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SURGERY THEORY Contents RUSTAMSADYKOV ELEMENTSOFSURGERY THEORY Contents 0 Background information 7 0.1 Homotopy groups 7 0.2 Bordisms and Cobordisms 10 0.3 Homology and Cohomology 12 1 Stable homotopy groups of spheres 15 1.1 Framed submanifolds in Euclidean spaces 16 1.2 Cobordisms of framed manifolds 18 1.3 The Pontryagin construction 20 S 1.4 Example: The stable group p0 22 1.5 Further reading 23 2 Immersion Theory 24 2.1 Ambient isotopy 25 2.2 The Global Compression Theorem 26 2.3 The Freudenthal suspension theorem 28 2.4 Further reading 30 3 Spherical surgeries 32 3.1 Surgery and cobordisms 33 3.2 Framed base of surgery 37 3.3 Reduction of homotopy groups 38 S S 3.4 Examples: Stable homotopy groups p1 and p2 39 3.5 Further reading 43 3 4 The Whitney trick 45 4.1 The Whitney weak embedding and immersion theorems 45 4.2 The Whitney trick 46 4.3 Strong Whitney embedding theorem 50 4.4 Strong Whitney immersion theorem 51 4.5 Solution to Exercises 53 4.6 Further reading 53 5 The Kervaire invariant and signature 55 5.1 The Wall representative and the invariant m 56 5.2 Homotopy spheres 61 5.3 Bilinear and quadratic forms 63 5.4 Main theorems 66 5.5 Further reading 67 6 Surgery on framed manifolds 70 6.1 Invariance of signature and Kervaire invariant 71 6.2 The main theorems for even dimensional manifolds 73 6.3 The main theorems for odd dimensional manifolds 76 6.4 Solutions to exercises 79 7 Characteristic classes of vector bundles 84 7.1 Vector bundles over topological spaces 84 7.2 Characteristic classes of vector bundles 85 7.2.1 The Euler class 86 7.2.2 Stiefel-Whitney classes 87 7.2.3 Chern classes 89 7.2.4 Pontryagin classes 89 7.3 The Thom theorem on orientable cobordism groups 90 7.4 Example: Pontryagin classes of almost framed manifolds 91 8 Exotic spheres: discovery of exotic spheres S7 93 8.1 Maps of SU(2) and Spin(4) to orthogonal groups 94 8.1.1 The map SU(2) ! SO(3) 94 8.1.2 The map Spin(4) ! SO(4) 94 8.2 Milnor’s fiber bundles 95 4 8.3 The quaternionic Hopf fibration 97 8.4 Invariants of Milnor’s fibrations 98 8.5 Exotic spheres of dimension 7 100 8.6 Further reading 100 9 Exotic spheres 102 9.1 The group of smooth structures on a sphere 102 9.2 The surgery long exact sequence for spheres 105 9.3 Calculation of the group Pm 106 9.4 Calculation of the group Nm. 108 9.5 Calculation of homomorphisms in the surgery exact sequence 110 9.6 Hirzebruch genera and computation of the L-genus 111 9.6.1 Multiplicative series 111 4n 9.6.2 Genera of manifolds M with trivial p1, ..., pn−1. 113 s 9.7 Example: The third stable homotopy group p3 114 9.8 Further reading 115 10 The h-cobordism theorem 116 10.1 Handle manipulation techniques 117 10.1.1 Handle rearrangement 117 10.1.2 Handle creation 118 10.1.3 Handle cancellation 119 10.1.4 Handle slides 119 10.1.5 Handle trading 120 10.2 The h-cobordism theorem 121 11 Surgery on maps of simply connected manifolds 127 11.1 Normal maps 127 11.2 Products in homology and cohomology 130 11.2.1 Tensor products 130 11.2.2 Naturality of the cup and cap products 131 11.2.3 Poincaré Duality 132 11.3 Degree one maps 133 11.4 The Browder-Novikov theorem 135 12 The surgery long exact sequence 137 5 13 (Co)homology with twisted coefficients 141 13.1 The Whitney trick in non-simply connected manifolds 142 13.2 Right and left Modules 144 13.3 Homology and cohomology with twisted coefficients 145 13.4 Products in (co)homology with twisted coefficients 147 13.5 Poincaré complexes 148 13.6 Homological algebra 150 q 13.7 The kernels Kq M and K M 151 14 Surgery on non-simply connected manifolds 153 14.1 Surgery below the middle range 153 14.2 Surgery on maps of manifolds of even dimension 154 14.2.1 Wall representatives 154 14.2.2 The intersection pairing l and self-intersection form m 156 14.2.3 Effect of surgery on homotopy groups 159 14.2.4 The main theorem 159 14.3 Surgery on maps of manifolds of odd dimension 160 14.3.1 Heegaard splittings 160 14.3.2 #-quadratic formations 163 14.4 Further reading 168 14.5 Solutions to Exercises 168 99 Additional Topics 171 99.1 Topics for Chapter 1 172 99.1.1 Hopf fibration 172 S 99.1.2 Stable homotopy group p1 173 99.1.3 The Pontryagin-Thom construction 174 99.1.4 The stable J-homomorphism 175 99.2 Topics for Chapter 2 177 99.2.1 The Multicompression theorem 177 99.2.2 The Smale-Hirsch Theorem 179 99.2.3 The Smale Paradox 181 99.2.4 Cobordisms of oriented immersed manifolds 182 99.2.5 The Local Compression Theorem 183 99.2.6 Solutions to Exercises 186 99.3 Topics for Chapter 3 189 99.3.1 Example: Cobordisms 189 99.3.2 Cairns-Whitehead technique of smoothing manifolds 189 6 99.4 Topics for Chapter 5 194 99.4.1 Bilinear forms on free modules 194 99.5 Topics for Chapter 6 195 99.5.1 Heegaard splittings 195 99.5.2 Alternative proof of the surgery theorem for framed manifolds of odd dimension 196 99.6 Topics for Chapter 14 199 99.6.1 The Whitehead torsion 199 99.6.2 The s-cobordism theorem 200 0 Background information 0.1 Homotopy groups Pointed topological spaces. A pointed topological space is a topological space X together with a point ∗ 2 X called the basepoint of X. A map of pointed topological spaces is required to take the basepoint in the domain to the basepoint in the target. The pointed union X _ Y of a pointed topological spaces X and Y is the space X t Y/∼ in which the basepoints of X and Y are identified. Example 0.1. Let Sn denote the standard unit sphere centered at ori- n+1 n gin in the Euclidean space R with basis e1, ..., en. The sphere S be- comes pointed if we pick its south pole, i.e., the end point of −en+1 to be the basepoint of Sn. The intersection of Sn with the hyperplane per- n−1 n n n−1 pendicular to e1 is a meridian S of S . The quotient space S /S is homeomorphic to the bouquet of spheres Sn _ Sn. A homotopy of a map f : X ! Y to a map g : X ! Y is a continuous family of maps Ft : X ! Y parametrized by t 2 [0, 1] such that F0 = f and F1 = g. When the spaces X and Y are pointed, every map Ft in the homotopy is required to be pointed as well. A space X is homotopy equivalent to a space Y if there are maps f : X ! Y and g : Y ! X such that g ◦ f is homotopic to the identity map of X, and f ◦ g is homotopy equivalent to the identity map of Y. CW complexes. Most topological spaces that we will encounter are Figure 1: A CW-complex is constructed CW complexes. A CW complex X is constructed inductively beginning by induction beginning with a (red) dis- 0 0 n crete set X , by attaching (blue) seg- with a discrete set X called the 0-skeleton of X. The n-skeleton X ments, then (grey) 2-dimensional cells − is obtained from the (n − 1)-skeleton Xn 1 by attaching a collection and so on. 8 elements of surgery theory n tDa of closed discs of dimension n with respect to attaching maps n n−1 n n−1 ja : ¶Da ! X . In other words, X consists of X and the discs n n n−1 Da with each x in the boundary ¶Da identified with ja(x) 2 X . n n We say that the interior of ¶Da is an open cell ea. The so-constructed space is a CW-complex X = [Xn. A set U in X is declared to be open (respectively, closed) if each intersection U \ Xn is open (respectively, closed). Homotopy groups. For n ≥ 0, the set of homotopy classes [ f ] of pointed maps f : Sn ! X into a pointed topological space is denoted by pnX. The set pnX is a group when n > 0 with the sum [ f ] + [g] represented by the composition f _g Sn −! Sn/Sn−1 ≈ Sn _ Sn −−! X, where the first map is one collapsing the meridian of Sn into a point. The set p0X is the set of path components of X. The group p1X is the fundamental group. The homotopy groups pnX with n > 1 are abelian. A space X is n-connected if the groups piX are trivial in the range i ≤ n. Thus, a non-empty space is −1-connected, a path connected space is 0-connected, while a simply connected space is 1-connected. A map f : X ! Y of topological spaces induces a homomorphism f∗ : piX ! piY of homotopy groups by [g] 7! [ f ◦ g]. Theorem 0.2 (Whitehead). Two path-connected CW-complexes X and Y are homotopy equivalent if and only if there exists a map f : X ! Y which induces an isomorphism of homotopy groups f∗ : piX ! piY for all i ≥ 0.
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