This page intentionally left blank CAMBRIDGE TRACTS IN MATHEMATICS GeneralEditors B. BOLLOBAS, W. FULTON, A. KATOK, F. KIRWAN, P. SARNAK 147 Floer homology groups in Yang–Mills theory Floer homology groups in Yang–Mills theory S. K. Donaldson Imperial College, London with the assistance of M. Furuta and D. Kotschick           The Pitt Building, Trumpington Street, Cambridge, United Kingdom    The Edinburgh Building, Cambridge CB2 2RU, UK 40 West 20th Street, New York, NY 10011-4211, USA 477 Williamstown Road, Port Melbourne, VIC 3207, Australia Ruiz de Alarcón 13, 28014 Madrid, Spain Dock House, The Waterfront, Cape Town 8001, South Africa http://www.cambridge.org ©Cambridge University Press 2004 First published in printed format 2002 ISBN 0-511-02924-1 eBook (Adobe Reader) ISBN 0-521-80803-0 hardback Contents 1 Introduction page 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 Lp theory 70 v vi Contents 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 U(2) and SO(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 D , D 168 1 2 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 HomF-complex and connected sums 206 8 Further directions 213 8.1 Floer homology for other 3-manifolds 213 Contents vii 8.2 The blow-up formula 219 Bibliography 231 Index 235