INTRODUCTION

CHAPTER Ⅰ Clifford Algebras，Spin Groups and Their Representations

1．Clifford algebras

2．The groups Pin and Spin

3．The algebras Clnand Clr,s

4．The classification

5．Representations

6．Lie algebra structures

7．Some direct applications to geometry

8．Some further applications to the theory of Lie groups

9．K-theory and the Atiyah-Bott-Shapiro construction

10．KR-theory and the (1,1)-Periodicity Theorem

CHAPTER Ⅱ Spin Geometry and the Dirac Operators

1．Spin structures on vector bundles

2．Spin manifolds and spin cobordism

3．Clifford and spinor bundles

4．Connections on spinor bundles

5．The Dirac operators

6．The fundamental elliptic operators

7．Clk-linear Dirac operators

8．Vanishing theorems and some applications

CHAPTER Ⅲ Index Theorems

1．Differential operators

2．Sobolev spaces and Sobolev theorems

3．Pseudodifferential operators

4．Elliptic operators and parametrices

5．Fundamental results for elliptic operators

6．The heat kernel and the index

7．The topological invariance of the index

8．The index of a family of elliptic operators

9．The G-index

10．The Clifford index

11．Multiplicative sequences and the Chern character

12．Thom isomorphisms and the Chern character defect

13．The Atiyah-Singer Index Theorem

14．Fixed-point formulas for elliptic operators

15．The Index Theorem for Families

16．Families of real operators and the Clk-index Theorem

17．Remarks on heat and supersymmetry

CHAPTER Ⅳ Applications in Geometry and Topology

1．Integrality theorems

2．Immersions of manifolds and the vector field problem

3．Group actions on manifolds

4．Compact manifolds of positive scalar curvature

5．Positive scalar curvature and the fundamental group

6．Complete manifolds of positive scalar curvature

7．The topology of the space of positive scalar curvature metrics

8．Clifford multiplication and K？hler manifolds

9．Pure spinors,complex structures，and twistors

10．Reduced holonomy and calibrations

11．Spinor cohomology and complex manifolds with vanishing first Chern class

12．The Positive Mass Conjecture in general relativity

APPENDIX A Principal G-bundles

APPENDIX B Classifying Spaces and Characteristic Classes

APPENDIX C Orientation Classes and Thom Isomorphisms in K-theory

APPENDIX D Spinc-manifolds

BIBLIOGRAPHY

INDEX

NOTATION INDEX