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A Basic Introduction to Surgery Theory

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A Basic Introduction to Surgery Theory A Basic Introduction to Surgery Theory Wolfgang L¨uck1 Fachbereich Mathematik und Informatik Westf¨alische Wilhelms-Universit¨atM¨unster Einsteinstr. 62 48149 M¨unster Germany October 27, 2004 1email: [email protected] www: http://www.math.uni-muenster.de/u/lueck/ FAX: 49 251 8338370 II 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 illu- minating 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¨unsterand the participants of the summer school who made very useful comments and suggestions, in par- ticular 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 IV Preface Contents 1 The s-Cobordism Theorem 1 Introduction . 1 1.1 Handlebody Decompositions . 4 1.2 Handlebody Decompositions and CW -Structures . 9 1.3 Reducing the Handlebody Decomposition . 12 1.4 Handlebody Decompositions and Whitehead Groups . 18 1.5 Miscellaneous . 20 2 Whitehead Torsion 23 Introduction . 23 2.1 Whitehead Groups . 24 2.2 Algebraic Approach to Whitehead Torsion . 27 2.3 The Geometric Approach to Whitehead Torsion . 34 2.4 Reidemeister Torsion and Lens Spaces . 38 2.5 Miscellaneous . 47 3 Normal Maps and the Surgery Problem 49 Introduction . 49 3.1 Poincar´eDuality . 50 3.2 The Spivak Normal Fibration . 56 3.2.1 Pontrjagin-Thom Construction . 57 3.2.2 Spherical Fibrations . 60 3.2.3 The Existence and Uniqueness of the Spivak Normal Fi- bration . 62 3.3 Normal Maps . 64 3.4 The Surgery Step . 69 3.4.1 Motivation for the Surgery Step . 69 3.4.2 Immersions and Embeddings . 73 3.4.3 The Surgery Step . 75 3.5 Miscellaneous . 78 4 The Algebraic Surgery Obstruction 79 Introduction . 79 4.1 Intersection and Selfintersection Pairings . 80 V VI CONTENTS 4.1.1 Intersections of Immersions . 80 4.1.2 Selfintersections of Immersions . 82 4.2 Kernels and Forms . 85 4.2.1 Symmetric Forms and Surgery Kernels . 85 4.2.2 Quadratic Forms and Surgery Kernels . 89 4.3 Even Dimensional L-Groups . 93 4.4 The Surgery Obstruction in Even Dimensions . 95 4.5 Formations and Odd Dimensional L-Groups . 99 4.6 The Surgery Obstruction in Odd Dimensions . 102 4.7 Variations of the Surgery Obstruction and the L-Groups . 104 4.7.1 Surgery Obstructions for Manifolds with Boundary . 105 4.7.2 Surgery Obstructions and Whitehead Torsion . 107 4.8 Miscellaneous . 111 5 The Surgery Exact Sequence 113 Introduction . 113 5.1 The Structure Set . 113 5.2 Realizability of Surgery Obstructions . 114 5.3 The Surgery Exact Sequence . 117 5.4 Miscellaneous . 119 6 Homotopy Spheres 123 Introduction . 123 6.1 The Group of Homotopy Spheres . 124 6.2 The Surgery Sequence for Homotopy Spheres . 127 6.3 The J-Homomorphism and Stably Framed Bordism . 130 6.4 Computation of bP n+1 . 133 6.5 Computation of Θn=bP n+1 . 137 6.6 The Kervaire-Milnor Braid . 137 6.7 Miscellaneous . 140 7 The Farrell-Jones and Baum-Connes Conjectures 143 7.1 G-Homology Theories . 143 7.2 Isomorphisms Conjectures and Assembly Maps . 148 7.2.1 Classifying Spaces of Families . 148 7.2.2 The Formulation of the Isomorphism Conjectures . 149 7.2.3 Conclusions from the Isomorphism Conjectures . 151 7.3 Characterizing Assembly Maps . 153 7.4 The Choice of the Families . 155 7.5 Miscellaneous . 158 8 Computations of K- and L-Groups for Infinite Groups 161 8.1 Equivariant Chern Characters . 161 8.2 Rational Computations . 167 8.3 Some Special Cases . 170 CONTENTS VII References 174 Notation 185 Index 187 Chapter 1 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 group π = π1(M0). Then ≥ 1. Let (W ; M0; f0;M1; f1) be an h-cobordism over M0. Then W is trivial over M0 if and only if its Whitehead torsion τ(W; M0) Wh(π) vanishes; 2 2. For any x Wh(π) there is an h-cobordism (W ; M0; f0;M1; f1) over M0 2 with τ(W; M0) = x Wh(π); 2 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 relative M0 of h-cobordisms over M0 to the Whitehead group Wh(π). Here are some explanations. An n-dimensional cobordism (sometimes also called just bordism) (W ; M0; f0;M1; f1) consists of a compact oriented n-dimen- sional manifold W , closed (n 1)-dimensional manifolds M0 and M1, a disjoint − decomposition @W = @0W @1W of the boundary @W of W and orientation ` preserving diffeomorphisms f0 : M0 @W0 and f1 : M − @W1. Here and ! 1 ! in the sequel we denote by M1− the manifold M1 with the reversed orientation and we use on @W the orientation with respect to the decomposition TxW = 2 Tx@W R coming from an inward normal field for the boundary. If we equip D with the⊕ standard orientation coming from the standard orientation on R2, the induced orientation on S1 = @D2 corresponds to the anti-clockwise orientation 1 on S . If we want to specify M0, we say that W is a cobordism over M0. If @0W = M0, @1W = M1− and f0 and f1 are given by the identity or if f0 and f1 are obvious from the context, we briefly write (W ; @0W; @1W ). Two cobordisms (W; M0; f0;M1; f1) and (W 0;M0; f00 ;M10 ; f10 ) over M0 are diffeomorphic relative M0 if there is an orientation preserving diffeomorphism F : W W 0 with F ! ◦ 1 2 CHAPTER 1. THE S-COBORDISM THEOREM f0 = f00 . We call an h-cobordism over M0 trivial, if it is diffeomorphic relative M0 to the trivial h-cobordism (M0 [0; 1]; M0 0 ; (M0 1 )−). Notice × × f g × f g that the choice of the diffeomorphisms fi 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 for i = 0; 1 are homotopy equivalences. We will later see that the Whitehead! group of the trivial group vanishes. Thus the s-Cobordism Theorem 1.1 implies Theorem 1.2 (h-Cobordism Theorem) Any h-cobordism (W ; M0; f0;M1; f1) over a simply connected closed n-dimensional manifold M0 with dim(W ) 6 is trivial. ≥ Theorem 1.3 (Poincar´eConjecture) The Poincar´eConjecture is true for a closed n-dimensional manifold M with dim(M) 5, namely, if M is simply ≥ n connected and its homology Hp(M) is isomorphic to Hp(S ) for all p Z, then M is homeomorphic to Sn. 2 Proof : We only give the proof for dim(M) 6. Since M is simply con- n ≥ nected and H (M) ∼= H (S ), one can conclude from the Hurewicz Theorem and Whitehead Theorem∗ [121,∗ Theorem IV.7.13 on page 181 and Theorem IV.7.17 n n on page 182] that there is a homotopy equivalence f : M S . Let Di M ! n ⊂n for i = 0; 1 be two embedded disjoint disks. Put W = M (int(D0 ) int(D1 )). Then W turns out to be a simply connected h-cobordism.− Hence` we can find n n n n n a diffeomorphism F :(@D0 [0; 1]; @D0 0 ; @D0 1 ) (W; @D0 ; @D1 ) ×n n × f g × f g ! which is the identity on @D0 = @D0 0 and induces some (unknown) dif- n n × f g feomorphism f1 : @D0 1 @D1 . By the Alexander trick one can extend n n × f gn ! n n f1 : @D0 = @D0 1 @D1 to a homeomorphism f1 : D0 D1 . Namely, any ×f gn !1 n 1 ! n n homeomorphism f : S − S − extends to a homeomorphism f : D D ! n 1 ! by sending t x for t [0; 1] and x S − to t f(x). Now define a homeomor- n· 2 n 2 n · phism h: D0 0 i0 @D0 [0; 1] i1 D0 1 M for the canonical inclusions n ×f g[ n × [ ×f g ! i : @D k @D [0; 1] for k = 0; 1 by h n = id, h n = F k 0 0 D0 0 @D0 [0;1] × f g ! × j ×{ g j × n and h n = f .
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