u1 and u2 be 2–forms on R

2 respectively,such that |ui| ≡ 1, i = 1, 2, pointwisely. Then clearly dui = 0, i = 1, 2,and u1 + u2 gives a self-dual 2–form with bounded pointwise norm. Then oneputs v = exp(−w)(u1 + u2) ∈ L2

w(Y ; Λ2+). Moreover v satisfies the equation

d(exp(w)v) = 0. Thus v is a nontrivial element in H2(AHS).

1.B ASD moduli space

Let E → Y be a G–vector bundle (where G = SO(3) or U(1)) such thatexcept on some compact subset K ⊂ Y , E|Y \K is trivial. One denotes thecorresponding principal G–bundle by P . For SO(3)–bundles, E is determinedby w2(E) ∈ H2(Y ; Z2). In later sections, we fix a trivialization of E|Y \K .Thus we fix an SO(3)–bundle with w2 and p1(E). Let A be a smooth connec-tion over E such that except some compact subset on Y , it satisfies the ASDequation:

FA + ∗FA = 0

where FA is the curvature form of A. Let us denote by R(Y ) the set of smoothconnections as above satisfying |FA|L

2(Y ) < ∞. By changing a trivialization,one may assume |a|L2(Y \K) <∞ where A = d+ a on Y \K . Later we alwaysassume this property. If A is the trivial connection except some compact subset,then it is an element in R(Y ).

Lemma 1.1 For A ∈ R(Y ), p1(A) = 14π2

∫Ytr(FA ∧ FA) is an integer.

In order to show this, one uses the following:

Sublemma 1.1 [14] Let D ⊂ Y be a δ–ball with any point as the center,where δ > 0 is sufficiently small. Then there exists another small ǫ > 0 with


790 Tsuyoshi Kato

the following property; suppose |FA|L2(D) < ǫ. Then there exists a gauge

transformation g over D , such that:

supx∈D|∇l(g∗(A) − d)|(x) ≤ Cl|FA|L2(D)

where Cl are constants, independent of A.

Proof of lemma Let us take any small ǫ > 0. Then there exists a compactsubset K ⊂ Y , and a smooth gauge transformation g ∈ C∞(Y ; Aut(E)) suchthat |g∗A − d|W 1(Y \K) < ǫ, where d is the trivial connection (Y is simply-connected at infinity). In order to get g , one needs to use the above sublemmaand the patching argument. We omit to describe the process (see [11]). Onemay assume that K is a compact submanifold with boundary. Let D(K) bea double of K . Then using a cut-off function ϕ around ∂K , one may extendA′ ≡ ϕ(g∗A) over D(K), by putting d over the other side of K ⊂ D(K).By the Chern–Weil theory, p1(A

′) is an integer. On the other hand, sincep1(A) − p1(A

′) may be arbitrarily small as above, one concludes that p1(A) isan integer. This completes the proof.

Let us put Ad(P ) = P ×G G, where G is the Lie algebra of G. (P ×G R3 = E ).

We also denote by G the trivial Lie G–bundle. Then we have the Atiyah–Hitchin–Singer complex (AHS complex) as:

0 −→ C∞c (Y ; Ad(P ))

dA−→ C∞c (Y ; Ad(P ) ⊗ Λ1)

d+

A−→ C∞c (Y ; Ad(P ) ⊗ Λ2

+) −→ 0

where d+A = (1+∗)◦dA . Using the L2

w adjoint operator, we get the next ellipticoperator:

P (A)w = (dA)∗w ⊕ d+A : W k+1

w (Y ; Ad(P ) ⊗ Λ1) →W kw(Y ; Ad(P ) ⊗ (Λ0 ⊕ Λ2

+)).

Let us fix A0 ∈ R(Y ). One defines an affine Hilbert space as:

Ak(P )0 = {A0 + a|a ∈W kw(Y ; AdP ⊗ T ∗Y )}, k ≥ 3.

Let us take g ∈ C1loc(Y ; AutE). By embedding as g ∈ C1

loc(Y ; Hom(E,E)),one may consider ∇A0

g ∈ C0(Y ; Hom(E,E)⊗T ∗Y ). Notice that if g is locallyW 4 , then it is of C1 class. Now one defines the weighted Sobolev gauge group:

Gl(P ) = {h ∈W lloc(Y ; Aut(E))|∇A0

h ∈W l−1w (Y ; Hom(E,E) ⊗ T ∗Y )}, l ≥ 4.

Gl(P )0 = {h ∈W lloc(Y ; Aut(E))|h − id ∈W l−1

w (Y ; Hom(E,E) ⊗ T ∗Y )}.(1.1)

First of all one has the following property of Gl(P ). Suppose one of thefollowing two conditions; (1) G = U(1), or (2) G = SO(3) and A|Y \K ∈W k

w(Y \K; Ad(P ) ⊗ Λ1) where the trivialization of E|Y \K is fixed. Then:



Lemma 1.2 For each h ∈ Gk+1(P ), there exists h ∈ G such that h − h ∈W k

w(Y \K; Aut(E)), where h is a constant gauge transformation.

Proof This follows from Kato’s inequality, |∇A0h| ≥ d|h| almost everywhere,

and admissibility condition (3).

The Lie algebras of Gl(P ) and Gl(P )0 are correspondingly as follows:

gl(P ) = {h ∈W lloc(Y ; Ad(P ))|∇A0

h ∈W l−1w (Y ; Ad(P ) ⊗ T ∗Y )},

gl(P )0 = W lw(Y ; Ad(P )).

(1.2)

Lemma 1.3 (1) Gk+1(P )0 acts on Ak(P )0 by g∗(A0+a) = g−1∇A0g+g−1ag .

(2) Suppose P is a U(1)–bundle with G = U(1). Then Gk+1(P ) acts onAk(P ).

The proof is standard, and we omit it.

Notice that Gk+1(P ) may not act on Ak(P )0 , since A0 may live only in L2(Y ).

Let us define:Mk(P ) = {A ∈ Ak(P )0 : FA + ∗FA = 0}.

Then Gk+1(P )0 acts on Mk(P ). Thus one gets the quotient space:

Mk(P ) = Mk(P )/Gk+1(P )0.

Remark 1.1 Mk(P ) is constructed after choosing a base connection A0 ∈R(Y ) and W k

w Hilbert spaces. It can be shown that this space is in fact inde-pendent of choice of k . However it would definitely depend on choice of A0 (A0

lies only in L2 with respect to the trivialization on the end). Thus one couldexpress the space as:

Mk(P ) = Mk(P,A0).

Let us find a linear space which corresponds to the tangent space of M(P )k .

Lemma 1.4 There exist natural continuous maps:

exp: gk+1(P ), (W k+1w (Y ; Ad(P ))) → Gk+1(P ) (Gk+1(P )0).

Moreover, let us put G′k+1(P )(ǫ) = {g ∈ Gk+1(P )||g − 1|W k+1

w (Y ) ≤ ǫ}. Then

for sufficiently small ǫ > 0, there exists log : Gk+1(P )0(ǫ) → W k+1w (Y ; AdP )

which inverts exp. One has a similar statement for exp |W k+1w (Y ; Ad(P )).


792 Tsuyoshi Kato

We also omit the proof.

Let us take any A ∈ Ak(P )0 , and consider the following continuous map:

dA : gk+1(P ) →W kw(Y ; AdP ⊗ T ∗Y )

by u→ ddt

(exp(tu)A)|t=0 . Locally this map is expressed as u→ du+ [u, a].

Lemma 1.5 Suppose an admissible pair (g,w) is given. Then for the AHScomplex, dA and d+

A are closed operators.

Proof Let us show that dA|Wk+1w (Y ; Ad(P )) has closed range. Let A′

0 ∈Ak(P ) be a smooth connection which coincides with A on some compact subsetK , and is trivial except on another compact subset. Then one has A = A′

0 +a,a ∈W k(Y ; AdP⊗T ∗Y ). Let ϕ be a cut-off function such that for some compactsubsets K ⊂ K ′ ⊂ K ′′ ⊂ Y , one has ϕ|K ′ ≡ 1 and ϕ|(K ′′)c ≡ 0. By theassumption, d : W k+1

w (Y ; Λ0) → W kw(Y ; Λ1) is invertible (satisfies |d(u)|W k

w ≥C|u|W k+1

w ). Then one sees d(1−ϕ)a is invertible by choosing sufficiently large

K ′ ⊂ K ′′ , using the Sobolev embedding, W kloc → C0 . Then one has the

following estimate:

|u|W k+1w ≤ C(|dA(u)|W k

w + |u|W kw)

≤ C(|dA(u)|W kw + |ϕu|W k

w + |(1 − ϕ)u|W kw)


w + |d(1−ϕ)a((1 − ϕ)u)|W kw)


w + |(1 − ϕ)d(1−ϕ)a(u)|W kw+

|[d(1−ϕ)a, (1 − ϕ)]u|W kw)


w + |(1 − ϕ)dA(u)|W kw+

|[d(1−ϕ)a, (1 − ϕ)]u|W kw + |(1 − ϕ)[ϕa +A′

0, u]|Wkw)

≤ C(|dA(u)|W kw + |ψu|W k

w)

(1.3)

where ψ is another compactly-supported cut-off function with ψ|Suppϕ ≡ 1.The last inequality shows that dA is a closed operator (see lemma 3.1).

Next let us consider d+A . Notice that we have now the adjoint operator (dA)∗w .

Then it is enough to see d+A(ker(dA)∗w ∩ W k+1

w ) ⊂ W kw(Y ; Ad(P ) ⊗ Λ2

+) hasclosed range (see lemma 1.6 and lemma 3.1). Let us put D = (dA)∗w ⊕ d+

A .Then using admissibility condition (2) and the above argument, one gets asimilar inequality |u|W k+1

w ≤ C(|D(u)|W kw + |ψu|W k

w) where a cut-off functionψ has compact support. In particular D : W k+1

w →W kw has closed range.



Let us take u ∈W k+1w (Y ; Ad(P ) ⊗ Λ1) with (dA)∗w(u) = 0. Then by the above

two facts, one has a similar inequality:

|u|W k+1w ≤ C(|d+

A(u)|W kw + |ψu|W k

w).

This shows that d+A also has closed range. This completes the proof.

Corollary 1.1 Let A ∈ Ak(P )0 . Then the AHS complex

0 −→W k+1w (Y ; Ad(P ))

dA−→W kw(Y ; Ad(P ) ⊗ Λ1)

d+

A−→W k−1w (Y ; Ad(P ) ⊗ Λ2

+) −→ 0

is a Fredholm complex of index −2p1(P ) + 3(dimH1(AHS) − dimH2(AHS)

).

The index computation uses the excision principle, or relative index theorem[20]. One can also use [2] and the method in section 5 here. One denotes theircohomology groups by H∗

A(AHS), ∗ = 0, 1, 2.

Let us consider the restriction:

dA : W k+1w (Y ; AdP ) →W k

w(Y ; AdP ⊗ T ∗Y ).

By the above corollary, one may consider the adjoint operator:

(dA)∗w : W kw(Y ; AdP ⊗ T ∗Y ) →W k−1

w (Y ; AdP ).

One understands (dA)∗w in the geometric way as follows. Let us take A ∈Ak(P )0 . Then one has the ‘tangent space’ of Ak(P )0/Gk+1(P )0 at [A]:

W kw(Y ; AdP ⊗ T ∗Y )

dA(W k+1w (Y ; AdP ))

∼= ker(dA)∗w : W kw(Y ; AdP ⊗ T ∗Y ) →W k−1

w (Y ; AdP ).

Let us take a smooth family of connections A(t) ∈ Ak(P )0 , t ∈ [0, 1]. Then onehas:

∂F+A(t)

∂t

∣∣∣∣∣t=0

= (dA(0) + ∗dA(0))B ≡ d+A(0)B

where B = dA(t)dt

∣∣∣t=0

∈W kw(Y ; AdP⊗T ∗Y ). Notice that near infinity, FA(t) can

be written as dA(t)+A(t)∧A(t). Each A(t) can be written as A(t) = A0 +at ,where at ∈ W k

w(Y ; AdP ⊗ Λ1). Since F+A0

= 0, one sees that F+At

is a smooth

family in W k−1w (Y ; AdP ⊗Λ2

+). Now one gets the ‘tangent space’ of Mk(P ) at[A] as:

ker (dA)∗w ⊕ d+A : W k

w(Y ; AdP ⊗ T ∗Y ) →W k−1w (Y ; AdP ⊗ (Λ0 ⊕ Λ2

+)).

In particular this space is isomorphic to H1A(AHS) which is of finite dimension.


794 Tsuyoshi Kato

Lemma 1.6 ker dA : W k+1w (Y ; Ad(P )) →W k

w(Y ; Ad(P ) ⊗ Λ1) is zero.

Proof This follows from Kato’s inequality, |∇Au| ≥ |d|u|| almost everywhere.Notice that there are also no L2 functions u with dA(u) = 0. This is alsoseen as follows: Suppose u ∈ L2(Y ; Ad(P )), |u|L2 = 1, satisfies dA(u) = 0.Then for g = exp(u) ∈ G, one has g∗(A) = A. Then it is known that forany path l between p and q in Y , one has Pl ◦ g(p) = g(q) ◦ Pl , where Pl

is the parallel translation along l . Let q → ∞. Then g(q) approaches tothe identity. On the other hand since Pl holds the inner product, one has〈Pl(g(p)(u)), g(q)(Pl(u))〉q ∼ 〈g(p)(u), u〉p . This shows that g(p) is near theidentity. This impossible. This completes the proof.

Let us consider Mk(P ) = Mk(P )0/Gk+1(P )0 , and show that under some as-sumptions, this space is a finite-dimensional smooth manifold. For A ∈ Ak(P )0 ,let us put:

CGkA = {A′ ∈ Ak(P )0|(dA)∗w(A′ −A) = 0}.

I : CGkA × Gk+1(P )0 → Ak(P )0; (A, g) 7→ g∗A.

Let us calculate dI at (A, id). If u ∈W k+1w (Y ; Ad(P )) and v ∈W k

w(Y ; Ad(P )⊗Λ1), one has:

d

dt(exp tu)∗(A+ tv)|t=0 = dAu+ v.

Thus dI is given by:

dI = 1 ⊕ dA : ker(dA)∗w∩Wkw(Y ; AdP ⊗ T ∗Y ) ×W k+1

w (Y ; AdP )

→W kw(Y ; AdP ⊗ T ∗Y ).

(1.4)

Since dA has closed range, it is clear that the above map is an isomorphism atevery (A, id).

Corollary 1.2 There are neighborhoods U ⊂ Ak(P )0 of A and V ⊂ Gk+1(P )0of id such that:

I : U ∩ CGkA × V → Ak(P )0

is a homeomorphism into its image. By restriction,

I : U ∩ CGkA(P ) ∩ Mk(P )0 × V → Mk(P )0

is also a homeomorphism into Mk(P )0 .

Now one has the following:



Proposition 1.2 Suppose d+A : W k

w(Y ; AdP ⊗ T ∗Y ) →W k−1w (Y ; AdP ⊗Λ2

+)

is a surjection for A ∈ Mk(P )0 . Then Mk(P ) = Mk(P )0/Gk+1(P )0 is a finite-dimensional smooth manifold at [A]. Its tangent space is naturally isomorphicto ker d+

A/ im dA .

Proof This follows from the general facts on group actions [14, page 48]. Fromthe above corollary, it is enough to verify that the above I gives a slice for theprojection:

Mk(P )0 → Mk(P ) = Mk(P )0/Gk+1(P )0.

These can be verified by bootstrapping as [14]. We omit to write it.

1.C Perturbation of Riemannian metrics

In order to construct smooth moduli spaces, one uses K Uhlenbeck’s genericmetric theorem. Let (g,w) be an admissible pair for Y . Let us choose asmooth map h : Y → [0,∞) with h(x) ≥ w

2 (x). Then one introduces thefollowing Banach manifold:

C = {φ ∈ C l(Gl(TM)) : lim supK(Σlj=0e

h∣∣∇j(φ∗g − g)

∣∣K = 0,K compact}.

Let us take φ ∈ C, and put g′ = φ∗g . Then one has the AHS complex withrespect to g′ :

0 −→W k+1w ((Y, g′); Λ0)

d−→W k

w((Y, g′); Λ1)d+

−→W k−1w ((Y, g′); Λ2

+) −→ 0.

One sees that this is a Fredholm complex with the same index as the unper-turbed one. Our aim in this section is to show the following:

Proposition 1.3 Suppose Y is indefinite. Then by a small perturbation ofthe Riemannian metric, one has no orbit of reducible connections in Mk(P ).In particular Mk(P ) is a smooth manifold.

In order to verify this, we follow [14]. Let us take any L2 ASD connection A0

over Y , and consider Ak(P )0 = {A0 + a|a ∈ W kw(Y ; Ad(P ) ⊗ Λ1). Then one

introduces a Gk+1(P )0 equivariant map:

P+ : Ak(P )0 × C →W k−1w ((Y, g); Λ2

+)

by P+(A,φ) = P+(g)(φ−1(FA)), where P+(g) is the projection to the self dualpart with respect to g . Notice that this map is well-defined since A0 is L2 ASDwith respect to g . Let us put Mk(P ) = P−1

+ (0).


796 Tsuyoshi Kato

Proposition 1.4 [14] Mk(P ) ∩ A∗k(P ) × C/Gk+1(P )0 is a smooth Banach

manifold.

Proof We sketch its proof. First we see that dP+ is surjective at any (A,ϕ)with P+(A,ϕ) = 0. Then it follows that Mk(P ) is a Banach manifold on whichGk+1(P )0 acts. Then as before by making a slice for the action, one gets theresult.

Let c be the Lie algebra of C. Then dP+ splits as:

dP+ = d1P+ ⊕ d2P+ : W kw(Y ; Ad(P ) ⊗ Λ1) ⊕ c →W k−1

w (Y ; Ad(P ) ⊗ Λ2+),

where d1P+(α) = P+(g)(ϕ−1(dA(α)))|(u, ϕ), and d2P+(r) = P+((ϕ−1)∗(r∗F )).We show that the differential of P+ is surjective – notice that

ker dA ∩W k+1w (Y ; Ad(P )) = 0.

Let us consider the AHS complex:

0 −→ W k+1w (Ad(P ))

dA−→W kw(Ad(P ) ⊗ Λ1)

d+

A−→W k−1w (Ad(P ) ⊗ Λ2

+) −→ 0.

Since this is Fredholm, one sees dP+ has finite codimension. Let us take arepresentative u ∈ coker dP+ with (d+

A)∗(u) = 0. Then one has (d+A)∗(ewu) = 0.

Then one has the equations, dA(FA) = d∗A(FA) = dA(v) = d∗A(v) = 0, wherev = ϕ∗(ewu). Then the same argument as [14, page 56] shows that on an opendense subset of Y , FA can be expressed as α⊗ a ∈ Λ2

+ ⊗ Ad(P ), with |a| = 1(pointwise norm) and dA(a) = 0. By the irreducibility, it follows a = 0, whichcontradicts to non triviality of p1(P ). This completes the proof.

Proposition 1.5 Let us fix A0 , an L2 ASD connection with respect to g .Then for a Baire set of φ ∈ C, there are no reducible connections in Ak(P )0with respect to φ∗(g).

Proof Suppose A0 is reducible and denote the corresponding U(1)–connectionby the same A0 . Let P be a G = U(1)–bundle. Then, similarly to before, oneputs:

Ak(P )0 = {A0 + a|a ∈W kw(Y ; Λ1)},

Gk+1(P )0 = {h ∈W k+1loc (Y ; Aut(E))|h − id ∈W k+1

w (Y ;T ∗Y ⊗ C)}.(1.5)

Then Gk+1(P )0 acts on Ak(P )0 . Now, as above, one considers P+ : Ak(P )0 ×C →W k−1

w ((Y, g′); Λ2+). As [14], if dP+ is not surjective, then it follows FA0

=0. This shows that dP+ is surjective.



Now one has a smooth Banach manifold Mk(P ) = P−1+ (0)/Gk+1(P )0 . Let

us consider a Fredholm map between Banach manifolds π : Mk(P ) → C. ItsFredholm index is the one of the following:

0 −→W k+1w (Y ; Λ0)

d−→W k

w(Y ; Λ1)d+

−→W k−1w (Y ; Λ2

+) −→ 0.

One has H0(AHS) = H1(AHS) = 0, and H2(AHS) ≥ b2+ . This is also the casewhen one perturbs the Riemannian metrics slightly. In particular, if b2+ > 0,then one has dimH1 − dimH2 < 0. Then the Sard–Smale theorem showsthat for a Baire set of C, there are no L2 ASD connections over nontrivial linebundles. This shows that there are no orbits of reducible ASD connections inMk(P ), for a Baire set of C. This completes the proof.

Let us put the projection π : Mk(P ) → C. Then the direct application of [14]to this case shows:

Corollary 1.3 [14] Suppose b2+ > 0. Then a Baire set of φ ∈ C exists suchthat Mk(φ) ≡ π−1(φ) are smooth finite dimensional manifolds.

2 A complete Riemannian metric on int W0

Roughly speaking the set of the anti-self-dual connections modulo equivalenceforms a finite-dimensional manifold, in the case when the base four–manifold isclosed. Then dimension is equal to the Fredholm index of the elliptic complex,the Atiyah–Hitchin–Singer complex. One may make a Fredholm operator for afour–manifold with boundary, using suitable complete Riemannian metrics onits interior. In this case one uses the weighted Sobolev spaces on the ellipticoperator. For our purpose this is a basic construction, and later we will useinfinitely many such open Riemannian manifolds. In fact by gluing out all ofthem iteratively, we will get another complete Riemannian manifold such thatthe Atiyah–Hitchin–Singer complex becomes Fredholm between the weightedSobolev spaces. The aim in later sections is to construct admissible pairs overCasson handles of bounded type.

2.A A specified embedding of W0 into RN

Let W0 be a compact manifold with boundary of arbitrary dimension n. Letus choose compact submanifolds with boundary M1,N0 ⊂ ∂W0 of the samedimension, with empty intersection, M1 ∩ N0 = ∅. Suppose M1 and N0 are


798 Tsuyoshi Kato

diffeomorphic to each other, and denote W0 = intW0 ∪M1 ∪N0 . Let us takea countable number of W0 and index them by W i

0 = W0 , i ∈ Z. Then one

constructs an open manifold Y by ∪W i0 identifying M i

1 with N i+10 :

Y = . . . W i0 ∪M i

1∼N i+1

0

W i+10 ∪ . . .

There is a natural smooth Z–action on Y , and we call Y a periodic open

manifold. There are obvious notions of periodic vector bundles over Y , andperiodic differential operators over E .

In order to define (weighted) Sobolev spaces over Y we need, first of all, acomplete Riemannian metric on intW0 which extends to the one on Y . Werecall an explicit picture of end-connected sums along M1 and N0 . Let uschoose smooth embeddings:

γ(γ′) : M1(N0) × [0,∞) → ∂W0 × [0,∞) ⊂ intW0.

Then we have the following equality (at this stage it is C∞ sense, by smoothingcorners):

intW0♮ intW0 = intW0\ im γ ∪ intW0\ im γ′

where we identify ∂M1 × [0,∞) ∪ M1 × {0} with ∂N0 × [0,∞) ∪ N0 × {0}.In order to make this construction compatible with the metric on intW0 , wechoose a particular style of Riemannian metric on int W0 as follows. Let uschoose an embedding:

I : W0 → RN+2

with the following six properties:

(1) Let us choose a small neighborhood of ∂W0 in W0 and identify it with∂W0 × [0, ǫ] ≡ W0(ǫ), ∂W0 = ∂W0 × {0}. Then for any τ ∈ [0, ǫ], we have thefollowing:

W0 ∩ RN+1 × [0, ǫ] = W0(ǫ), W0(ǫ) ∩ ∂W0 × τ ⊂ R

N+1 × τ.

(2) For a smaller ǫ′ < ǫ, p({W0(ǫ′) ∩ ∂W0 × τ}) ⊂ R

N+1 is independent ofτ ∈ [0, ǫ′], where p : R

N+2 → RN+1 is the obvious projection.

(3) I|W0\W0(ǫ) is a C∞ embedding (ie without corners).

(4) Let M1 and N0 be tubular neighborhoods of M1 and N0 in ∂W0 re-spectively. As before we identify small neighborhoods of ∂M1 and ∂N0 with∂M1 × (0, δ] ≡M1(δ) and ∂N0 × (0, δ] ≡ N0(δ) respectively. Then we have thefollowing:

(a) M1 and N0 are diffeomorphic to M1 and N0 respectively,



(b) ∂W0 ∩ RN × {0} = M1, ∂W0 ∩ R

N × {1} = N0 ,

(c) for 0 < τ < δ , M1(δ)∩RN×τ and N0(δ)∩R

N×{1−τ} are diffeomorphicto ∂M1 and ∂N0 respectively.

(5) W0(ǫ) ∩ RN × τ × [0, ǫ] and W0(ǫ) ∩ R

N × {1 − τ} × [0, ǫ], τ ∈ (0, δ], arediffeomorphic to ∂M1 × [0, ǫ] ∪M1 and ∂N0 × [0, ǫ] ∪N0 respectively.

Moreover W0(ǫ) ∩ RN × τ × [0, ǫ] ⊂ R

N × {0} × [0, ǫ] and W0(ǫ) ∩ RN × {1 −

τ} × [0, ǫ] ⊂ RN × {0} × [0, ǫ] are independent of τ .

(6) Let us denote M1(τ) = W0(ǫ) ∩ RN × {τ} × [0, ǫ] = {(a, b, c) ∈ M1(δ) ∩

RN × R × R}. We have similar notations for N0 . Then we have N0(τ) =

M♯1(τ) ≡ {(a, 1 − b, c)|(a, b, c) ∈M1(τ)}, τ ∈ (0, δ].

Let us denote W ′0 = W0\W0 ∩ R

N × {0, 1} × [0, ǫ]. Then I|W ′0 is a smooth

embedding with corners. Let us denote by g0 the Riemannian metric on W ′0

induced from the standard one on RN+2 through I . Notice important proper-

ties of g0 ; for ǫ′ < ǫ:

g0|W0(ǫ′) = (g0|∂W0) × dy2, g0|M 1((0, δ]) ∩W0(ǫ

′) = (g0|∂M1) × dx2 × dy2

where we have the coordinates (t, x, y) ∈ RN × R × R. We have a similar

expression for g0|N0(δ).

Now we define the desired Riemannian metric on intW0 as follows. Let us put:

W0 = W ′0 ∪ [∂W0\{M1 ∪N0}] × (−∞, 0] ∪

[∂M1 × (−∞, ǫ] ∪ M1 × {ǫ}] × (−∞, 0] ∪

[∂N0 × (−∞, ǫ] ∪ N0 × {ǫ}] × [1,∞).

Clearly W0 is diffeomorphic to intW0 . On the other hand, by the above re-mark, there is a natural extension of g0 to g|W0 . This is the desired completeRiemannian metric. Later we will use the following notations. Let us chooselarge t≫ 0 and s≫ 0. Then we denote:

W0(t) = [∂W0\{M1 ∪N0}] × {−t}

∪ ∂M1 × {−t} × (−∞, 0]

∪ ∂N0 × {−t} × [1,∞), (2.1)

M1(s) = {∂M1 × (−∞, ǫ] ∪M1 × {ǫ}} × {−s}.

We denote M1([S0, S1]) = ∪s∈[S0,S1]M1(s). We have a similar notation N0(s).


800 Tsuyoshi Kato

2.B Casson handles as Riemannian manifolds

Let us generalize the previous construction over the triple (W0,M1,N0) as fol-lows. Suppose we have n+ 1 submanifolds M0,M1, . . . ,Mn in ∂W0 such thatthey are disjoint from each other, Mi∩Mj = ∅. Then by choosing an appropri-ate embedding I : W0 → R

N , one can construct a complete Riemannian metricon intW0 as 2.A. Then one expresses this as Wn :

Wn = W ′0 ∪ [∂W0\ ∪i Mi] × (−∞, 0] ∪i [∂Mi × (−∞, ǫ] ∪Mi × {ǫ}] × (−∞, 0].

Let Σ be a Riemann surface of genus g ≥ 1, and Σ be its solid Riemann surface.Thus Σ is a three–manifold with boundary Σ. Let us put Wg = Σ× [0, 1], andtake g embeddings of S1 × D2 into ∂Wg which represents a generating setof π1(Wg). We denote their images by M1, . . . ,Mg . One may assume thatall Mi and Mj are disjoint. A kinky handle with k kinks is expressed as(Wk, N0,M1, . . . ,Mk), where N0 is another embedding of S1 ×D2 into ∂Wk .The isotopy type of N0 is determined by the sign (±, · · · ±) (k times). Thereare exactly k+1 diffeomorphism types of kinky handles with k kinks, since thediffeomorphism type of the kinky handle is invariant under permutation of thesigns.

A Casson handle is constructed by taking end connected sums of kinky handlesinfinitely many times. It is expressed by an infinite tree (embedded in R

2 ) withadditional information. Let us introduce the following:

Definition 2.1 Let T be an infinite tree with only one end point ∗. T is saidto be a signed tree if every edge e is assigned one of + or −.

Let n(v) + 1 be an integer which is the number of edges of T with commonend-point v . A signed tree T is said to be of bounded multiplicity if there existsa constant C with C ≥ n(v) for any v ∈ T .

Any tree with one end-point ∗ admits a natural distance from ∗. One willdenote it by | |. Let T be a signed tree with one end-point ∗. Let us takeany vertex v ∈ T . Then there exists a unique edge e on T such that one of theend-point of e is v , and the other v′ satisfies |v′| < |v|. To specify this e, one

denotes it as e(v). Let us denote finite subtrees by Tv = ∪n(v)j=0 ej , where all the

edges ej has v as one end-point, and e0 = e(v). Recall that each ej is assignedwith ±. Since ej , 1 ≤ j ≤ n(v), are all assigned one of ±, the set {e1, . . . , en(v)}determines an isotopy class of smooth embedding N0

∼= S1 × D2 → ∂Wn(v) ,

where Wn(v) is diffeomorphic to Σn(v) × [0, 1] and e(v) corresponds to N0 .This class determines a diffeomorphism type of a kinky handle. Thus every



Tv corresponds to data (Wn(v),N0,M1, . . . ,Mn(v)), a kinky handle with n(v)kinks as described above. In particular one may assign a complete Riemannianmanifold Wv for every vertex v ∈ T .

Let us take an edge e with end-points v and v′ , and assign a large numberS ≫ 0. Then one can make end-connected sums of Wv with Wv′ as:

Wv′♮e,SWv ≡ Wv′\Me([S,∞))♮Wv\Nv([S,∞)).

One will also denote the set (T, S) by T , and also call it a signed tree. For every

e, one may connect Wv and Wv′ as above. By iterating these end-connectedsums for every edge, one gets a complete Riemannian four–manifold CH(T ).The diffeomorphism type of this space, is the Casson handle corresponding toT .

We denote (W0, N0,M1) as the simplest kinky handle. Then Y (S) is a periodicRiemannian four–space:

Y (S) = ∪j∈ZWj0 ([−S, S]), CH(R+) = Y (S)0 = ∪j∈NW

j0 ([−S, S])

where W j0 ([−S, S]) = W j

0 \N0([S,∞)) ∪M1([S,∞)), and one enumerates the

same W0 by W j0 , j ∈ Z. Even though CH(R+) has very complicated smooth

structure, Y (S) is diffeomorphic to R4 , and so it would not appropriate to

write it as CH(R).

Let (Wl, N0,M1, . . . ,Ml) be a kinky handle with l kinks. Then one defines thefollowing another Riemannian manifolds. Let us choose any 1 ≤ j ≤ l , sayj = 1. Now one defines:

V (S, l)0 = Wl\N0([S,∞)) ∪lk=1 Mk([S,∞)),

Y (S, l, 1)0 = V (S, l)0 ∪lj=2 ∪Mj(S)∼N0(S)Y (S)0\N0([S,∞)),

Y (S, l, 1) = Y (S, l, 1)0/N0(S) ∼M1(S),

Y (S, l, 1)0 = ∪j∈N ∪Mj1(S)∼N

j+1

0(S)

Y (S, l, 1)j0,

Y (S, l, 1) = ∪j∈Z ∪Mj1(S)∼N

j+1

0(S)

Y (S, l, 1)j0.

(2.2)

There is an infinite signed tree with a base point (Tl,1)0 such that Y (S, l, 1)0 isdiffeomorphic to CH((Tl,1)0).

Let us take l half-lines R+, . . .R+ (l times) where on each R+ the same signis given. Signs on two different half-lines may be mutually different. Then oneidentifies all R+ at 0. The result is a connected signed infinite tree with a basepoint. One denotes it by T l

0 . (Tl,1)0 can be expressed as R+ ∪j∈N (T l−10 )j ,


802 Tsuyoshi Kato

where (T l−10 )j are the same (T l−1

0 ) indexed by j ∈ N, and on R+ the samesign is given.

Let us define:

Y (S, l,m, 1)0 ≡ V (S, l)0 ∪lj=2 ∪M

j1(S)∼N1

0(S)Y (S,m, 1)0,

Y (S, l,m, 1) = Y (S, l,m, 1)0/N0(S) ∼M1(S),

Y (S, l,m, 1)0 = ∪j∈N ∪Mj1(S)∼N

j+1

0(S)

Y (S, l,m, 1)j0,

Y (S, l,m, 1) = ∪j∈Z ∪Mj1(S)∼N

j+1

0(S)

Y (S, l,m, 1)j0.

(2.3)

Similarly one has Y (S, l,m, 1)0 = CH((Tl,m,1)0). Let us take signed infinitetrees (Tm,1)0, . . . , (Tm,1)0 (l times). Then one identifies all (Tm,1)0 at 0. Theresult is a connected signed infinite tree with a base point. One denotes it by(Tm,1)

l0 . As above, (Tl,m,1)0 can be expressed as R+ ∪j∈N [(Tm,1)

l−10 ]j , where

[(Tm,1)l−10 ]j are the same (Tm,1)

l−10 indexed by j ∈ N, and on R+ the same

sign is given. Similarly one uses a notation (Tl,m,1)k0 , which one identifies k

(Tl,m,1)0 at 0. Then using this, one can obtain (Tk,l,m,1)0 by a similar method.

By iteration, one obtains Y (S, l,m, n, 1) and Y (S, l,m, n, 1)0 = CH((Tl,m,n,1)0)as above. Inductively one can obtain CH((Tn1,...,nk,1)0) by iteration. One calls(Tn1,...,nk,1)0 a homogeneous tree of bounded type.

Let n = {n1, n2, . . . , nl, . . . } be an infinite set of positive integers. One can iter-ate the previous construction infinitely many times. Then one gets a completeRiemannian manifold:

CH((Tn)0) = CH((Tn1,n2,...)0)

where we call (Tn)0 a homogeneous tree. If there is a bound C with nj ≤ Cfor all j , then we call it a homogeneous tree of bounded multiplicity.

3 Some properties of elliptic operators over W0

3.A Spectral decomposition

Throughout this paper, one uses the following lemmas. Let P be an order–1 el-liptic differential operator over a complete Riemannian manifold W of boundedgeometry. Let w be a weight function over W and weighted Sobolev k–normson W by |u|W k

w = (Σl≤k


1

2 . Thus one has a bounded mapP : W k+1

w (W ) → W kw(W ). Let us denote by P ∗

w the formal adjoint operatorwith respect to the weighted L2

w inner product.



Lemma 3.1 Suppose Spec∆ = P ∗w ◦P is contained in [λ2,∞) for some λ > 0.

Then P : W k+1w →W k

w has closed range.

Similarly P ∗w : W k+1

w → W kw also has closed range.

Proof Let us consider the first statement. One verifies this by induction. Letk = 0 and let {ui} be a sequence in W 1

w with |P (ui)|L2w → 0. Since one has

the estimate |P (u)|L2w ≥ λ|u|L2

w , one sees |ui|L2w → 0. Moreover by the elliptic

estimate:|u|W k+1

w ≤ Ck(|P (u)|W kw + |u|W k

w), k ≥ 0

for some constants Ck , one concludes |ui|W1w → 0. This shows the result for

k = 0.

We show by induction that if a sequence {ui} in W k+1w satisfies |P (ui)|W

kw → 0,

then |ui|Wkw → 0. Then by the elliptic estimate, one gets the conclusion.

Suppose the result holds for k ≤ k0 , and take a sequence {ui} in W k0+2w with

|P (ui)|Wk0+1w → 0. Then since |P (ui)|W

k0w also converges to 0, one knows that

|ui|Wk0w converges to 0 by the induction hypothesis. By the elliptic estimate,

one gets the result for k0 + 1. This completes the proof of the first statement.

Next let us consider the second statement. Let H⊥ ⊂ L2w be a closed subspace

which consists of the orthogonal complement of H ≡ P (W 1w) ⊂ L2

w . Notice thefollowing relation:

∆(W k+2w ) = P ∗

w(H ∩W k+1w ) ⊂W k

w.

Now let {vi} be a sequence in H ∩W 1w with |P ∗

w(vi)|L2w → 0. Since one may

express vi = P (ui) for ui ∈ W 2w , one has |∆(ui)|L

2w → 0. Thus one has

|ui|L2w → 0. Then by Cauchy–Schwartz, one has the estimate |P (ui)|

2L2w ≤

|∆(ui)|L2w|ui|L

2w . This shows |vi|L

2w → 0. By the elliptic estimate, one sees

|vi|W1w → 0. Then by proceeding similarly as above, one verifies the conclusion

by induction. This completes the proof.

Let W and w be as above, and consider an elliptic complex:

0 −→ E0d0−→ E1

d1−→ E2 −→ 0

over vector bundles on W .

Lemma 3.2 (1) Suppose d0 : W k+1w (E0) → W k

w(E1) has closed range, andker d0 = 0. Then there exists a constant C such that:

|(d0)∗w ◦ d0(u)|W

kw(E0) ≥ C|u|W k+2

w (E0)


804 Tsuyoshi Kato

for any u. In particular (d0)∗w ◦ d0 gives an isomorphism.

(2) Suppose the corresponding bounded complex:

0 −→W k+2w (E0)

d0−→W k+1w (E1)

d1−→ W kw(E2) −→ 0

satisfies the following; (1) d0 and d1 have closed range, (2) the first cohomologygroup ker d1/ im d0 = 0, and (3) ker d0 = 0.

Then for P = (d0)∗w ⊕ d1 and ∆ = P ∗

w ◦ P , one has Spec∆ ⊂ [λ,∞) for somepositive constant λ > 0. Moreover (d1)

∗w has closed range.

Proof (1) Suppose the contrary, and take a sequence {ui} ⊂W 2w with |(d0)

∗w◦

d0(ui)|L2w → 0. Then by Cauchy–Schwartz one has a convergent sequence

|d0(ui)|L2w → 0. Then by the elliptic estimate, one has |d0(ui)|W

1w → 0. By

closedness and the open mapping theorem, this shows existence of non-trivialkernel. This is a contradiction. The rest of the proof follows by induction.

Part (2) follows from (1) and lemma 3.1. This completes the proof.

Let us use the above notations. Let us put H = L2w(E1) ∩ (ker d1)

⊥ . Let usconsider restriction of ∆ = (d1)

∗w ◦ d1|H .

Corollary 3.1 Suppose d0 and d1 has closed range as above. Then Spec∆|His contained in [λ,∞) for some positive λ > 0.

3.B Analysis over W0

Let us take the complete Riemannian manifold W0 in 2.A. Let us take a smoothfunction w : W0 → [0,∞). Then as before one has weighted Sobolev k norms

on W0 by |u|W kw = (Σl≤k


1

2 . Then we have the Atiyah–Hitchin–Singer complex (AHS complex) as:

0 −→W k+2w (W0; Λ

0)d

−→W k+1w (W0; Λ

1)d+−→ W k

w(W0; Λ2+) −→ 0

where d+ = (1 + ∗) ◦ d.

Let M be a complete Riemannian 3 manifold without boundary, but possiblybe non-compact. We have in mind M = W0(t),N0(s) or M1(s). Let g be aRiemannian metric on M , and denote also by g the product metric on M ×R.Let Λ1(M × R) and Λ2

+(M × R) be the exterior differentials on 1 forms andself-dual 2–forms with respect to g . Then we have the natural identification:

Λ1(M × R) = p∗(Λ1(M)) ⊕ p∗(Λ0(M)),



Λ2+(M × R) = p∗(Λ1(M))

where p : M × R →M is the projection. The isomorphisms are given by:

u+ v dt ↔ (u, v), ∗Mu+ u ∧ dt ↔ u.

Let us put X = M × R. Using the L2 adjoint operator (here we do not useany weight functions), we get the elliptic operator P = d∗ ⊕ d+ : Λ1(X) →Λ0(X) ⊕ Λ2

+(X). Using the above identification, this is expressed as:

P = d∗ ⊕ d+ : p∗(Λ0(M) ⊕ Λ1(M)) → p∗(Λ0(M) ⊕ Λ1(M)).

Let us use t as the coordinate on R. Then by a straightforward calculation,one has the following expression:

P = −d

dt+

(∗Md dd∗ 0

)≡ −

d

dt+Q.

Notice that Q is an elliptic self adjoint differential operator on L2(M ; Λ0(M)⊕Λ1(M)). Notice that if M is closed, then Q is not invertible (one has constantfunctions). When M is non compact, 0 may be contained in the spectrum ofQ, even if there are no kernels for Q.

Let us introduce a weight function τ on X as follows. Let M be possiblynon-compact as above. Let us choose a positive number δ > 0 which willbe specified later, and choose a smooth proper function τ ′ : R → [0,∞) withτ ′|R\[−1, 1](t) = δ|t|. Then we define τ : M × R → [0,∞) by τ(m, t) ≡ τ ′(t).Let us introduce weighted Sobolev k norms on X by:

|u|W kτ =

(Σl≤k

∫exp(τ)|∇lu|2vol

) 1

2

.

We denote these Sobolev weighted spaces by W kτ or L2

τ (for k = 0). Then onedefines an isometry:

I : L2(X, p∗(Λ0 ⊕ Λ1)) → L2τ (X, p

∗(Λ0 ⊕ Λ1))

by I(u) = exp(− τ2 )u. Then one has the equality:

I−1PI = P +1

2

dτ

dt: W k+1(X; p∗(Λ0 ⊕ Λ1)) →W k(X; p∗(Λ0 ⊕ Λ1)).

Let d∗τ be the L2τ adjoint operator, i.e. 〈u, d(v)〉|L2

τ = 〈d∗τ (u), v〉|L2τ , and put

Pτ = d∗τ ⊕ d+ . Then we have the following expression:

I−1PτI = −d

dt+

(∗Md d

d∗ −dτdt

)+

1

2

dτ

dt≡ −

d

dt+Qτ .


806 Tsuyoshi Kato

We recall the definition of W0(t) or M1(s),N0(s) in the last paragraph in 2.A.

Practically we will apply the next lemma for M = W0(t). Let us consider theoperator:

Qδ =

(∗Md+ δ

2 d

d∗ − δ2

): W k+1(M,Λ1 ⊕ Λ0) →W k(M,Λ1 ⊕ Λ0).

Notice that Qδ coincides with Qτ above for t ≥ 1.

Lemma 3.3 Suppose M is a closed Riemannian manifold. Then for a smallδ > 0 and k ≥ 0, Qδ gives an isomorphism.

Proof Let us take (u, v) ∈ W k+1(M,Λ1 ⊕ Λ0), and put Qδ(u, v) = (x, y).Then we have: (

∗d+δ

2

)u+ dv = x, d∗u−

δ

2v = y.

For convenience, we recall ∗2u = (−1)p(n−p)u, d∗u = (−1)np+n+1 ∗ d ∗ u for ap–form u on an n–dimensional manifold. Then from the above, we have:

d∗ ∗ du+δ

2d∗u+ d∗dv = d∗x =

((δ

2

)2

+ d∗d

)v +

δ

2y.

Thus we can solve v from (x, y) as follows:

v =

((δ

2

)2

+ d∗d

)−1(d∗x+

δ

2y

).

Next we have the following equality:

∗2du+δ

2∗ u+ ∗dv = ∗x = du+

δ

2∗ u+ ∗dv.

d∗du+δ

2d∗ ∗ u+d∗ ∗ dv = d∗ ∗ x

= d∗du+δ

2∗ du = d∗du+

δ

2

(x−

δ

2u− dv

).

(3.1)

Then from this, we have the following equation:(d∗d−

(δ

2

)2)u = −

δ

2(x− dv) + d∗ ∗ x,

dd∗u =δ

2dv + dy.

(3.2)



Thus one can again solve u as:

u =

(∆ −

(δ

2

)2)−1(

dy + d∗ ∗ x−δ

2x+ δ dv

).

It follows that there is a constant C = C(M) such that Spec |Qδ| ≥ C( δ2)4 .

Notice the following inequalities:

|(u, v)|W k+1 ≤C

(δ

2

)−2∣∣∣∣∣

(∆ ±

(δ

2

)2)

(u, v)

∣∣∣∣∣Wk−1

≤C

(δ

2

)−4 (|x|W k + |y|W k

)≤ C

(δ

2

)−4

|Qδ(u, v)|Wk.

(3.3)

In fact one may assume that C only depends on sup0≤l≤2 supx∈M |∇lR|(x),where R is the curvature form with respect to the Levi–Civita connection.This completes the proof.

Let (Pτ )∗τ be the formal adjoint of the Sobolev weighted spaces. Then oneconsiders the same analysis for (Pτ )∗τ . Notice the equality:

(I−1 ◦ Pτ ◦ I)∗ = I−1 ◦ (Pτ )∗τ ◦ I.

Thus one can analyze an operator of the form P ∗τ = d

dt+ Qδ , since Qδ is

self-adjoint over M .

Let us take a smooth (non-proper) function:

τ : W0 → [0,∞)

which is horizontally constant, namely for some positive constant δ > 0 wehave τ |W0(t) = δt for t ≥ 0. Using τ , one defines weighted Sobolev spaces

W kτ ((W0, g),Λ∗). One has the expression Pτ |M1([0,∞)) = − d

ds+Qτ ′ where:

Qτ ′ =

(∗d dd∗τ ′ 0

): W k+1

τ ′ (M1(s); Λ1 ⊕ Λ0) →W k

τ ′(M1(s); Λ1 ⊕ Λ0)

is an elliptic operator over M1(s), and τ ′ = τ |M1(s) is its restriction. It followsthat Qτ ′ has closed range with empty kernel. We do not use this fact later.In this case one can not see whether Pτ defined above is Fredholm or not,since one has no simple statement as above for the weighted adjoint operator(Qτ ′)∗τ ′ |W k

τ ′(M1(s); Λ1).

Let us take another weight function µ : W0 → [0,∞), with µ|M1(s),N0(s) ≡δ′s, where we choose a sufficiently small 0 < δ′ ≪ δ . Let us put:

w = τ + µ : W0 → [0,∞).


808 Tsuyoshi Kato

We call w a weight function with weight (δ, δ′). Let us use w as the weightfunction. Then one obtains a bounded operator:

Pw = d∗w ⊕ d+ : W k+1w (W0; Λ

1) →W kw(W0; Λ

0 ⊕ Λ2+).

Passing through the isometry I : L2 → L2w , I(u) = exp(−w

2 )u, Pw has thefollowing expression on M1([0,∞)) and N0([0,∞)):

I−1 ◦ Pw ◦ I = −∂

∂s+

(∗d dd∗τ ′ 0

)+δ′

2

(1 00 −1

)≡ −

∂

∂s+Qv

w.

On the other hand one has another expression on W0([0,∞)):

I−1 ◦ Pw ◦ I = −∂

∂t+

(∗d+ δ

2 d

d∗µ − δ2

)= −

∂

∂t+Qh

w.

3.C Decay estimate

Let M be a (possibly non-compact) n–dimensional Riemannian manifold of

bounded geometry. In practice M stands for W0(t) (or N0(s),M1(s)). Let ustake a smooth function µ : M → [0,∞). Let E and F be vector bundles overM , and take an elliptic operator:

Q : W k+1µ (M ;E) →W k

µ (M ;F ).

Let us choose τ : M×R → R+ by τ(m, t) = δ|t| for |t| ≥ 1, and put w = τ+µ.Then one considers the following operator:

P = −d

dt+Q : W k+1

w (M × R, p∗E) →W kw(M × R; p∗F ).

Let P ∗w be the formal adjoint operator with respect to the weighted inner prod-

uct. Then P ∗wP admits a spectral decomposition. For u ∈W k

w(M × R, p∗(E)),we denote Specu ∈ [0, ǫ] if u lies in the image of the spectral projectionP([0, ǫ]).

Suppose the above Q is a Fredholm operator, and is self adjoint with respectto the weighted L2

µ inner product.

Proposition 3.1 [24, 13, 28] For sufficiently small δ > 0, the above P isFredholm. Moreover if Q is invertible, then one may choose δ = 0. In particularP is also invertible.

Suppose Q : W k+1µ (M ;E) → W k

µ (M ;F ) has closed range with kerQ = 0. Letus put ∆ = Q∗

µ ◦ Q, and choose a positive constant λ0 > 0 with Spec∆ ∈



[λ20,∞). Then under the above condition, one has the following; for any

small ǫ ≪ λ0 , there exist positive constants C, δ0 > 0 such that for anyu ∈ W k+l(M × R, p∗E) with Specu ∈ [0, ǫ] for P ∗

w ◦ P , we have the followingdecay estimate:

|u( , t)|L2µ(M) ≤ C exp(−δ0|t|)|u|L

2µ(M × [0,∞)), t ≥ 0.

Proof The first statement follows from the decay estimate. Next let us takeu as above, and denote its slice on M × s by us . Let us put a smooth functionf : R → [0,∞) by f(s) =

∫Ms

exp(µ)|us|2 . Then by differentiating, one has the

inequalities:

f ′′(s) = 2

[∫

Ms

exp(µ)〈u′s, u′s〉 +

∫

Ms

exp(µ)〈u′′s , us〉

]

≥ 2

[∫

Ms

exp(µ)〈u′s, u′s〉 +

∫

Ms

exp(µ)〈(∆ − ǫ)us, us〉

]

≥ 2

∫

Ms

exp(µ)〈(∆ − ǫ)us, us〉 ≥ 2(λ0 − ǫ)f(s).

(3.4)

Thus one has differential inequalities f ′′ ≥ λ0f . From this, one can get thedesired estimate. This completes the proof.

Let Y be a complete Riemannian four–manifold such that except for a compactsubset K ⊂ Y , Y \K is isometric to Y (0) × [0,∞) where Y (0) is a closedmanifold. Let us equip a weight function τ : Y → [0,∞) by τ |Y (0) × t = δt.Let I : L2(Y ) ∼= L2

τ (Y ) be the isometry.

Corollary 3.2 [24] The corresponding:

P = I−1PτI : W k+1(Y ; Λ1) →W k(Y ; Λ0 ⊕ Λ2+)

gives a Fredholm map.

Proof This is well known, but for convenience we recall the proof. Let us takea cut-off function ϕ|Y with ϕ|Y (0) × [1,∞) ≡ 1, and ϕ|K ≡ 0. Let us chooseany u ∈W k+1(Y ; Λ1). Then one may regard ϕu ∈W k+1(Y (0) × R; Λ1). Thusone has the estimate:

|u|W k+1 ≤ |ϕu|W k+1 + |(1 − ϕ)u|W k+1

≤ C{|ϕP (u)|W k + |[ϕ,P ]u|W k + |(1 − ϕ)Pu|W k

+ |[(1 − ϕ), P ]u|W k + |(1 − ϕ)u|W k}.

(3.5)

This shows that P has closed range with finite-dimensional kernel. One has asimilar estimate for P ∗ . This shows the result.


810 Tsuyoshi Kato

3.D First cohomology

Let w = τ + µ be as in the last paragraph of 3.B.

Lemma 3.4 For a sufficiently small choice of δ′ ≪ δ , d : W k+1w (W0; Λ

0) →

W kw(W0; Λ

1) has closed range with ker d = 0.

Proof Clearly ker d = 0.

As lemma 3.3, passing through the isometry I , I−1 ◦ d ◦ I|W0([0,∞)) can beexpressed as:

d

dt+

(d,−

δ

2

): W k+1(W0([0,∞)) →W k(W0([0,∞); Λ0 ⊕ Λ1).

Clearly this has closed range. One has the same property on N0([0,∞)) andM1([0,∞)).

Now let Hi ⊂ W k+1(W0), i = 0, 1, 2, 3 be closed subsets whose supports lie

in N0([0,∞)), M1([0,∞)), W0([0,∞)) and K respectively, where K ⊂ W0 is

some compact subset. Then one may assume ∪iHi = W k+1(W0). Now by theopen mapping theorem and above, every image d(Hi) has closed range. Thiscompletes the proof.

Let us consider W0 and the weight function w on it with the weight constants(δ, δ′). Let us recall that we have a bounded complex:

0 −→ W k+2w (W0)

d−→W k+1

w (W0; Λ1)

d+−→W k

w(W0; Λ2+) −→ 0

where the weight function is w = τ + µ.

It is clear ker d0 = 0. Suppose ω ∈ W 1w(W0; Λ

1) with d+(ω) = 0. Then byintegration by parts, one has 0 =

∫W0

|d+ω|2 = 1

2

∫W0

|dω|2 . In particular we

have dω = 0. The following sublemma shows H1(AHS) = 0.

Let W = W0 or Y (S) in 2.B. Recall that one has a weight function τ over

W0 . There is a natural extension of τ on Y (S). Let µ : Y (S) → [0,∞) be

another weight function defined as µ|W n0 ([−S, S])(x) = δ′(|n| + t(x)), where

t : W 00 ([−S, S]) → [0, 1] satisfies t|N0(S) ≡ 0 and t|M1(S) ≡ 1.

Let w = τ +µ be a weight function on W . In order to make explicit the weightconstants, sometimes one uses notations τ(δ) and µ(δ′). Later one will showthat d : W k+1

w (Y (S)) → W kw(Y (S); Λ1) has closed range. Assuming this, one

has the following:



Sublemma 3.1 Let f ∈ C∞(W ) satisfies |df |L2w(W ) < ∞. Then there are

constants C and k = k(f) such that:

|f − k|L2w <∞.

Proof The situation differs from [27, lemma 5.2], in that on the end, we have

non compact slices, W0(t),M1(s), and N0(s). In order to verify this, one takestwo steps. First one sees a weaker version of the sublemma.

Step 1: Let us choose any δ0 < δ and any δ′0 < δ′ so that the pair (δ0, δ′0) is

sufficiently near (δ, δ′). Let us put w = τ(δ) + µ(δ′) and w′ = τ(δ0) + µ(δ′0).Then we claim that f − k ∈ L2

w′ .

By lemma 3.2(1) and lemma 3.4, one has an estimate:∫

W

exp(w)|u|2 ≤ C

∫

W

exp(w)|du|2.

This shows that one has a Hilbert space, which is a completion of C∞cp (W ) by

the norm |u|2 =∫W

exp(w)|du|2 . Let us take any f ∈ C∞(W ) with df ∈ L2w .

Using a functional s : H → R, by s(v) =∫W

exp(w)(|dv|2 −2〈dv, df〉), one getsa unique u = (d∗wd)

−1(d∗wdf) ∈ H with d∗wd(f−u) = 0. We show that g = f−uis in L2

w′ . From dg ∈ L2w , one has estimates

∫W ([t,t+1]) |dg|

2 ≤ exp(−δt)ǫt ,

where ǫt → 0. Notice that dg satisfies the elliptic equation (d∗w ⊕ d)dg = 0.Thus by the local Sobolev embedding and the above estimate, one gets:

supW0(t)

|dg| ≤ C exp(− δt2 )ǫt.

In particular one has the next estimate:

|g(m, t)| ≤ C1 +

∫ t

0C2 exp

(−δs

2

)ǫsds,

where C1 = |g(m, 0)|. This shows that there is a constant Cm such thatlimt→∞ supW (m,t) |g − Cm| = 0. It is clear that Cm is independent of m, andwe write the same constant by C .

Let us make a similar procedure on s direction along M1(s) and N0(s). Thenone finds other constants C ′ and C ′′ such that g − C ′|M1(m, s) and g −C ′′|N0(m, s) vanish at infinity for every m. Again it is clear that C = C ′ = C ′′ .

Now one finds a constant c with |dg|(x) ≤ c exp(−w(x)2 ). Then by integration,

one has the inequality:

|g − C|(x) ≤ c exp

(−w(x)

2

).


812 Tsuyoshi Kato

Notice that for any α > 0, wα is integrable over W . Combining with this andthe above inequality, one gets the first claim.

Step 2: For any sufficiently small pair (δ, δ′), one has a well-defined first co-homology group H1(δ, δ′) = ker d ∩ L2

w(W ; Λ1)/d(W 1w(W )). Then one has a

natural map:

i : H1(δ, δ′) → H1(δ0, δ′0).

We claim that i is an injection. This is enough for the proof of the sublemma.

Let us put ∆(δ, δ′) = (d∗w ⊕ d)∗w(d∗w ⊕ d) on L2w(W ; Λ1), w = w(δ, δ′). Then

the spectrum of ∆(δ, δ′) is discrete near 0. Thus for (δ0, δ′0) sufficiently near

(δ, δ′), one has dimH1(δ, δ′) ≥ dimH1(δ0, δ′0).

Suppose these dimensions are different. Then one chooses another δ1 < δ0and δ′1 < δ′0 . If dimH1(δ0, δ

′0) = dimH1(δ1, δ

′1), then one puts δ0 = δ and

δ′ = δ′0 , and gets the conclusion. If these dimensions are different, one hasdimH1(δ0, δ

′0) > dimH1(δ1, δ

′1). If they are different, then one must have

dimH1(δ, δ′) ≥ 2. Let us take another δ2 < δ1 < δ0 and δ′2 similarly.

One iterates this process until one gets the equality on dimension. If this is notthe case, one has an infinite sequence (δi, δ

′i). By choice, one may assume that

this sequence converges to (δ∞, δ′∞) with 2δ0(

′) − δ(′) < δ∞(′) < δ(′). Then∆(δ∞, δ

′∞) must have a continuous spectrum near 0. This is a contradiction.

This completes the proof.

Remark 3.1 When δ′ = 0 (no weight in the horizontal direction), one canstill find some constant C such that g − C vanishes at infinity. Let us takeany (m′, t) ∈ W n

0 ([−S, S]) ⊂ Y (S). Then we show that if n is sufficientlylarge, then |g(m′, t) − Cm| is sufficiently small for Cm as above. With respectto n, let us choose a sufficiently large T . Then since one has the estimate|dg|Y (S)(T ) ≤ C exp(− δT

2 ), one may assume |g(m,T )−g(m′, T )| is sufficientlysmall. On the other hand, |g(m′, t) − g(m′, T )| is less than ǫn , where ǫn → 0as n→ ∞. Thus |g(m′, t) − Cm| is sufficiently small.

Lemma 3.5 Let W = Y (S). H1(δ, 0) is naturally isomorphic to H1(δ, δ′).

Proof Let α ∈ H1(δ, 0) be a harmonic representative. Then from corollary4.1 below, and by the same method as [27, lemma 5.3], one can verify α ∈L2

w(δ,δ′)(Y (S); Λ1). This completes the proof.



4 Fourier–Laplace transforms between open mani-

folds

4.A Fourier–Laplace transforms

Our main application is the analysis over (W0,N0,M1) = kinky handles. Recallthat for the simplest kinky handle, W0 is diffeomorphic to S1 × D3 . In thissection, one considers (W0 = S1 × D3,M1,N0), where M1,N0 are all diffeo-morphic to S1 ×D2 , and the embeddings are given as follows. Let us regard∂W0 = S1×D2

+ ∪S1×D2− , and consider two knots µ ⊂ S1×D2

+ , γ ∈ S1 ×D2−

as follows;

(1) µ represents a Whitehead link diagram (see [22, page 79, figure 3.3]) and

(2) γ = S1 × 0 ⊂ S1 ×D2− .

Notice that µ is null-homotopic, and γ represents a generator of π1(W0). Letus take tubular neighborhoods of µ and γ in S1 ×D2

± respectively. We denotethem as µ and γ . Then we choose µ = M1 and γ = N0 . Let us denote thequotient space by Y = W0/M1 ∼ N0 . By a simple calculation, one knows thefollowing:

Sublemma 4.1 H1(Y ; R) = R and H2(Y ; R) = H3(Y ; R) = 0.

In this section, one introduces the Fourier–Laplace transform over Y , which isan open analogue of the one in [27]. In order to do this, one will only require theabove topological conditions. In particular one can apply the method in thissection for cylindrical manifolds Y which includes the simplest Casson handles.

Now by the previous construction, one has a complete Riemannian manifoldW0 . Let us take infinitely many W0 , and for convenience, we enumerate thesame W0 as W i

0 , i ∈ Z. Let us choose a sufficiently large S , and introducenotations:

W0([−S, S]) = W0\{M1([S,∞)) ∪N0([S,∞))},

Y (S) = W0([−S, S])/N0(S) ∼M1(S).(4.1)

One has a natural extension of notation Y (S)(t) = W0(t)/N0(S, t) ∼M1(S, t),

t ∈ [0,∞). Recall that one has a weight function τ : W0 → [0,∞) by τ |W0(t) =δt. One can make a natural extension of τ on Y (S). We will use the same τfor this.


814 Tsuyoshi Kato

Now one has a periodic Riemannian manifold as:

Y (S) = · · · ∪M−n1

(S)∼N−n+1

0(S) W

−n+10 ([−S, S])

∪M−n+1

1(S)∼N−n+2

0(S) W

−n+20 ([−S, S]) . . .

(4.2)

There is a free and isometric Z–action on Y (S), and we denote the action 1 ∈ Z

by T .

Let us take vector bundles Ei → Y (S) (for i = 1, 2) and a differential operatorD between them. Then it lifts to the following one:

D : C∞cp (E1) → C∞

cp (E2)

where Ei → Y (S) are the natural lifts.

Let us take any ψ ∈ C∞cp (E1) and z ∈ C

∗ . Then we define the Fourier–Laplacetransform of ψ by:

ψz( ) = Σ∞n=−∞z

n(T nψ)( ).

Let us define another vector bundle over Y (S) × C∗ as:

E′1 ≡ [E1 ⊗R C]/Z → Y (S) × C

∗

where 1 ∈ Z sends (ρ, λ) ∈ E1 ⊗R C to (Tρ, zλ). One may regard E′1 as a

family of vector bundles {E′1(z)} over Y (S), parameterized by z ∈ C

∗ .

Now by restriction ψ|W 00 ([−S, S]), ψz defines a smooth section over Y (S) of

E′1(z), where W 0

0 ([−S, S]) ⊂ Y (S). Thus one may regard ψz ∈ C∞cp (E′

1(z)).

The Fourier–Laplace inversion formula is as follows. Let us take a smoothsection η ∈ C∞

cp (E′1) with ηz ∈ C∞

cp (E′1(z)). Then for any s ∈ (0,∞), the

following:

T nη(x) ≡1

2πi

∫

|z|=s

z−nηz(π(x))dz

z

defines a smooth section over E1 → Y (S), where π : Y (S) → Y (S) is the

projection and x ∈ W 00 ([−S, S]). By Cauchy’s formula, these are inverses of

each other, independently of s.

Let D be as above. Passing through the Fourier–Laplace transform, one hasanother differential operator between (E0)

′ and (E1)′ by:

Dzψz ≡ (Dψ)z.

Notice that at z = 1, (Ei)′ , i = 0, 1, are isomorphic to Ei → Y (S) re-

spectively. In fact every (Ei)′(z) is isomorphic to Ei as follows. Let us take



t : W0([−S, S]) → [0, 1], a smooth map such that t ≡ 0 near N0(S), and t ≡ 1near M1(S). By taking [0,∞) ⊂ C as a branched line, one may define log z .Let us put zt ≡ exp(t log z), and consider ztψz . This gives a C–valued sectionover E′

0 . One calculates Dz in terms of this identification. The result is as:

D(z) = D + zt[D, z−t] : C∞cp (E0) → C∞

cp (E1).

4.B Elliptic complexes over periodic covers

Let {Ei,Di}i=0,1,2 be an elliptic complex over Y (S). It is clear that it gives abounded one as:

0 −→W k+2τ (E0)

D0−→W k+1τ (E1)

D1−→ W kτ (E2) −→ 0.

Proposition 4.1 The above Di is an acyclic Fredholm complex, ie all coho-mologies H i vanish, if for all z ∈ C = {w ∈ C

∗; |w| = 1}, {Diz : W k+i

τ →W k+i−1

τ } are also so.

Proof For any u ∈ W kτ (Y (S); Ei), uz has the following property; it lies in

W kτ ((Ei)z;Y (S)z) for all z with 0 < |z| ≤ 1.

When |z| = 1, the inner product on Ei has a natural extension of a Hermitianmetric on Ei(z). Recall that Di : W

k+1τ (Y (S);Ei(z)) → W k

τ (Y (S);Ei+1(z))has closed range. Let (Di)

∗τ be the formal adjoint operator. This differen-

tial operator is defined independently of z , with |z| = 1 over Ei(z). By theassumption, one has an isomorphism:

D = D0 ⊕ (D1)∗τ : W k+1

τ (Y (S);E0(z) ⊕ E2(z)) →W kτ (Y (S);E1(z)).

We denote by D−1 its inverse. On the other hand one has a natural lift of Das:

D : W k+1w (Y (S); E0 ⊕ E2) →W k

w(Y (S); E1).

Here we show that there is another bounded operator:

R : W kw(Y (S); E1) → W k+1

w (Y (S); E0 ⊕ E2)

with R ◦D is the identity. From this, one knows that D has closed range. Inparticular D0 has closed range.

Next one proceeds similarly for D∗τ = (D0)

∗τ ⊕D1 . Then one also knows that

its lift D∗τ has closed range. Clearly it follows that D1 has also closed range.


816 Tsuyoshi Kato

Now let us take ψ ∈ W k+1w (Y (S); E0 ⊕ E2). Then one considers b(ψ) =

D−1(ψ) ∈ W k+1τ (Y (S);E1(z)). This element also has the property in the first

paragraph of the proof.

Then one defines R(ψ) ∈W k+1w (Y (S); E1) by:

T nR(ψ)|W 00 ([−S, S])(x) =

1

2πi

∫

C

z−n(D−1(ψ)(π(x)))dz

z.

We claim that this assignment:

R : W kw(Y (S); E1) → W k+1

w (Y (S); E0 ⊕ E2)

gives a bounded operator. Then by the definition of Di over Y (S), it satisfiesR ◦D = 1.

Let us put C = {z = exp(iθ) ∈ C}, ψn|W00 ([−S, S]) = T n(ψ). By regarding

b(ψ)z ∈ W k+1τ (W 0

0 ([−S, S])), one has a Fourier expansion b(ψ) = Σnznϕn ,

ϕn+1|N0(S) = ϕn|M1(S). Notice the equality D(ϕn) = ψn . One can regardthe value of the inner product 〈b(ψ), b(ψ)〉|W k

τ (Y (S);E∗(z)) as a real-valuedfunction of θ ∈ [0, 2π]. Then one has the equality:

1

2π

∫ 2π

0dθ|b(ψ)|2W k+1

τ (Y (S);E∗(z)) =

1

2π

∫ 2π

0dθΣn,m exp(iθ(n−m))〈ϕn, ϕm〉|W k+1

τ (W 00 ([−S, S])) = Σn|ϕn|

2W k+1τ .

(4.3)Combining with the inequality (D−1 are bounded operators for all z ∈ C)|b(ψ)|W k+1

τ (Y (S);E∗(z)) ≤ C|ψ|W kτ for every θ , one gets the estimate:

Σn|ϕn|Wk+1τ (W 0

0 ([−S, S])) ≤ CΣn|ψn|Wkτ (W 0

0 ([−S, S])).

From the equality, T n(ψ) = ψn , one has the desired estimate:

|R(ψ)|2W k+1τ (Y (S); E∗) = Σn|ϕn|

2W k+1τ (W 0

0 ([−S, S])) ≤ CΣn|D(ϕn)|2W kτ

= CΣn|Tn(ψ)|2W k

τ (W 00 ([−S, S])) = C|ψ|2W k

τ .


4.C Computation of parametrized cohomology groups

In 4.C, one will verify the assumption in proposition 4.1.

Sublemma 4.2 Di : Wk+1τ (Y (S);Ei(z)) → W k

τ (Y (S);Ei+1(z)) has closedrange.



Proof It is true for z = 1. Recall Y (S) = W0([−S, S])/N0(S) ∼ M1(S). LetHi ⊂W k+1

τ (Y (S);Ei(z)) be closed subsets satisfying:

(1) H1 ∪H2 = W k+1τ (Y (S);Ei(z)) and

(2) SuppH1 ⊂ W0([−(S − 1), S − 1]) and SuppH2 ⊂ N0([−S,−(S − 1)]) ∪M1([S − 1, S]).

By regarding H1 ⊂ W k+1τ (Y (S);Ei(1)), it follows that Di(H1) is a closed

subspace of W kτ (Y (S);Ei+1(z)). Let u ∈ H2 . Then one may associate u ∈

W k+1τ (Y (S);Ei(1)) by u|N0([−S,−(S − 1)]) = u and u|M1([S − 1, S]) = z−1u.

Clearly Di(H2) = Di(H2) is a closed subspace of W kτ (Y (S);Ei+1(1)). Since

this assignment is isometric, it follows that H2 is also a closed subspace ofW k

τ (Y (S);Ei+1(z)).


Proposition 4.2 Let {D∗} be an AHS complex. Suppose it gives a Fredholmcomplex and all the cohomology groups vanish, H i(W k

τ (Y (S); Λ∗);D∗) = 0,i = 0, 1, 2. Then for C(1) = {z : |z| = 1}, one has H∗(W k

τ (Y (S); Λ∗); (D∗)z) =0 for all z ∈ C(1) and i = 0, 1, 2.

Proof The first part is essentially [27, page 390]. Let u ∈W kτ (Y (S); Λ∗

z) withd0(u) = 0. Then passing through the identification zt : Λ0

z∼= Λ0 , one has

d(z−tu) = 0 where z−tu ∈W kτ (Y (S); Λ0). Clearly this shows u = 0.

Next let us take α ∈ W kτ (Y (S); Λ1

z) with d+(α) = 0, d∗τ (α) = 0. Then by

integration by parts, one has d(α) = 0. α is zero in H1(W0; R) (see the proof

of lemma 5.1). Thus one may express α = d(f) on W0 for f ∈ C∞(W0([−S, S]))and d∗τd(f) = 0. Let i : N0(S) ∼= M1(S) be the identification. Then one hasi∗(f) = zf+const . If one takes f so that it vanishes at infinity, then const = 0.Thus one has the equality:

0 =

∫

W0([−S,S])exp(τ)〈f, d∗τd(f)〉vol = −

∫exp(τ)|df |2.

This shows df = 0. Let O ⊂ C(1) be the subset satisfying a property thatany α ∈W k

τ (Y (S); Λ1z) with d+(α) = 0 satisfies dα = 0. The above implies O

is non-empty, open and closed in C(1). Thus H∗(W kτ (Y (S); Λ∗); (D∗)z) = 0,

∗ = 0, 1 for all z ∈ C(1).

Now let us put D = (D0)∗τ ⊕D1 , and consider ∆ = D∗

τD on L2τ (Y (S); Λ1(z)).

Then there is a constant λ > 0 with λz > λ, where λz is the bottom of ∆.Suppose H2 6= 0 at z = exp(iθ0), but H2 = 0 for all z = exp(iθ), 0 ≤ θ < θ0 .


818 Tsuyoshi Kato

Then by choosing some θ , it follows that there exists u ∈ W 1τ (Y (S); Λ2

z), z =exp(iθ) with (1) |u|W 1

τ = 1 and (2) |(D1)∗τ (u)|L2

τ < ǫ.

Since D1 is surjective, one finds α ∈ W 2τ (Y (S); Λ1

z) with D1(α) = u and(D0)

∗τ (α) = 0. Then there is a constant C independent of z such that:

|u|W 1τ ≤ C|α|W 2

τ ≤ C|(D1)∗τD1(α)|L2

τ .

This is a contradiction. This shows that the second cohomology also vanishesfor all z ∈ C(1). This completes the proof.

4.D Computation of cohomology groups with z = 1

Now we compute the cohomology H i(W kτ (Y (S); Λ∗);D∗) of the AHS complex

for the simplest kinky handles (W0,N0,M1). Notice that the end of Y (S) isisometric to M × [0,∞) for a compact Riemannian three manifold M . We willdenote Mt = M × {t} ⊂ Y (S).

Proposition 4.3 Let Y (S) be as above. Then for all i = 0, 1, 2, the coho-mologies H i(W k

τ (Y (S); Λ∗);D∗) vanish.

Proof Clearly H0(AHS) = 0. Let us consider H1(AHS). Let us take anyrepresentative u. By integration by parts, one finds du = 0. One may expressu = w + f dt around the end Y (S) ∼= M × [0,∞), where w does not containdt component. Then through this isometry, one considers wt ∈ Λ1(M) fort ∈ [0,∞). Let d3 be the differential over M . Then for every t, one hasd3wt = 0. Thus one may express wt = Σigi(t)αi + dµt , where {αi}i consistsof the orthogonal basis of H1(M ; R). Here one may assume d∗3(αi) = 0. Bytaking into account of dt component of du, one has the following equation:

Σigi(t)′αi + d3µ(t)′ = d3ft.

This shows gi(t)′ ≡ 0. Since u ∈ L2

τ , one concludes gi(t) ≡ 0. Then one hasanother equality, ft = µ(t)′ + c(t), where c(t) are constants depending on t.Thus one gets u = d3µt + (µ′t + c(t))dt.

Sublemma 4.3 If d3µt ∈ W kτ , then for a smooth family of constants d(t),

one has µt − d(t) ∈W kτ .

Proof of sublemma Let us put d(t) = (volM )−1∫Mµtvol . Then we show

that this family is the desired one. Notice that one has the following bound:|d3µt|L

2τ ≥ C|µt − d(t)|L2

τ for some positive constant C . In particular one has



µt−d(t) ∈ L2τ . Notice that one has the equality µt−d(t) = (∆0)

−1d∗d(µt−d(t)),and (∆0)

−1d∗ is a translation invariant bounded operator. Thus we get:

µt − d(t)

dt=

d

dt[(∆0)

−1d∗3d3(µt − d(t))] = (∆0)−1d∗

d

dt[d3µt].

From the last term, one sees that µt−d(t)dt

∈ L2τ . By a similar consideration, one

gets the result. This completes the proof of the sublemma.

Proof of proposition (continued) Then by replacing µt by µt − d(t), onemay assume µt ∈ W k

τ . Thus dµt = d3µt + µ′tdt ∈ W k−1τ . On the other hand,

let us consider the equality ft = µ′t + c(t). Then it follows c(t) ∈ W k−1τ . Let

us put C(t) =∫ t

0 c(s)ds −∫∞0 c(t). Then by [27, lemma 5.2] (see sublemma

3.1), one finds that C(t) ∈ W kτ , and dC(t) = c(t). In particular we have

u| endY (S) = df , f ∈W kτ . Then using a cut-off function, one may represent u

by a compactly supported smooth 1 form, which is itself exact by a compactlysupported smooth function, since H1

cp(Y (S); R) = 0. This shows H1(AHS) =0.

Next we consider H2(AHS). Let us take a representative u ∈ H2(AHS). Onemay choose u so that it satisfies d(eτu) = e−τd(ueτ ) = 0. Since H2(Y (S); R) =0, one may express eτu = dµ, µ ∈ C∞(Λ1). Let us denote µ| endY (S) =β + f dt, where β does not contain dt component. Then we have the followingrelation:

dµ| endY (S) = d3βt + (d3f − β′t)dt = d3βt + ∗3d3βt ∧ dt

where ∗3 is the star operator over M . Let us decompose βt = β1t + β2

t , whereβ1

t (β2t ) does (not) consists of a closed form over M . Then from the last two

terms, one finds d3ft = (β1t )′ . In particular one may represent:

ewu| endY (S) = dβ2t = d3β

2t − (β2

t )′ ∧ dt = d3β2t + ∗3d3β

2t ∧ dt.

By the decomposition, one finds a positive constant C such that:

|d3β2t |W

k−1(Mt) ≥ C|β2t |W

k(Mt).

Now we have the next relations (put µ = β2t on the end):

eτu = dµ, |µt|Wk(Mt) ≤ C|eτu|W k−1(Mt).

Sublemma 4.4 If δ > 0 is sufficiently small with respect to S , then eτu ∈L2(Y (S); Λ2

+).


820 Tsuyoshi Kato

Proof of sublemma Let us consider eτu| end = dµ. One may assume that forevery t, µt ∈ C∞(Λ1(Mt)) lies on the orthogonal complement of ker d. Then∗3d is invertible on (ker d)⊥ by corollary 3.1. Moreover it is self-adjoint withrespect to the L2 inner product. Since µ satisfies the equation, ( ∂

∂t+ ∗3d3)µ =

0, one has the exponential decay estimate for µ. More precisely there existconstants C > 0, λ0 > 0 which are independent of δ , such that:

|µ|L2(Mt) ≤ exp(−λ0t) sup{|µ|L2(Ms); 0 ≤ s ≤ 2t}.

Notice that a priori, µ satisfies the following bound of its growth |µ|L2(Mt) ≤C exp(t δ

2 )|u|L2(Mt). Combining these estimates, one gets the exponential es-timate for µ. Then one has also the exponential decay estimate for dµ on theend. This completes the proof of the sublemma.

Proof of proposition (continued) Let ϕt : Y (S) → [0, 1] be a cut-off func-tion such that ϕ|M × [t+ 1,∞) ≡ 0, ϕ|(M × [t,∞))c ≡ 1. Let us fix a large S ,and let δ > 0 satisfy the above condition. By the above estimate on W k norm,one finds ϕtµ ∈ L2(Y (S); Λ2

+) for all t. Then for any small ǫ > 0, one finds alarge t so that:

∫|ewu|2 =

∣∣∣∣∫d(ϕtµ) ∧ d(ϕtµ) +

∫|ewu|2

∣∣∣∣ < ǫ.

This shows u ≡ 0, and we have shown H2(AHS) = 0 for this pair (δ, S). Thiscompletes the proof of the proposition.

Remark 4.1 Suppose for all δ′ ∈ [δ, δ0], the differential of the AHS complexhas closed range, where δ is sufficiently small. Then H∗(AHS) also vanishesfor all δ′ . This is seen as follows. Notice that for any choice of the constants,one knows H0(AHS) = H1(AHS) = 0. We want to calculate H2(AHS) whenwe vary δ′ > 0. Let us take an isomorphism:

I : W ∗τ(δ)(Y (S); Λ∗) ∼= W ∗

τ(δ′)(Y (S); Λ∗)

where we denote τ(δ) to express the weight constant. Passing through I , onehas {W ∗

τ(δ)(Y (S); Λ∗); I−1 ◦d∗ ◦I}, a continuous family of Fredholm complexes.In particular, the indices of these complexes are invariant. Since for all cases,one has H0(AHS) = H1(AHS) = 0, one concludes H2(AHS) = 0.

Remark 4.2 Suppose a cylindrical four–manifold Y has nonzero H2(Y ; R).Then in this case, one has a bound:

dimH2(AHS) ≤ 2 dimH2(Y ; R).

This directly follows from the proof of proposition 4.3.



Corollary 4.1 Let us choose a sufficiently small δ > 0. Then one has aninvertible operator:

Pw : W k+1w (Y (S); Λ1) ∼= W k

w(Y (S); Λ0 ⊕ Λ2+).

5 An asymptotic method to compute cohomology

5.A Pw over Y (S, 2)

Let (W0, N0,M1) be a simplest kinky handle, and W0 be the Riemannian mani-fold constructed before. In the previous section, one has the invertible operator:

Pτ : W k+1τ (Y (S); Λ1) → W k

τ (Y (S); Λ0 ⊕ Λ2+).

By the Fourier–Laplace transform, one gets an invertible Fredholm Pτ overY (S) where:

Y (S) = · · · ∪ W i−10 ([−S, S]) ∪ W i

0([−S, S]) ∪ . . . i ∈ Z.

Let T2 be the periodic tree as R∪n∈ZR0 , where R0 is half the real line. The aimhere is to show that Pτ is invertible over the periodic cover of Y (S, 2) definedbelow which corresponds to T2 . Let (W2,N0,M1,M2) be a kinky handle with

two kinks, and W2 be the corresponding complete Riemannian manifold. Thusone has three ends in the horizontal direction, N0([0,∞)),Mj([0,∞)), j = 1, 2.Let us choose a large S , and put:

Y (S)0 = W 10 ((−∞, S]) ∪ W 2

0 ([−S, S]) ∪ . . . ,

Y (S, 2)′ = W2\N0([S,∞)) ∪M1([S,∞))/N0(S) ∼M1(S),

Y (S, 2) = Y (S, 2)′\M2([S,∞)) ∪ Y (S)0\N0([S,∞)).

(5.1)

One may equip a weight function τ on Y (S, 2) as before.

Lemma 5.1 d∗ : W k+1τ (Y (S, 2); Λ∗) → W k

τ (Y (S, 2); Λ∗+1) has closed rangewith H0(AHS) = H1(AHS) = 0.

Proof Closedness follows from the excision method used before. Let us take(Ws = S1 ×D3, N0 = S1 ×D2), where N0 represents a generator of π1(Ws).

By the construction in 2.A, one can get a complete Riemannian manifold Ws

with one end N0([0,∞)) along the horizontal direction. Let us put:

Y (S, 2)(0) = Y (S, 2)′\M2([S,∞)) ∪ Ws\N0([S,∞)).


822 Tsuyoshi Kato

Notice that Y (S, 2)(0) is diffeomorphic to Y (S, 2)′ , but Riemannian metricsare different. (Y (S, 2)(0) has cylindrical end.) By corollary 3.2, the differentialof the AHS complex has closed range.

Let Hi ⊂W k+1τ (Y (S, 2); Λ∗) be closed subsets with:

(1) H1 ∪H2 = W k+1τ (Y (S, 2); Λ∗),

(2) SuppH1 ⊂ Y (S, 2)′\M2([S,∞)) and SuppH2 ⊂ Y (S)0\N0([S − 1,∞)).

One regards H1 ⊂ W k+1τ (Y (S, 2)(0); Λ∗) and H2 ⊂ W k+1

τ (Y (S); Λ∗). Fromthis, it follows that d∗(Hi) are both closed subspace of W k

τ (Y (S, 2); Λ∗+1).

Let us see kerPw = 0. kerPw is isomorphic to the first cohomology H1(AHS).One can easily check H1(Y (S, 2); R) = R. Let us take a nonzero element w .Then this has the property 〈w,C〉 6= 0 for any loop C representing a generatorof π1(Y (S, 2)). One may choose C sufficiently near to infinity while having abounded length.

Let us take u ∈ H1(AHS), and consider its class [u] ∈ H1(Y (S, 2); R). Itfollows [u] = 0 , since one can make 〈u,C〉 arbitrarily small by choosing C asabove. Thus one may express u = df for f ∈ C∞(Y (S, 2)). One may assumef ∈ L2

w(Y (S, 2)), by subtracting some constant. This shows u = 0. Thus wehave shown the result.

5.B Some continuity of H1(AHS)

Let us introduce another weight function:

µ : Y (S) → [0,∞)

by µ|Wn(s ≤ S)(x) = |n|+ t(x). We choose another small constant δ′ (we takeit so that δ′ ≪ δ , where δ is the weight constant for τ ). Then we introduceanother weight w = δ′µ+ τ and weighted Sobolev k spaces W k

w , where:

(|u|W kw)2 =

∫

Y

exp(δ′µ+ τ)(Σkl=0|∇

lu|2).

Since one knows H∗(AHS) = 0 over Y (S), it follows that there exists a positiveconstant C > 0 such that |Pw|, |(Pw)∗w| ≥ C , for any small δ′ ∈ [0, δ′0].

Let Y be a complete Riemannian manifold of bounded geometry. Let us takea family of smooth maps:

w(δ′) : Y → [0,∞), w(δ′′)(x) ≤ w(δ′)(x), δ′′ ≤ δ′ ∈ [0, δ], x ∈ Y.

Suppose that for all w(δ′), the corresponding AHS complex between weightedSobolev spaces are Fredholm. In any case it is immediate to see H0(AHS) = 0.Here one has some continuity property as:



Lemma 5.2 H1(AHS) = 0 for δ′ = 0 when H1(AHS) = 0 for δ′ > 0.

Proof Let us denote ker(d+)∗w(δ′) = H2(δ′) (the second cohomology H2(AHS)

when it is defined). Let us choose sufficiently small constants:

δ ≫ δ′ > δ′′ ≥ 0.

Then one has H2(δ′′) ∼= H2(δ′). This follows from that by varying these weightsfrom w(δ′) to w(δ′′), one gets a family of Fredholm complexes between theweighted Sobolev spaces (these spaces also vary with respect to the deformationof the weights). For every value of weights, one has H0(AHS) = H1(AHS) = 0.Thus one gets the above statement.

Sublemma 5.1 If H2(δ′′) 6= 0, then one has also H2(δ′) 6= 0.

Proof of sublemma Let us denote w = w(δ′) and w′ = w(δ′′). Let us takev ∈ L2

w′ with d∗(exp(w′)v) = 0. Then one puts u = exp(−w + w′)v . One mayassume u ∈ L2

w . Since u satisfies d∗(exp(w)u) = 0, one gets the result.

Proof of lemma (continued) Recall that by putting w = w(δ′), one alsohas a Fredholm complex {W ∗

w(Y (S, 2); Λ∗), d∗}. Then by varying a parameterδ′ ∈ [0, δ], one has a family of Fredholm complexes:

{W ∗w(δ′)(Y (S, 2); Λ∗), d∗}.

From the above proof, one has the inclusion H2(δ′′) ⊂ H2(δ′) for all 0 ≤ δ′′ ≤δ′ . Now let us see H2(δ′) = H2(0) and H1(0) = 0. Suppose H1(0) 6= 0. Thenby invariance of Fredholm indices, one must have dimH2(0) > dimH2(δ′).This contradicts the above. This completes the proof.

5.C Computation of H2(AHS)

To show that Pτ : W k+1τ (Y (S, 2); Λ1) → W k

τ (Y (S, 2); Λ0 ⊕ Λ2+) is invertible,

one uses an asymptotic method. Roughly speaking one approximates Y (S, 2)by a family of Riemannian manifolds with cylindrical ends. Then the spectrumof Pτ over each cylindrical manifold has a uniform lower bound on 1 forms.From this one gets a lower bound of (d+)∗τ over Y (S, 2). In the presence ofH2(Y ; R), one can get a uniform bound of dimH2(AHS). The approximationof spaces corresponds to the one of infinite tree by its finite subtrees. Thismethod completely works for higher stages in 5.D.


824 Tsuyoshi Kato

Computation of dimH2(AHS) uses information of H1(AHS) on both approx-imation spaces and the limit space. The former is obtained by proposition 4.3and the latter by sublemma 3.1. On the other hand for all cases H0(AHS) = 0.Then using this and proposition 4.3, one can apply the asymptotic method inthis section to verify H1(AHS) = 0 for the limit space. This will be anothermethod to compute H1(AHS) without using sublemma 3.1.

Let us take (Ws = S1 ×D3, N0 = S1 ×D2) where N0 represents a generator ofπ1(Ws). Then by the construction in 2.A, one gets a complete Riemannian man-

ifold Ws with one end N0([0,∞)) along horizontal direction. Let (W0,N0,M1)be the simplest kinky handle as before. Let us put:

Y (S)0(n)′ = W 10 ([−S, S]) ∪M1

1(S)∼N2

0(S) W

20 ([−S, S])

· · · ∪Mn−1

1(S)∼Nn

0(S) W

n0 ([−S, S]),

Y (S)0(n) = Y (S)0(n)′ ∪Mn1

(S)∼N0(S) Ws\N0([S,∞)),

Y (S)′′0 = . . . W−n0 ([−S, S]) ∪M−n

1(S)∼N−n+1

0(S) W

−n+10 ([−S, S]) . . .

∪Mn−1

1(S)∼Nn

0(S) W

n0 ([−S, S]) ∪Mn

1(S)∼N0(S) Ws\N0([S,∞)).

(5.2)

Notice Y (S)0(∞)′ = Y (S)0 . Let us put Y (S, 2)(n)′ = Y (S, 2)′ ∪M2(S)∼N00(S)

Y (S)0(n)′ and:

Y (S, 2)(n) = Y (S, 2)′ ∪M2(S)∼N00(S) Y (S)0(n).

Y (S, 2)(n) is a complete Riemannian manifold with cylindrical end. Let theweight function be τ : Y (S, 2)(n) → [0,∞), with weight δ as before.

Proposition 5.1 One gets isomorphisms:

Pτ : W k+1τ (Y (S, 2)(n); Λ1) ∼= W k

τ (Y (S, 2)(n); Λ0 ⊕ Λ2+).

Proof By proposition 4.3, the conclusion is true when one uses a sufficientlysmall weight constant δ′ . On the other hand, by the first part of the proofof lemma 5.1, the differential of the AHS complex has closed range for allδ′ ∈ (0, δ]. From this, one gets H0(AHS) = H1(AHS) = 0 for all δ′ . To seeH2(AHS) = 0, one can use the same method as proposition 4.2. This completesthe proof.

In particular there are constants Cn such that for ∆ = (Pτ )∗τ ◦ Pτ , one has thebounds |u|W k+2

τ (Y (S, 2)(n)) ≤ Cn|∆(u)|W kτ (Y (S, 2)(n)). Also one has bounds:

|u|W k+2τ (Y (S, 2)) ≤ C|∆(u)|W k

τ (Y (S, 2)),

|u|W k+2τ (Y (S)′′0) ≤ C|∆(u)|W k

τ (Y (S)′′0).



Sublemma 5.2 There is a lower bound Cn ≤ C for all n.

Proof For any small ǫ > 0, there exists a large n with the following property:for any u ∈ W k+2

τ (Y (S, 2)(n)) with |u|W k+2τ = 1, there is at least one n′ ,

0 ≤ n′ ≤ n with |u|W k+2τ (W n′

0 ([−S, S]) < ǫ. Let ϕ(n′) be a cut-off function

on Y (S, 2)(n) with ϕ(n′)|(W n′

0 ([−S, S]))c ≡ 1 and Y (S, 2)(n)\Supp dϕ(n′) hastwo components. Then one may express ϕ(n′)u = u1 + u2 . One may regardu1 ∈W k+2

τ (Y (S, 2)), u2 ∈W k+2τ (Y (S)′′0). In particular one has the estimates:

|u|W k+2τ ≤ |u1|W

k+2τ + |u2|W

k+2τ + |(1 − ϕ(n′))u|W k+2

τ

≤ C{|∆(u1)|Wkτ + |∆(u2)|W

kτ } + ǫ ≤ C|∆(u)|W k

τ + Cǫ(5.3)

where C is independent of ǫ and u. This gives the result.

Now suppose H2(AHS) is nonzero over Y (S, 2), and take u ∈ ker(d+)∗τ ∩W k+1

τ (Y (S, 2); Λ2+) with |u|W k+1

τ = 1. For each n, let us take a cut-off func-tion ϕ(n) on Y (S, 2) with ϕ(n)|Y (S, 2)(n − 1)′ ≡ 1, ϕ(n)|(Y (S, 2)(n)′)c ≡0. Then one may regard ϕ(n)u ∈ W k+1

τ (Y (S, 2)(n); Λ2+). Thus there exists

vn ∈ W k+2τ (Y (S, 2)(n); Λ1) with d∗τ (vn) = 0, d+(vn) = ϕ(n)u. By the above

sublemma, one has the uniform estimates:

1 − ǫ ≤ |ϕ(n)u|W k+1τ = |d+(vn)|W k+1

τ

≤ C|vn|Wk+2τ ≤ C|d+(vn)|W k+1

τ ≤ C|ϕ(n)u|W k+1τ . (∗)

(5.4)

By the above (∗), one has the uniform estimates C ≤ |vn|Wk+2τ ≤ C ′ . Let us

take a sequence 1 ≫ ǫ1 ≫ ǫ2 ≫ · · · → 0. Then for each n, there exists a largeN = N(n) and L(n) ≫ 0 such that at least one |vN |W k+2

τ (W n′

0 ([−S, S])) isless than ǫn for N − L(n) ≤ n′ ≤ N . Let us take a subsequence {vN(n)}n anddenote it by {vn}n . For simplicity of the notation, one may assume n′ = n−1.

Using this, one can verify ker(d+)∗τ = 0 over Y (S, 2). Let us put ∆ = dd∗τ ⊕(d+)∗τd+ on W k+2

τ (Y (S, 2); Λ1). From above, one has the estimate:

|dd∗τ (ϕ(n − 1)vn)|W kτ

≤ C{|dvn|Wkτ (W n−1

0 ([−S, S])) + |vn|Wkτ (W n−1

0 ([−S, S]))} ≤ Cǫn(5.5)


826 Tsuyoshi Kato

where C is independent of n. Then one has the estimates:

|∆(ϕ(n − 1)vn)|W kτ ≤ C[|(d+)∗τ (ϕ(n − 1)ϕ(n)u)|W k

τ

+ |(d+)∗τ [d+, ϕ(n − 1)]vn|Wkτ (Suppdϕ(n − 1))] + Cǫn

≤ C[|ϕ(n− 1)ϕ(n)(d+)∗τ (u)|W kτ

+ |[(d+)∗τ , ϕ(n − 1)ϕ(n)]u|W kτ (∪j=n−1,nW

j0 ([−S, S]))

+ |(d+)∗τ [d+, ϕ(n − 1)]vn|Wkτ (W n−1

0 ([−S, S])) + Cǫn

≤ |[(d+)∗τ , ϕ(n − 1)ϕ(n)]u|W kτ + |(d+)∗τ [d+, ϕ(n − 1)]vn|W

kτ + Cǫn.

(5.6)

One may assume that the last term is arbitrarily small. This shows thatthere is a sequence {wn ∈ W k+2

τ (Y (S, 2); Λ1)} with C ≤ |wn|Wk+2τ ≤ C ′

and |P (wn)|W k+1τ converges to zero. This is a contradiction. This shows

ker(d+)∗τ = 0 over Y (S, 2).

Let (Wt, N0) = (D4, S1 ×D2) be the standard disk. Let Y = Wt\N0([S,∞))∪Y (S)0\N0([S,∞)). In this case, one has H0(AHS) = H1(AHS) = 0 (notice Yis simply connected). But H2(Y ; R) = R. In this case, one has the following:

Corollary 5.1 Suppose Y has nonzero H2(Y ; R). Then one has a bounddimH2(AHS) ≤ 2 dimH2(Y ; R).

Proof Suppose dimH2(AHS) ≥ 2 dimH2(Y ; R) + 1 = b2 + 1. Let us take L2τ

orthogonal vectors u1, . . . , ub2+1, · · · ∈ L2τ (Y : Λ2

+) ∩ ker(d+)∗τ with |ui|L2τ = 1.

One can make a family of cylindrical manifolds Y (n) by the above method. Bya straightforward calculation, one has an upper bound of dimH2(Y (n); R) byb2 . By remark 4.2, it follows that dimH2(AHS) has also an upper bound overY (n). Now let ϕ(n) be as above. Then for any small ǫ > 0, there exists a largen0 such that for all n ≥ n0 , one has:

|(d+)∗τ (ϕ(n)ui)|L2τ ≤ ǫ, |ϕ(n)ui|L

2τ ≥ 1 − ǫ,

〈ϕ(n)ui, ϕ(n)uj〉|L2τ ≤ ǫ, i = 1, . . . , b2 + 1.

(5.7)

Let v1, . . . , vl be an orthonormal basis of H2(AHS) over Y (n), l ≤ b2 . Thenone may express ϕ(n)ui = Σja

ijvj +d+(αi), a

ij ∈ R. Let ai = (ai

1, . . . , ail) ∈ R

l .Then the set of vectors a1, . . . , ab2+1 would satisfy |ai| ≥ 1− ǫ and 〈ai, ai′〉 ≤ ǫwith respect to the standard norm in R

l . Since l < b2 + 1, it is impossible tofind such set. This completes the proof.



5.D Fourier–Laplace transforms on higher stages

Let us put:

W2([−S, S]) = W2\N0([S,∞)) ∪j=1,2 Mj([S,∞)),

Y (S, 2)0 = W2 ∪M2(S)∼N0(S) Y (S)0\N0([S,∞)),

Y (S, 2) = . . . Y (S, 2)−n0 ∪

M−n1

(S)∼N−n+1

0(S) Y (S, 2)−n+1

0 . . .

Y (S, 2)0 = Y (S, 2)10 ∪M11(S)∼N2

0(S) Y (S, 2)20 ∪M2

1(S)∼N3

0(S) Y (S, 2)30 . . .

(5.8)

where (Y (S, 2)n0 ,Mnj (S), Nn

0 (S)) is the same triple (Y (S, 2)0,Mj(S),N0(S))

as before. Notice that Y (S, 2)0 is diffeomorphic to CH(T 02 ), where T 0

2 =R+ ∪n∈N R+ . Now by the Fourier–Laplace transform, one gets the following:

Corollary 5.2 Pτ : W k+1τ (Y (S, 2); Λ1) → W k

τ (Y (S, 2); Λ0 ⊕Λ2+) is invertible.

Now one defines: T3 = R ∪n∈Z T02 and T 0

3 = R+ ∪n∈N T02 . Similarly T4 =

R ∪n∈Z T03 . One inductively defines Tj , j = 1, 2, . . . as:

Tj+1 = R ∪n∈Z T0j , T 0

j+1 = R+ ∪n∈N T0j

where one puts T1 = R, T 01 = R+ .

One can construct the corresponding spaces. Let (W2,N0,M1,M2) be a kinkyhandle with two kinks. One has already defined Y (S), Y (S, 2). Let us defineinductively Y (S, j) as follows:

Y (S, j)0 = W2([−S, S]) ∪M2(S)∼N0(S) Y (S, j − 1)0,

Y (S, j) = Y (S, j)0/N0(S) ∼M1(S),

Y (S, j) = . . . Y (S, j)−n0 ∪M−n

1(S)∼N−n+1

0(S) Y (S, j)−n+1

0 . . .

Y (S, j)0 = Y (S, j)1 ∪M11(S)∼N2

0(S) Y (S, j)2 ∪M2

1(S)∼N3

0(S) Y (S, j)3 . . .

(5.9)

One may express CH(T 0j ) = Y (S, j)0 . The previous method works for all

Y (S, j) iteratively.

Proposition 5.2 Pτ over Y (S, j) are all Fredholm with H∗(AHS) = 0 for∗ = 0, 1, 2 and j = 1, 2, . . . .

In the notation in 2.B, one expresses T 0j = (T2,2,...,2,1)0 ((2, . . . , 2) j− 1 times).

Let n1, . . . , nk ∈ {1, 2, . . . } be a set of positive integers. Then using kinkyhandles with nj kinks, one has a natural extension, and gets (Tn1,...,nk,1)0 whichis a signed infinite tree. For Riemannian metrics on CH((Tn1,...,nk,1)0), see 2.B.


828 Tsuyoshi Kato

Let n = {n1, . . . , nl−1, 1} be a set of positive integers, and denote the cor-responding homogeneous tree of bounded type by (Tn)0 . By the previousmethod, one gets a complete Riemannian metric and a weight function on everyCH((Tn)0). Recall that one has constructed complete Riemannian metrics andweight functions on Y (S, n).

Proposition 5.3 Pτ : W k+1τ (Y (S, n); Λ1) → W k

τ (Y (S, n); Λ0 ⊕ Λ2+) gives an

isomorphism for any n = {n1, . . . , nl−1, 1}.

Proof One has shown the result for l = 2. Suppose the result is true forl ≤ l0 . Let us put n = {n1, . . . , nl0+1, 1}. Then by Fourier–Laplace trans-form and excision method used before, one knows Pτ : W k+1

τ (Y (S, n); Λ1) →W k

τ (Y (S, n); Λ0 ⊕ Λ2+) gives a closed operator with H∗(AHS) = 0 for ∗ = 0, 1.

One may follow the same process to see H2(AHS) = 0 as 5.C. Thus one hasshown the result for l0 + 1. This completes the induction step.

In practical applications, one considers open four–manifolds composed of one0–handle attached with Casson handles. Recall that k(S2×S2)\pt is homotopy-equivalent to some wedges of S2 . In particular H1

cp(M ; R) = H3(M ; R) = 0.Recall also that it has a link picture by k disjoint union of Hopf links with0–framings. Let (Wt,M1, . . . ,M2k) = (D4, S1 ×D2, . . . , S1 ×D2) express thelink diagram of k(S2 × S2)\pt.

Let (T1)0, . . . , (T2k)0 be signed homogeneous trees of bounded type. Let usconsider an open four–manifold S obtained by attaching CH((Ti)0) along(D4, S1 ×D3, . . . , S1 ×D3). Then by the previous procedure, one can equip acomplete Riemannian metric on S :

S ≡ Wt♮ ∪l CH((Tl)0).

As before one can also equip a weight function τ , and the AHS complex overS .

Corollary 5.3 The differential of AHS complex has closed range over S withH0(AHS) = H1(AHS) = 0 and dimH2(AHS) ≤ 2 dimH2(S; R).

By the work of Freedman [15], the end of S admits a topological color, ∼=S3 × [0,∞). In fact S is homeomorphic to k(S2 × S2)\pt. Now we havecompleted the verification that any open four–manifold with a tree-like end ofbounded type can admit an admissible pair (g, τ) on it.



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