Floer homology groups in Yang-Mills theory

Donaldson S.K.0511029241, 0521808030

Table of contents :
Cover……Page 1
Half-title……Page 3
Title……Page 5
Copyright……Page 6
Contents……Page 7
1 Introduction……Page 11
2.1 Yang–Mills theory over compact manifolds……Page 17
2.2 The case of a compact 4-manifold……Page 19
2.3 Technical results……Page 20
2.4 Manifolds with tubular ends……Page 23
2.5.1 Initial discussion……Page 24
2.5.2 The Chern–Simons functional……Page 26
2.5.3 The instanton equation……Page 30
2.5.4 Linear operators……Page 33
2.6 Appendix A: local models……Page 37
2.7 Appendix B: pseudo-holomorphic maps……Page 40
2.8 Appendix C: relations with mechanics……Page 43
3.1 Separation of variables……Page 50
3.1.1 Sobolev spaces on tubes……Page 55
3.2 The index……Page 57
3.2.1 Remarks on other operators……Page 61
3.3 The addition property……Page 63
3.3.1 Weighted spaces……Page 68
3.3.2 Floer’s grading function; relation with the Atiyah, Patodi, Singer theory……Page 74
3.3.3 Refinement of weighted theory……Page 78
3.4 … theory……Page 80
4 Gauge theory and tubular ends……Page 86
4.1 Exponential decay……Page 87
4.2 Moduli theory……Page 92
4.3 Moduli theory and weighted spaces……Page 97
4.4 Gluing instantons……Page 101
4.4.1 Gluing in the reducible case……Page 110
4.5.1 Convergence in the general case……Page 113
4.5.2 Gluing in the Morse–Bott case……Page 118
5.1 Compactness properties……Page 123
5.2 Floer’s instanton homology groups……Page 132
5.3 Independence of metric……Page 133
5.4 Orientations……Page 140
5.5 Deforming the equations……Page 144
5.5.1 Transversality arguments……Page 149
5.6 U(2) and SO(3) connections……Page 155
6.1 The conceptual picture……Page 161
6.2 The straightforward case……Page 168
6.3 Review of invariants for closed 4-manifolds……Page 171
6.4 Invariants for manifolds with boundary and………Page 175
7.1 The maps………Page 178
7.2 Manifolds with………Page 179
7.2.1 The case………Page 181
7.2.2 The case………Page 184
7.3.1 Algebro-topological interpretation……Page 186
7.3.2 An alternative description……Page 189
7.3.3 The reducible connection……Page 193
7.3.4 Equivariant theory……Page 198
7.3.5 Limitations of existing theory……Page 206
7.4.1 Surgery and instanton invariants……Page 211
7.4.2 The HomDigamma-complex and connected sums……Page 216
8.1 Floer homology for other 3-manifolds……Page 223
8.2 The blow-up formula……Page 229
Bibliography……Page 241
Index……Page 245