INTRODUCTION 3
CHAPTER Ⅰ Clifford Algebras，Spin Groups and Their Representations 7
1．Clifford algebras 7
2．The groups Pin and Spin 12
3．The algebras Clnand Clr,s 20
4．The classification 25
5．Representations 30
6．Lie algebra structures 40
7．Some direct applications to geometry 44
8．Some further applications to the theory of Lie groups 49
9．K-theory and the Atiyah-Bott-Shapiro construction 58
10．KR-theory and the (1,1)-Periodicity Theorem 70
CHAPTER Ⅱ Spin Geometry and the Dirac Operators 77
1．Spin structures on vector bundles 78
2．Spin manifolds and spin cobordism 85
3．Clifford and spinor bundles 93
4．Connections on spinor bundles 101
5．The Dirac operators 112
6．The fundamental elliptic operators 135
7．Clk-linear Dirac operators 139
8．Vanishing theorems and some applications 153
CHAPTER Ⅲ Index Theorems 166
1．Differential operators 167
2．Sobolev spaces and Sobolev theorems 170
3．Pseudodifferential operators 177
4．Elliptic operators and parametrices 188
5．Fundamental results for elliptic operators 192
6．The heat kernel and the index 198
7．The topological invariance of the index 201
8．The index of a family of elliptic operators 205
9．The G-index 211
10．The Clifford index 214
11．Multiplicative sequences and the Chern character 225
12．Thom isomorphisms and the Chern character defect 238
13．The Atiyah-Singer Index Theorem 243
14．Fixed-point formulas for elliptic operators 259
15．The Index Theorem for Families 268
16．Families of real operators and the Clk-index Theorem 270
17．Remarks on heat and supersymmetry 277
CHAPTER Ⅳ Applications in Geometry and Topology 278
1．Integrality theorems 280
2．Immersions of manifolds and the vector field problem 281
3．Group actions on manifolds 291
4．Compact manifolds of positive scalar curvature 297
5．Positive scalar curvature and the fundamental group 302
6．Complete manifolds of positive scalar curvature 313
7．The topology of the space of positive scalar curvature metrics 326
8．Clifford multiplication and K？hler manifolds 330
9．Pure spinors,complex structures，and twistors 335
10．Reduced holonomy and calibrations 345
11．Spinor cohomology and complex manifolds with vanishing first Chern class 357
12．The Positive Mass Conjecture in general relativity 368
APPENDIX A Principal G-bundles 370
APPENDIX B Classifying Spaces and Characteristic Classes 376
APPENDIX C Orientation Classes and Thom Isomorphisms in K-theory 384
APPENDIX D Spinc-manifolds 390
BIBLIOGRAPHY 402
INDEX 417
NOTATION INDEX 425