1 Introduction ................................................. 1
2 Basic material ............................................... 7
2.1 Yang-Mills theory over compact manifolds ................ 7
2.2 The case of a compact 4-manifold ........................ 9
2.3 Technical results ...................................... 10
2.4 Manifolds with tubular ends ............................ 13
2.5 Yang-Mills theory and 3-manifolds ...................... 14
2.5.1 Initial discussion .............................. 14
2.5.2 The Chern-Simons functional ..................... 16
2.5.3 The instanton equation .......................... 20
2.5.4 Linear operators ................................ 23
2.6 Appendix A: local models ............................... 27
2.7 Appendix B: pseudo-holomorphic maps .................... 30
2.8 Appendix C: relations with mechanics ................... 33
3 Linear analysis ............................................. 40
3.1 Separation of variables ................................ 40
3.1.1 Sobolev spaces on tubes .......................... 45
3.2 The index .............................................. 47
3.2.1 Remarks on other operators ....................... 51
3.3 The addition property .................................. 53
3.3.1 Weighted spaces ................................. 58
3.3.2 Floer's grading function; relation with the
Atiyah, Patodi, Singer theory ................... 64
3.3.3 Refinement of weighted theory ................... 68
3.4 Lρ theory .............................................. 70
4 Gauge theory and tubular ends ............................... 76
4.1 Exponential decay ...................................... 77
4.2 Moduli theory .......................................... 82
4.3 Moduli theory and weighted spaces ...................... 87
4.4 Gluing instantons ...................................... 91
4.4.1 Gluing in the reducible case ................... 100
4.5 Appendix A: further analytical results ................ 103
4.5.1 Convergence in the general case ................ 103
4.5.2 Gluing in the Morse-Bott case .................. 108
5 The Floer homology groups .................................. 113
5.1 Compactness properties ................................ 113
5.2 Floer's instanton homology groups ..................... 122
5.3 Independence of metric ................................ 123
5.4 Orientations .......................................... 130
5.5 Deforming the equations ............................... 134
5.5.1 Transversality arguments ........................ 139
5.6 1/(2) and 50(3) connections ........................... 145
6 Floer homology and 4-manifold invariants ................... 151
6.1 The conceptual picture ................................ 151
6.2 The straightforward case .............................. 158
6.3 Review of invariants for closed 4-manifolds ........... 161
6.4 Invariants for manifolds with boundary and b+ > 1 ..... 165
7 Reducible connections and cup products ..................... 168
7.1 The maps D1, D2 ....................................... 168
7.2 Manifolds with b+ = 0,1 ............................... 169
7.2.1 The case b+ = 1 ................................ 171
7.2.2 The case b+ = 0 ................................ 174
7.3 The cup product ....................................... 176
7.3.1 Algebro-topological interpretation ............. 176
7.3.2 An alternative description ..................... 179
7.3.3 The reducible connection ....................... 183
7.3.4 Equivariant theory ............................. 188
7.3.5 Limitations of existing theory ................. 196
7.4 Connected sums ........................................ 201
7.4.1 Surgery and instanton invariants ............... 201
7.4.2 The Hom -complex and connected sums ........... 206
8 Further directions ......................................... 213
8.1 Floer homology for other 3-manifolds .................. 213
8.2 The blow-up formula ................................... 219
Bibliography .................................................. 231
Index ......................................................... 235
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