Introduction .................................................... 1
1 Foundations .................................................. 8
1.1 Smooth manifolds ........................................ 9
1.2 Smooth maps, tangent vectors, submanifolds ............. 16
1.3 Fibre bundles .......................................... 23
1.4 Integration of smooth vector fields .................... 26
1.5 Manifolds with boundary ................................ 29
1.6 Notes on Chapter 1 ..................................... 34
2 Geometrical tools ........................................... 36
2.1 Riemannian metrics ..................................... 37
2.2 Geodesies .............................................. 39
2.3 Tubular neighbourhoods ................................. 45
2.4 Diffeotopy extension theorems .......................... 49
2.5 Tubular neighbourhood theorem .......................... 53
2.6 Corners and straightening .............................. 59
2.7 Cutting and glueing .................................... 63
2.8 Notes on Chapter 2 ..................................... 67
3 Differentiable group actions ................................ 68
3.1 Lie groups ............................................. 68
3.2 Smooth actions ......................................... 72
3.3 Proper actions and slices .............................. 74
3.4 Properties of proper actions ........................... 78
3.5 Orbit types ............................................ 81
3.6 Actions with few orbit types ........................... 87
3.7 Examples of smooth proper group actions ................ 90
3.8 Notes on Chapter 3 ..................................... 92
4 General position and transversality ......................... 94
4.1 Nul sets ............................................... 95
4.2 Whitney's embedding theorem ............................ 96
4.3 Existence of non-degenerate functions .................. 98
4.4 Jet spaces and function spaces ........................ 100
4.5 The transversality theorem ............................ 105
4.6 Multitransversality ................................... 111
4.7 Generic singularities of maps ......................... 114
4.8 Normal forms .......................................... 122
4.9 Notes on Chapter 4 .................................... 125
5 Theory of handle decompositions ............................ 129
5.1 Existence ............................................. 129
5.2 Normalisation ......................................... 137
5.3 Homology of handles and manifolds ..................... 139
5.4 Modifying decompositions .............................. 143
5.5 Geometric connectivity and the h-cobordism theorem .... 149
5.6 Applications of h-cobordism ........................... 153
5.7 Complements ........................................... 159
5.8 Notes on Chapter 5 .................................... 165
6 Immersions and embeddings .................................. 167
6.1 Fibration theorems ................................... 167
6.2 Geometry of immersions ................................ 169
6.3 The Whitney trick ..................................... 176
6.4 Embeddings and immersions in the metastable range ..... 184
6.5 Notes on Chapter 6 .................................... 193
7 Surgery .................................................... 195
7.1 The surgery procedure: a single surgery ............... 196
7.2 Surgery below the middle dimension .................... 199
7.3 Bilinear and quadratic forms .......................... 202
7.4 Poincare complexes and pairs .......................... 207
7.5 The even dimensional case ............................. 212
7.6 The odd dimensional case .............................. 216
7.7 Homotopy theory of Poincare complexes ................. 220
7.8 Homotopy types of smooth manifolds .................... 225
7.9 Notes on Chapter 7 .................................... 234
8 Cobordism .................................................. 237
8.1 The Thorn construction ................................ 239
8.2 Cobordism groups and rings ............................ 243
8.3 Techniques of bordism theory .......................... 248
8.4 Bordism as a homology theory .......................... 252
8.5 Equivariant cobordism ................................. 259
8.6 Classifying spaces, Ω*O, Ω*U ........................... 262
8.7 Calculation of Ω*SO and Ω*SU ........................... 269
8.8 Groups of knots and homotopy spheres .................. 281
8.9 Notes on Chapter 8 .................................... 292
Appendix A Topology .......................................... 296
A.l Definitions ........................................... 296
A.2 Topology of metric spaces ............................. 298
A.3 Proper group actions .................................. 303
A.4 Mapping spaces ........................................ 306
Appendix В Homotopy theory ................................... 314
B.l Definitions and basic properties ...................... 314
B.2 Groups and homogeneous spaces ......................... 319
B.3 Homotopy calculations ................................. 323
B.4 Further techniques .................................... 327
References .................................................... 331
Index of notations ............................................ 340
Index ......................................................... 345
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