Floer Homology Groups in Yang-Mills Theory (Cambridge Tracts in Mathematics, Series Number 147)

Annotation This monograph gives a thorough exposition of Floer's seminal work during the 1980s from a contemporary viewpoint. The material contained here was developed with specific applications in mind. However, it has now become clear that the techniques used are important for many current areas of research. An important example would be symplectic theory and gluing problems for self-dual metrics and other metrics with special holonomy. The author writes with the big picture constantly in mind. As well as a review of the current state of knowledge, there are sections on the likely direction of future research. Included in this are connections between Floer groups and the celebrated Seiberg-Witten invariants. The results described in this volume form part of the area known as Donaldson theory. The significance of this work is such that the author was awarded the prestigious Fields Medal for his contribution 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 The Concept Of Floer Homology Was One Of The Most Striking Developments In Differential Geometry. It Yields Rigorously Defined Invariants Which Can Be Viewed As Homology Groups Of Infinite-dimensional Cycles. The Ideas Led To Great Advances In The Areas Of Low-dimensional Topology And Symplectic Geometry And Are Intimately Related To Developments In Quantum Field Theory. The First Half Of This Book Gives A Thorough Account Of Floer's Construction In The Context Of Gauge Theory Over 3 And 4-dimensional Manifolds. The Second Half Works Out Some Further Technical Developments Of The Theory, And The Final Chapter Outlines Some Research Developments For The Future - Including A Discussion Of The Appearance Of Modular Forms In The Theory. The Scope Of The Material In This Book Means That It Will Appeal To Graduate Students As Well As Those On The Frontiers Of The Subject. S.k. Donaldson With The Assistance Of M. Furuta And D. Kotschick. Includes Bibliographical References (p. 231-233) And Index. The seminal work of Floer has now been placed in a contemporary setting. The author of this monograph writes with the big picture constantly in mind, reviewing current knowledge and predicting future directions. This forms part of the work for which Simon Donaldson was awarded the prestigious Fields Medal In 1985 Andreas Floer discovered new topological invariants of certain 3-manifolds, the 'Floer homology groups'.