劳森的这本《自旋几何》系统讲述了自旋流形，自旋场，狄拉克算子，这些都在现代数学中扮演越来越重要的角色。这些理论的深层次的应用需要具对Atiyah-Singer指标定理有较深的了解，所以书中详细讲述了定理的证明和相关预备知识。大量的例子和应用结合微分几何、拓扑、和数学物理使书的内容更加丰富。Clifford代数及其应用是本书一贯使用的技巧，Clifford乘法和Dirac算子的性质被用于替代标准张量微积分。这些独特的技巧加上几何中的标准椭圆算子为曲率的计算带来新的见解。

PREFACE

ACKNOWLEDGMENTS

INTRODUCTION

CHAPTER I Clifford Algebras, Spin Groups and Their Representations

 §1.Clifford algebras

 §2.The groups Pin and Spin

 §3.The algebras Cln and 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

CHA§'eR 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.Thorn 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 ClOt-index Theorem

 §17.Remarks on heat and supersymmetry

CHAPTER IV Applications in Geometry and Topology

 §1.lntegrality 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/ihler manifolds

 §9.Pure spinors, complex structures, and twistors

 §10.Reduced hoionomy 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 Thorn lsomorphisms in K-theory

APPENDIX D Spin-manifolds

BIBLIOGRAPHY

INDEX

NOTATION INDEX