A Basic Introduction to Surgery Theory

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Wolfgang L¨ uck

¹

Fachbereich Mathematik und Informatik Westf¨ alische Wilhelms-Universit¨ at M¨ unster

Einsteinstr. 62 48149 M¨ unster

Germany October 27, 2004

1email: lueck@math.uni-muenster.de

www: http://www.math.uni-muenster.de/u/lueck/

FAX: 49 251 8338370

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II

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Preface

This manuscript contains extended notes of the lectures presented by the au- thor at the summer school “High-dimensional Manifold Theory” in Trieste in May/June 2001. It is written not for experts but for talented and well educated graduate students or Ph.D. students who have some background in algebraic and differential topology. Surgery theory has been and is a very successful and well established theory. It was initiated and developed by Browder, Kervaire, Milnor, Novikov, Sullivan, Wall and others and is still a very active research area. The idea of these notes is to give young mathematicians the possibility to get access to the field and to see at least a small part of the results which have grown out of surgery theory. Of course there are other good text books and survey articles about surgery theory, some of them are listed in the references.

The Chapters 1 and 2 contain interesting and beautiful results such as the s-Cobordism Theorem and the classification of lens spaces including their illuminating proofs. If one wants to start with the surgery machinery immediately, one may skip these chapters and pass directly to Chapters 3, 4 and 5. As an application we present the classification of homotopy spheres in Chapter 6.

Chapters 7 and 8 contain material which is directly related to the main topic of the summer school.

I thank the members of the topology group at M¨unster and the participants of the summer school who made very useful comments and suggestions, in particular C.L. Douglas, J. Verrel, T. A. Kro, R. Sauer and M. Szymik. I thank the ICTP for its hospitality and financial support for the school and the conference.

M¨unster, October 27, 2004 Wolfgang L¨uck

III

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IV Preface

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The s-Cobordism Theorem

Introduction

In this chapter we want to discuss and prove the following result

Theorem 1.1 (s-Cobordism Theorem) Let M₀ be a closed connected ori- ented manifold of dimension n≥5 with fundamental groupπ=π₁(M₀). Then 1. Let (W;M0, f0, M1, f1) be an h-cobordism over M0. Then W is trivial overM0if and only if its Whitehead torsionτ(W, M0)∈Wh(π)vanishes;

2. For any x∈Wh(π)there is an h-cobordism (W;M₀, f₀, M₁, f₁)over M₀ withτ(W, M₀) =x∈Wh(π);

3. The function assigning to an h-cobordism (W;M0, f0, M1, f1) over M0

its Whitehead torsion yields a bijection from the diffeomorphism classes relativeM0 ofh-cobordisms overM0 to the Whitehead groupWh(π).

Here are some explanations. Ann-dimensional cobordism (sometimes also called just bordism) (W;M₀, f₀, M₁, f₁) consists of a compact orientedn-dimensional manifoldW, closed (n−1)-dimensional manifoldsM₀andM₁, a disjoint decomposition ∂W =∂0W`∂1W of the boundary ∂W ofW and orientation preserving diffeomorphisms f0: M0 → ∂W0 and f1:M₁⁻ → ∂W1. Here and in the sequel we denote byM₁⁻ the manifoldM1 with the reversed orientation and we use on ∂W the orientation with respect to the decompositionTxW = Tx∂W⊕Rcoming from an inward normal field for the boundary. If we equipD² with the standard orientation coming from the standard orientation on R², the induced orientation onS¹=∂D² corresponds to the anti-clockwise orientation on S¹. If we want to specify M0, we say that W is a cobordism over M0. If

∂0W =M0,∂1W =M₁⁻ andf0 andf1are given by the identity or iff0andf1

are obvious from the context, we briefly write (W;∂₀W, ∂₁W). Two cobordisms (W, M₀, f₀, M₁, f₁) and (W⁰, M₀, f₀⁰, M₁⁰, f₁⁰) over M₀ arediffeomorphic relative M₀ if there is an orientation preserving diffeomorphismF: W →W⁰ withF◦

1

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f₀ =f₀⁰. We call anh-cobordism overM₀trivial, if it is diffeomorphic relative M₀ to the trivial h-cobordism (M₀×[0,1];M₀× {0},(M₀ × {1})⁻). Notice that the choice of the diffeomorphisms f_i do play a role although they are often suppressed in the notation. We call a cobordism (W;M₀, f₀, M₁, f₁) an h-cobordism, if the inclusions∂iW →W fori= 0,1 are homotopy equivalences.

We will later see that the Whitehead group of the trivial group vanishes.

Thus thes-Cobordism Theorem 1.1 implies

Theorem 1.2 (h-Cobordism Theorem) Anyh-cobordism(W;M0, f0, M1, f1) over a simply connected closedn-dimensional manifoldM0 withdim(W)≥6 is trivial.

Theorem 1.3 (Poincar´e Conjecture) The Poincar´e Conjecture is true for a closedn-dimensional manifold M with dim(M)≥5, namely, if M is simply connected and its homologyH_p(M)is isomorphic toH_p(Sⁿ)for allp∈Z, then M is homeomorphic to Sⁿ.

Proof : We only give the proof for dim(M) ≥ 6. Since M is simply connected andH_∗(M)∼=H_∗(Sⁿ), one can conclude from the Hurewicz Theorem and Whitehead Theorem [121, Theorem IV.7.13 on page 181 and Theorem IV.7.17 on page 182] that there is a homotopy equivalencef:M →Sⁿ. Let D_iⁿ ⊂M fori= 0,1 be two embedded disjoint disks. PutW =M−(int(Dⁿ₀)`int(D₁ⁿ)).

Then W turns out to be a simply connectedh-cobordism. Hence we can find a diffeomorphism F: (∂Dⁿ₀ ×[0,1], ∂Dⁿ₀ × {0}, ∂D₀ⁿ× {1}) → (W, ∂Dⁿ₀, ∂D₁ⁿ) which is the identity on ∂D₀ⁿ =∂D₀ⁿ× {0} and induces some (unknown) dif- feomorphismf1: ∂Dⁿ₀ × {1} → ∂Dⁿ₁. By the Alexander trick one can extend f1:∂Dⁿ₀ =∂Dⁿ₀×{1} →∂Dⁿ₁ to a homeomorphismf1:Dⁿ₀ →Dⁿ₁. Namely, any homeomorphism f: Sⁿ⁻¹ →Sⁿ⁻¹ extends to a homeomorphismf:Dⁿ →Dⁿ by sendingt·xfort∈[0,1] andx∈Sⁿ⁻¹tot·f(x). Now define a homeomor- phismh:Dⁿ₀×{0}∪i0∂D₀ⁿ×[0,1]∪i1Dⁿ₀×{1} →M for the canonical inclusions i_k: ∂Dⁿ₀ × {k} →∂D₀ⁿ×[0,1] for k = 0,1 by h|D₀ⁿ×{0} = id,h|∂Dⁿ₀×[0,1] =F and h|Dⁿ₀×{1} = f₁. Since the source of his obviously homeomorphic to Sⁿ, Theorem 1.3 follows.

In the case dim(M) = 5 one uses the fact that M is the boundary of a contractible 6-dimensional manifoldW and applies thes-cobordism theorem to W with an embedded disc removed.

Remark 1.4 Notice that the proof of the Poincar´e Conjecture in Theorem 1.3 works only in the topological category but not in the smooth category. In other words, we cannot conclude the existence of a diffeomorphismh: Sⁿ→M. The proof in the smooth case breaks down when we apply the Alexander trick. The construction off given by coningf yields only a homeomorphismf and not a diffeomorphism even if we start with a diffeomorphismf. The mapf is smooth outside the origin ofDⁿ but not necessarily at the origin. We will see that not every diffeomorphism f:Sⁿ⁻¹ → Sⁿ⁻¹ can be extended to a diffeomorphism Dⁿ → Dⁿ and that there exist so called exotic spheres, i.e. closed manifolds

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Introduction 3 which are homeomorphic toSⁿ but not diffeomorphic toSⁿ. The classification of these exotic spheres is one of the early very important achievements of surgery theory and one motivation for its further development (see Chapter 6).

Remark 1.5 In some sense the s-Cobordism Theorem 1.1 is one of the first theorems, where diffeomorphism classes of certain manifolds are determined by an algebraic invariant, namely the Whitehead torsion. Moreover, the Whitehead group Wh(π) depends only on the fundamental groupπ=π1(M0), whereas the diffeomorphism classes of h-cobordisms overM0 a priori depend on M0 itself.

Thes-Cobordism Theorem 1.1 is one step in a program to decide whether two closed manifolds M and N are diffeomorphic what is in general a very hard question. The idea is to construct an h-cobordism (W;M, f, N, g) with van- ishing Whitehead torsion. Then W is diffeomorphic to the trivialh-cobordism overM what implies thatM andN are diffeomorphic. So thesurgery program would be:

1. Construct a homotopy equivalencef:M →N;

2. Construct a cobordism (W;M, N) and a map (F, f,id) : (W;M, N) → (N×[0,1], N× {0}, N× {1});

3. Modify W and F relative boundary by so called surgery such that F becomes a homotopy equivalence and thus W becomes an h-cobordism.

During these processes one should make certain that the Whitehead torsion of the resultingh-cobordism is trivial.

The advantage of this approach will be that it can be reduced to problems in homotopy theory and algebra which can sometimes be handled by well-known techniques. In particular one will get sometimes computable obstructions for two homotopy equivalent manifolds to be diffeomorphic. Often surgery theory has proved to be very useful when one wants to distinguish two closed manifolds which have very similar properties. The classification of homotopy spheres (see Chapter 6) is one example. Moreover, surgery techniques can be applied to problems which are of different nature than of diffeomorphism or homeomorphism classifications.

In this chapter we want to present the proof of the s-Cobordism Theorem and explain why the notion of Whitehead torsion comes in. We will encounter a typical situation in mathematics. We will consider an h-cobordism and try to prove that it is trivial. We will introduce modifications which we can apply to a handlebody decomposition without changing the diffeomorphism type and which are designed to reduce the number of handles. If we could get rid of all handles, the h-cobordism would be trivial. When attempting to cancel all handles, we run into an algebraic difficulty. A priori this difficulty could be a lack of a good idea or technique. But it will turn out to be the principal obstruction and lead us to the definition of the Whitehead torsion and Whitehead group.

The rest of this Chapter is devoted to the proof of thes-cobordism Theorem 1.1. Its proof is interesting and illuminating and it motivates the definition of

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Whitehead torsion. But we mention that it is not necessary to go through it in order to understand the following chapters.

1.1 Handlebody Decompositions

In this section we explain basic facts about handles and handlebody decompositions.

Definition 1.6 Then-dimensional handle of indexqor brieflyq-handle isD^q× Dⁿ⁻^q. Its coreisD^q× {0}. The boundary of the coreisS^q⁻¹× {0}. Its cocore is{0} ×Dⁿ⁻^q and its transverse sphere is{0} ×Sⁿ⁻^q⁻¹.

Let(M, ∂M)be ann-dimensional manifold with boundary∂M. Ifφ^q:S^q⁻¹× Dⁿ⁻^q→∂M is an embedding, then we say that the manifoldM + (φ^q) defined by M ∪φ^qD^q ×Dⁿ⁻^q is obtained from M by attaching a handle of index q by φ^q.

ObviouslyM+ (φ^q) carries the structure of a topological manifold. To get a smooth structure, one has to use the technique of straigthening the angle to get rid of the corners at the place, where the handle is glued toM. The boundary

∂(M + (φ^q)) can be described as follows. Delete from ∂M the interior of the image of φ^q. We obtain a manifold with boundary together with a diffeomorphism from S^q⁻¹×Sⁿ⁻^q⁻¹ to its boundary induced by φ^q|S^q−1×S^n−q−1. If we use this diffeomorphism to glueD^q×Sⁿ⁻^q⁻¹to it, we obtain a closed manifold, namely,∂(M + (φ^q)).

Let W be a compact manifold whose boundary ∂W is the disjoint sum

∂0W`∂1W. Then we want to construct W from ∂0W ×[0,1] by attaching handles as follows. Notice that the following construction will not change∂0W =

∂0W× {0}. Ifφ^q:S^q⁻¹×Dⁿ⁻^q →∂1W is an embedding, we get by attaching a handle the compact manifold W1 = ∂0W ×[0,1] + (φ^q) which is given by W ∪φ^qD^q×Dⁿ⁻^q. Its boundary is a disjoint sum ∂0W1`

∂1W1, where∂0W1

is the same as∂0W. Now we can iterate this process, where we attach a handle to∂1W1. Thus we obtain a compact manifold with boundary

W =∂₀W×[0,1] + (φ^q₁¹) + (φ^q₂²) +. . .+ (φ^q_r^r),

whose boundary is the disjoint union∂₀W`∂₁W, where∂₀W is just∂₀W×{0}. We call such a description of W as above a handlebody decomposition of W relative∂₀W. We get from Morse theory [54, Chapter 6], [82, part I].

Lemma 1.7 Let W be a compact manifold whose boundary∂W is the disjoint sum∂0W`∂1W. ThenW possesses a handlebody decomposition relative∂0W, i.e. W is up to diffeomorphism relative∂0W =∂0W× {0} of the form

W =∂0W×[0,1] + (φ^q₁¹) + (φ^q₂²) +. . .+ (φ^q_r^r).

If we want to show that W is diffeomorphic to∂0W ×[0,1] relative∂0W =

∂₀W×{0}, we must get rid of the handles. For this purpose we have to find pos- sible modifications of the handlebody decomposition which reduce the number of handles without changing the diffeomorphism type ofW relative∂₀W.

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1.1. HANDLEBODY DECOMPOSITIONS 5 Lemma 1.8 (Isotopy lemma) Let W be ann-dimensional compact manifold whose boundary∂W is the disjoint sum∂₀W`∂₁W. Ifφ^q, ψ^q:S^q⁻¹×Dⁿ⁻^q →

∂₁W are isotopic embeddings, then there is a diffeomorphismW+ (φ^p)→W+ (ψ^q)relative ∂₀W.

Proof : Leti: S^q⁻¹×Dⁿ⁻^q×[0,1]→∂₁W be an isotopy fromφ^q toψ^q. Then one can find a diffeotopy H:W ×[0,1] → W with H₀ = id_W such that the composition ofH withφ^q×id_[0,1] isiandHis stationary on∂₀W [54, Theorem 1.3 in Chapter 8 on page 184]. ThusH1:W →W is a diffeomorphism relative

∂0W and satisfies H1◦φ^q = ψ^q. It induces a diffeomorphism W + (φ^p) → W + (ψ^q) relative∂0W.

Lemma 1.9 (Diffeomorphism lemma) LetW resp. W⁰ be a compact manifold whose boundary ∂W is the disjoint sum∂₀W`∂₁W resp. ∂₀W⁰`∂₁W⁰. LetF:W →W⁰be a diffeomorphism which induces a diffeomorphismf₀:∂₀W →

∂₀W⁰. Letφ^q:S^q⁻¹×Dⁿ⁻^q→∂₁W be an embedding. Then there is an embed- dingφ^q:S^q⁻¹×Dⁿ⁻^q →∂₁W⁰and a diffeomorphismF⁰:W+(φ^q)→W⁰+(φ^q) which inducesf0 on∂0W.

Proof : Putφ^q =F◦φ^q.

Lemma 1.10 Let W be an n-dimensional compact manifold whose boundary

∂W is the disjoint sum ∂0W`

∂1W. Suppose that V =W + (ψ^r) + (φ^q) for q≤r. Then V is diffeomorphic relative ∂0W toV⁰ =W+ (φ^q) + (ψ^r) for an appropriate φ^q.

Proof : By transversality and the assumption (q−1) + (n−1−r) < n−1 we can show that the embedding φ^q|S^q−1×{0}: S^q⁻¹× {0} → ∂₁(W + (ψ^r)) is isotopic to an embedding which does not meet the transverse sphere of the handle (ψ^r) attached by ψ^r [54, Theorem 2.3 in Chapter 3 on page 78]. This isotopy can be embedded in a diffeotopy on∂1(W+ (ψ^r)). Thus the embedding φ^q: S^q⁻¹×Dⁿ⁻^q→∂1(W+ (ψ^r)) is isotopic to an embedding whose restriction toS^q⁻¹× {0}does not meet the transverse sphere of the handle (ψ^r). Since we can isotope an embeddingS^q⁻¹×Dⁿ⁻^q →W+(ψ^r) such that its image becomes arbitrary close to the image ofS^q⁻¹× {0}, we can isotope φ^q:S^q⁻¹×Dⁿ⁻^q →

∂1(W + (ψ^r)) to an embedding which does not meet a closed neighborhood U ⊂ ∂1(W + (ψ^r)) of the transverse sphere of the handle (ψ^r). There is an obvious diffeotopy on∂1(W+ (ψ^r)) which is stationary on the transverse sphere of (ψ^r) and moves any point on∂₁(W+ (ψ^r)) which belongs to the handle (ψ^r) but not toU to a point outside the handle (ψ^r). Thus we can find an isotopy of φ^q to an embeddingφ^p which does not meet the handle (ψ^r) at all. Obviously W + (ψ^r) + (φ^q) andW + (φ^q) + (ψ^r) agree. By the Isotopy Lemma 1.8 there is a diffeomorphism relative ∂₀W from W + (ψ^r) + (φ^q) to W + (ψ^r) + (φ^q).

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Example 1.11 Here is a standard situation, where attaching first a q-handle and then a (q+ 1)-handle does not change the diffeomorphism type of an n- dimensional compact manifoldWwith the disjoint union∂₀W`∂₁W as boundary∂W. Let 0≤q≤n−1. Consider an embedding

µ:S^q⁻¹×Dⁿ⁻^q∪_Sq−1×S₊^n−1−qD^q×S₊ⁿ⁻¹⁻^q →∂₁W,

where S₊ⁿ⁻¹⁻^q is the upper hemisphere in Sⁿ⁻¹⁻^q = ∂Dⁿ⁻^q. Notice that the source of µ is diffeomorphic to Dⁿ⁻¹. Let φ^q: S^q⁻¹ ×Dⁿ⁻^q → ∂₁W be its restriction to S^q⁻¹×Dⁿ⁻^q. Let φ^q+1₊ : S₊^q ×Sⁿ₊⁻^q⁻¹ → ∂1(W + (φ^q)) be the embedding which is given by

S₊^q ×S₊ⁿ⁻^q⁻¹=D^q×S₊ⁿ⁻^q⁻¹⊂D^q×Sⁿ⁻^q⁻¹=∂(φ^q)⊂∂1(W + (φ^q)).

It does not meet the interior ofW. Letφ^q+1₋ :S₋^q ×S₊ⁿ⁻¹⁻^q →∂₁(W∪(φ^q)) be the embedding obtained fromµby restriction toS₋^q ×Sⁿ₊⁻¹⁻^q =D^q×S₊ⁿ⁻¹⁻^q. Thenφ^q+1₋ andφ^q+1₊ fit together to yield an embeddingψ^q+1:S^q×Dⁿ⁻^q⁻¹= S₋^q ×S₊ⁿ⁻^q⁻¹∪_Sq−1×S₊^n−q−1 S₊^q ×S₊ⁿ⁻^q⁻¹ → ∂1(W + (φ^q)). Then it is not difficult to check that W + (φ^q) + (ψ^q+1) is diffeomorphic relative ∂0W to W since up to diffeomorphism W + (φ^q) + (ψ^q+1) is obtained from W by taking the boundary connected sum ofW andDⁿ along the embeddingµofDⁿ⁻¹= S₊ⁿ⁻¹=S^q⁻¹×Dⁿ⁻^q∪_Sq−1×S^n−1−q₊ D^q×S₊ⁿ⁻¹⁻^q into ∂1W.

This cancellation of two handles of consecutive index can be generalized as follows.

Lemma 1.12 (Cancellation Lemma) Let W be an n-dimensional compact manifold whose boundary∂W is the disjoint sum∂₀W`∂₁W. Letφ^q:S^q⁻¹× Dⁿ⁻^q→∂₁W be an embedding. Letψ^q+1:S^q×Dⁿ⁻¹⁻^q→∂₁(W+ (φ^q))be an embedding. Suppose thatψ^q+1(S^q× {0})is transversal to the transverse sphere of the handle (φ^q) and meets the transverse sphere in exactly one point. Then there is a diffeomorphism relative ∂0W fromW toW+ (φ^q) + (ψ^q+1).

Proof : Given any neighborhood U ⊂ ∂(φ^q) of the tranverse sphere of (φ^q), there is an obvious diffeotopy on∂1(W+ (φ^q)) which is stationary on the transverse sphere of (φ^q) and moves any point on ∂1(W + (φ^q)) which belongs to the handle (φ^q) but not to U to a point outside the handle (φ^q). Thus we can achieve that ψ^q+1 maps the lower hemisphereS₋^q × {0} to points outside (φ^q) and is on the upper hemisphere S₊^q × {0} given by the obvious inclusion D^q× {x} →D^p×Dⁿ⁻^q = (φ^q) for somex∈Sⁿ⁻^q⁻¹and the obvious identifica- tion ofS^q₊×{0}withD^q×{x}. Now it is not hard to construct an diffeomorphism relative∂0W fromW+ (φ^q) + (ψ^q+1) toW modelling the standard situation of Example 1.11.

The Cancellation Lemma 1.12 will be our only tool to reduce the number of handles. Notice that one can never get rid of one handle alone, there must always be involved at least two handles simultaneously. The reason is that the

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1.1. HANDLEBODY DECOMPOSITIONS 7 Euler characteristicχ(W, ∂₀W) is independent of the handle decomposition and can be computed by P

q≥0(−1)^q·p_q, wherep_q is the number ofq-handles (see Section 1.2).

We call an embedding S^q ×Dⁿ⁻^q → M for q < n into an n-dimensional manifoldtrivial if it can be written as the composition of an embeddingDⁿ→ W and a fixed standard embeddingS^q×Dⁿ⁻^q →Dⁿ. We call an embedding S^q → M for q < n trivial if it can be extended to a trivial embedding S^q× Dⁿ⁻^q →M. We conclude from the Cancellation Lemma 1.12

Lemma 1.13 Let φ^q:S^q⁻¹×Dⁿ⁻^q →∂1W be a trivial embedding. Then there is an embedding φ^q+1:S^q ×Dⁿ⁻¹⁻^q → ∂1(W + (φ^q)) such that W and W + (φ^q) + (φ^q+1)are diffeomorphic relative ∂0W.

Consider a compactn-dimensional manifoldW whose boundary is the disjoint union∂0W`

∂1W. In view of Lemma 1.7 and Lemma 1.10 we can write it

W ∼= ∂₀W×[0,1] +

p0

X

i=1

(φ⁰_i) +

p1

X

i=1

(φ¹_i) +. . .+

pn

X

i=1

(φⁿ_i), (1.14) where ∼= means diffeomorphic relative∂0W.

Notation 1.15 Put for−1≤q≤n W_q := ∂₀W ×[0,1] +

p0

X

i=1

(φ⁰_i) +

p1

X

i=1

(φ¹_i) +. . .+

p_q

X

i=1

(φ^q_i);

∂₁W_q := ∂W_q−∂₀W × {0};

∂₁^◦Wq := ∂1Wq−

pq+1

a

i=1

φ^q+1_i (S^q×int(Dⁿ⁻¹⁻^q)).

Notice for the sequel that ∂^◦₁Wq ⊂∂1Wq+1.

Lemma 1.16 (Elimination Lemma) Fix an integer q with 1 ≤ q ≤n−3.

Suppose that pj = 0forj < q, i.e. W looks like W = ∂₀W×[0,1] +

p_q

X

i=1

(φ^q_i) +

p_q+1

X

i=1

(φ^q+1_i ) +. . .+

p_n

X

i=1

(φⁿ_i).

Fix an integer i0 with 1 ≤ i0 ≤ pq. Suppose that there is an embedding ψ^q+1:S^q×Dⁿ⁻¹⁻^q →∂^◦₁Wq with the following properties:

1. ψ^q+1|S^q×{0} is isotopic in∂1Wq to an embeddingψ^q+1₁ :S^q× {0} →∂1Wq

which meets the transverse sphere of the handle(φ^q_i₀)transversally and in exactly one point and is disjoint from to the transverse sphere of φ^q_i for i6=i₀;

2. ψ^q+1|S^q×{0}is isotopic in∂₁W_q+1to a trivial embeddingψ^q+1₂ :S^q×{0} →

∂₁W_q+1.

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ThenW is diffeomorphic relative∂₀W to a manifold of the shape

∂₀W×[0,1]+ X

i=1,2,...,pq,i6=i0

(φ^q_i)+

p_q+1

X

i=1

(φ^q+1_i )+(ψ^q+2)+

p_q+2

X

i=1

(φ^q+2_i )+. . .+

p_n

X

i=1

(φⁿ_i).

Proof : Sinceψ^q+1|S^q×{0}is isotopic toψ^q+1₁ andψ^q+1₂ is trivial, we can extend ψ^q+1₁ andψ₂^q+1to embeddings denoted in the same wayψ^q+1₁ :S^q×Dⁿ⁻^q⁻¹→

∂1Wq and ψ^q+1₂ :S^q ×Dⁿ⁻¹⁻^q → ∂₁^◦Wq+1 with the following properties [54, Theorem 1.5 in Chapter 8 on page 180]: ψ^q+1is isotopic toψ₁^q+1 in∂1Wq,ψ₁^q+1 does not meet the transverse spheres of the handles (φ^q_i) fori6=i₀,ψ₁^q+1|S^q×{0}

meets the transverse sphere of the handle (φ^q_i

0) transversally and in exactly one point, ψ^q+1 is isotopic to ψ^q+1₂ within∂₁W_q+1 and ψ^q+1₂ is trivial. Because of the Diffeomorphism Lemma 1.9 we can assume without loss of generality that there are no handles of index ≥q+ 2, i.e. pq+2 = pq+3 =. . . = pn = 0. It suffices to show for appropriate embeddingsφ^q+1_i andψ^q+2 that

∂₀W ×[0,1] +

p_q

X

i=1

(φ^q_i) +

p_q+1

X

i=1

(φ^q+1_i )

∼= ∂0W×[0,1] + X

i=1,2,...,pq,i6=i0

(φ^q_i) +

p_q+1

X

i=1

(φ^q+1_i ) + (ψ^q+2),

where∼= means diffeomorphic relative∂₀W. Because of Lemma 1.13 there is an embedding (ψ^q+2) satisfying

∂0W×[0,1] +

p_q

X

i=1

(φ^q_i) +

p_q+1

X

i=1

(φ^q+1_i )

∼= ∂0W ×[0,1] +

p_q

X

i=1

(φ^q_i) +

p_q+1

X

i=1

(φ^q+1_i ) + (ψ^q+1₂ ) + (ψ^q+2).

We conclude from the Isotopy Lemma 1.8 and the Diffeomorphism Lemma 1.9 for appropriate embeddingsψ^q+1_k fork= 1,2

∂0W ×[0,1] +

p_q

X

i=1

(φ^q_i) +

p_q+1

X

i=1

(φ^q+1_i ) + (ψ^q+1₂ ) + (ψ^q+2)

∼= ∂0W×[0,1] +

pq

X

i=1

(φ^q_i) +

pq+1

X

i=1

(φ^q+1_i ) + (ψ^q+1) + (ψ₁^q+2)

∼= ∂₀W×[0,1] +

p_q

X

i=1,2,...,p_q,i6=i₀

(φ^q_i) + (φ^q_i

0) + (ψ^q+1) +

p_q+1

X

i=1

(φ^q+1_i ) + (ψ^q+2₂ ).

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1.2. HANDLEBODY DECOMPOSITIONS ANDCW-STRUCTURES 9 We get from the Diffeomorphism Lemma 1.9 and the Cancellation Lemma 1.12 for appropriate embeddingsφ^q+1_i andψ^q+2₃

∂0W ×[0,1] +

pq

X

i=1,2,...,p_q,i6=i₀

(φ^q_i) + (φ^q_i

0) + (ψ^q+1) +

pq+1

X

i=1

(φ^q+1_i ) + (ψ₂^q+2)

∼= ∂0W×[0,1] +

pq

X

i=1,2,...,pq,i6=i0

(φ^q_i) +

pq+1

X

i=1

(φ^q+1_i ) + (ψ^q+2₃ ).

This finishes the proof of the Elimination Lemma 1.16.

1.2 Handlebody Decompositions and CW -Struc- tures

Next we explain how we can associate to a handlebody decomposition (1.14) a CW-pair (X, ∂0W) such that there is a bijective correspondence between the q-handles of the handlebody decomposition and the q-cells of (X, ∂0W). The key ingredient is the elementary fact that the projection (D^q×Dⁿ⁻^q, S^q⁻¹× Dⁿ⁻^q)→(D^q, S^q⁻¹) is a homotopy equivalence and actually – as we will explain later – a simple homotopy equivalence.

Recall that a (relative)CW-complex (X, A) consists of a pair of topological spaces (X, A) together with a filtration

X₋₁=A⊂X₀⊂X₁⊂. . .⊂X_q⊂X_q+1⊂. . .⊂ ∪q≥0X_q=X

such that X carries the colimit topology with respect to this filtration and for any q≥0 there exists a pushout of spaces

`

i∈IqS^q⁻¹

`

i∈Iqφ^q_i

−−−−−−→ Xq−1



 y



 y

`

i∈I_qD^q −−−−−−→`

i∈IqΦ^q_i

Xq

The map φ^q_i is called the attaching map and the map (Φ^q_i, φ^q_i) is called the characteristic map of the q-cell belonging to i ∈ Iq. The pushouts above are not part of the structure, only their existence is required. Only the filtration {Xq |q≥ −1} is part of the structure. The path components ofXq−Xq−1 are called theopen cells. The open cells coincide with the sets Φ^q_i(D^q−S^q⁻¹). The closure of an open cell Φ^q_i(D^q−S^q⁻¹) is calledclosed cell and turns out to be Φ^q_i(D^q).

Suppose thatX is connected with fundamental groupπ. Letp:Xe →X be the universal covering ofX. PutXfq =p⁻¹(Xq) andAe=p⁻¹(A). Then (X,e A)e inherits a CW-structure from (X, A) by the filtration {Xf_q | q ≥ −1}. The

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cellular Zπ-chain complexC_∗(X,e A) has ase q-thZπ-chain module the singular homologyH_q(Xf_q,X]_q₋₁) withZ-coefficients and the π-action coming from the deck transformations. Theq-th differentiald_q is given by the composition

H_q(Xf_q,X]_q₋₁)−→^∂^q H_q₋₁(X]_q₋₁)−→ⁱ^q H_q₋₁(X]_q₋₁,X]_q₋₂),

where ∂q is the boundary operator of the long exact sequence of the pair (Xfq,X]q−1) and iq is induced by the inclusion. If we choose for each i ∈ Iq

a lift (Φf^q_i,φf^q_i) : (D^q, S^q⁻¹)→(Xf_q,X]_q₋₁) of the characteristic map (Φ^q_i, φ^q_i), we obtain a Zπ-basis {bi | i ∈ In} for Cn(X,e A), if we definee bi as the image of a generator in H_q(D^q, S^q⁻¹)∼=Zunder the map H_q(fΦ^q_i,fφ^q_i) :H_q(D^q, S^q⁻¹)→ Hq(Xfq,X]q−1) = Cq(X,e A). We calle {bi | i ∈ In} the cellular basis. Notice that we have made several choices in defining the cellular basis. We call two Zπ-bases {αj |j ∈J} and {βk | k∈ K} forCq(X,e A)e equivalent if there is a bijection φ:J →K and elements j ∈ {±1} and γj ∈ π for j ∈ J such that _j·γ_j·α_j =β_φ(j). The equivalence class of the basis{b_i |i∈I_n} constructed above does only depend on the CW-structure on (X, A) and is independent of all further choices such as (Φ^q_i, φ^q_i), its lift (fΦ^q_i,fφ^q_i) and the generator of Hn(Dⁿ, Sⁿ⁻¹).

Now suppose we are given a handlebody decomposition (1.14). We construct by induction overq=−1,0,1, . . . , na sequence of spacesX₋1=∂0W ⊂X0⊂ X1 ⊂ X2 ⊂ . . . ⊂ Xn together with homotopy equivalences fq: Wq → Xq

such that f_q|W_q−1 = f_q₋₁ and (X, ∂₀W) is a CW-complex with respect to the filtration {X_q | q =−1,0,1, . . . , n}. The induction beginning f₁: W₋₁ =

∂₀W×[0,1]→X₋₁=∂₀W is given by the projection. The induction step from (q−1) toqis done as follows. We attach for each handle (φ^q_i) fori= 1,2, . . . , p_q a cellD^q toX_q₋₁by the attaching map f_q₋₁◦φ^q_i|S^q−1×{0}. In other words, we defineX_q by the pushout

`^pq

i=1S^q⁻¹

`_pq

i=1f_q−1◦φ^q_i|Sq−1×{0}

−−−−−−−−−−−−−−−−→ X_q₋₁



 y



 y

`^pq

i=1D^q −−−−→ Xq

Recall thatWq is the pushout

`pq

i=1S^q⁻¹×Dⁿ⁻^q

`_pq

i=1φ^q_i

−−−−−→ Wq−1



 y



 y

`pq

i=1D^q×Dⁿ⁻^q −−−−→ Wq

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1.2. HANDLEBODY DECOMPOSITIONS ANDCW-STRUCTURES 11 Define a spaceY_q by the pushout

`^pq

i=1S^q⁻¹

`_pq

i=1φ^q_i|Sq−1×{0}

−−−−−−−−−−−−→ W_q₋₁



 y



 y

`^pq

i=1D^q −−−−→ Yq

Define (g_q, f_q₋₁) : (Y_q, W_q₋₁)→(X_q, X_q₋₁) by the pushout property applied to homotopy equivalences given by f_q₋₁: W_q₋₁ → X_q₋₁ and the identity maps on S^q⁻¹ and D^q. Define (hq,id) : (Yq, Wq−1) → (Wq, Wq−1) by the pushout property applied to homotopy equivalences given by the obvious inclusions S^q⁻¹ →S^q⁻¹×Dⁿ⁻^q and D^q →D^q ×Dⁿ⁻^q and the identity on Wq−1. The resulting maps are homotopy equivalences of pairs since the upper horizontal arrows in the three pushouts above are cofibrations (see [18, page 249]). Choose a homotopy inverse (h⁻_q¹,id) : (Wq, Wq−1)→(Yq, Wq−1). Definefq by the com- positiongq◦h⁻_q¹.

In particular we see that the inclusionsWq →W areq-connected since the inclusion of the q-skeleton Xq → X is always q-connected for a CW-complex X.

Denote byp: fW →W the universal covering withπ=π₁(W) as group of deck transformations. Let Wfq be the preimage of Wq under p. Notice that this is the universal covering for q ≥2 since each inclusion Wq → W induces an isomorphism on the fundamental groups. LetC_∗(fW ,∂]₀W) be theZπ-chain complex whoseq-th chain group isHq(Wfq,W^q−1) and whoseq-th differential is given by the composition

Hq(Wfq,W^q−1)−→^∂^p Hq(W^q−1)−→ⁱ^q Hq−1(W^q−1,W^q−2),

where∂q is the boundary operator of the long homology sequence associated to the pair (Wf_p,W^_p₋₁) and i_q is induced by the inclusion. The map f:W →X induces an isomorphism of Zπ-chain complexes

C_∗(fe) :C_∗(fW ,∂]₀W) −→^∼⁼ C_∗(X,e ∂]₀W). (1.17) Each handle (φ^q_i) determines an element

[φ^q_i]∈Cq(fW ,∂]0W) (1.18) after choosing a lift (Φf^q_i,fφ^q_i) : (D^q ×Dⁿ⁻^q, S^q⁻¹×Dⁿ⁻^q) → (Wfq,^Wq−1) of its characteristic map (Φ^q_i, φ^q_i) : (D^q ×Dⁿ⁻^q, S^q⁻¹ ×Dⁿ⁻^q) → (Wq, Wq−1), namely, the image of the preferred generator inHq(D^q×Dⁿ⁻^q, S^q⁻¹×Dⁿ⁻^q)∼= H0({∗}) =Zunder the mapHq(Φf^q_i,fφ^q_i). This element is only well-defined up to multiplication with an elementγ ∈π. The elements{[φ^q_i]|i= 1,2, . . . , pq} form a Zπ-basis forC_q(fW ,∂]₀W). Its image under the isomorphism (1.17) is a cellular Zπ-basis.

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If W has no handles of index ≤ 1, i.e. p₀ = p₁ = 0, one can express C_∗(fW ,∂]0W) also in terms of homotopy groups as follows. Fix a base point z ∈∂0W and a liftze∈∂]0W. All homotopy groups are taken with respect to these base points. Letπ_∗(W_∗, W_∗−1) be theZπ-chain complex, whoseq-thZπ- module isπ_q(W_q, W_q₋₁) forq≥2 and zero forq≤1 and whoseq-th differential is given by the composition

π_q(W_q, W_q₋₁)−→^∂^q π_q₋₁(W_q₋₁)−−−−−→^π^q−1⁽ⁱ⁾ π_q₋₁(W_q₋₁, W_q₋₂).

The Zπ-action comes from the canonical π₁(Y)-action on the groupπ_q(Y, A) [121, Theorem I.3.1 on page 164]. Notice thatπ_q(Y, A) is abelian for any pair of spaces (Y, A) forq≥3 and is abelian also forq= 2 ifAis simply connected or empty. Forq≥2 the Hurewicz homomorphism is an isomorphism [121, Corol- lary IV.7.11 on page 181] πq(Wfq,^Wq−1)→ Hq(Wfq,^Wq−1) and the projection p: fW → W induces isomorphisms πq(Wfq,^Wq−1) → πq(Wq, Wq−1). Thus we obtain an isomorphism ofZπ-chain complexes

C_∗(fW ,∂]₀W)−→^∼⁼ π_∗(W_∗, W_∗−₁). (1.19) Fix a pathw_i in W from a point in the transverse sphere of (φ^q_i) to the base pointz. Then the handle (φ^q_i) determines an element

[φ^q_i] ∈ π_q(W_q, W_q₋₁). (1.20) It is represented by the obvious map (D^q × {0}, S^q⁻¹× {0}) → (W_q, W_q₋₁) together with wi. It agrees with the element [φ^q_i] ∈ Cq(fW ,∂]0W) defined in (1.18) under the isomorphism (1.19) if we use the lift of the characteristic map determined by the pathwi.

1.3 Reducing the Handlebody Decomposition

In the next step we want to get rid of the handles of index zero and one in the handlebody decomposition (1.14).

Lemma 1.21 Let W be ann-dimensional manifold forn≥6 whose boundary is the disjoint union ∂W = ∂₀W`∂₁W. Then the following statements are equivalent

1. The inclusion∂₀W →W is1-connected;

2. We can find a diffeomorphism relative∂₀W W ∼= ∂₀W ×[0,1] +

p₂

X

i=1

(φ²_i) +

p₃

X

i=1

(φ³_i) +

p_n

X

i=1

(φⁿ_i).

A Basic Introduction to Surgery Theory

Wolfgang L¨ uck

Fachbereich Mathematik und Informatik Westf¨ alische Wilhelms-Universit¨ at M¨ unster

Einsteinstr. 62 48149 M¨ unster

Germany October 27, 2004

Preface

Contents

Chapter 1

The s-Cobordism Theorem

Introduction

1.1 Handlebody Decompositions

1.2 Handlebody Decompositions and CW -Struc- tures

1.3 Reducing the Handlebody Decomposition