Spin Geometry (PMS-38), Volume 38

H. Blaine Lawson; Marie-Louise Michelsohn

Published by Princeton University Press 1990

Spin Geometry (PMS-38), Volume 38

H. Blaine Lawson and Marie-Louise Michelsohn

Volume 20 in the series Princeton Mathematical Series

https://doi.org/10.1515/9781400883912

Overview
Contents

About this book

This book offers a systematic and comprehensive presentation of the concepts of a spin manifold, spinor fields, Dirac operators, and A-genera, which, over the last two decades, have come to play a significant role in many areas of modern mathematics. Since the deeper applications of these ideas require various general forms of the Atiyah-Singer Index Theorem, the theorems and their proofs, together with all prerequisite material, are examined here in detail. The exposition is richly embroidered with examples and applications to a wide spectrum of problems in differential geometry, topology, and mathematical physics. The authors consistently use Clifford algebras and their representations in this exposition. Clifford multiplication and Dirac operator identities are even used in place of the standard tensor calculus. This unique approach unifies all the standard elliptic operators in geometry and brings fresh insights into curvature calculations. The fundamental relationships of Clifford modules to such topics as the theory of Lie groups, K-theory, KR-theory, and Bott Periodicity also receive careful consideration. A special feature of this book is the development of the theory of Cl-linear elliptic operators and the associated index theorem, which connects certain subtle spin-corbordism invariants to classical questions in geometry and has led to some of the most profound relations known between the curvature and topology of manifolds.

Topics

Frontmatter Publicly Available Download PDF	i
Contents Publicly Available Download PDF	vii
Preface Requires Authentication Unlicensed Licensed Download PDF	ix
Acknowledgments Requires Authentication Unlicensed Licensed Download PDF	xii
Introduction Requires Authentication Unlicensed Licensed Download PDF	1
I. Clifford Algebras, Spin Groups and Their Representations Requires Authentication Unlicensed Licensed Download PDF	7
II. Spin Geometry and the Dirac Operators Requires Authentication Unlicensed Licensed Download PDF	77
III. Index Theorems Requires Authentication Unlicensed Licensed Download PDF	166
IV. Applications in Geometry and Topology Requires Authentication Unlicensed Licensed Download PDF	278
Appendix A. Principal G-Bundles Requires Authentication Unlicensed Licensed Download PDF	370
Appendix B. Classifying Spaces and Characteristic Classes Requires Authentication Unlicensed Licensed Download PDF	376
Appendix C. Orientation Classes and Thom Isomorphisms in K-Theory Requires Authentication Unlicensed Licensed Download PDF	384
Appendix D. Spin‘-Manifolds Requires Authentication Unlicensed Licensed Download PDF	390
Bibliography Requires Authentication Unlicensed Licensed Download PDF	402
Index Requires Authentication Unlicensed Licensed Download PDF	417
Notation Index Requires Authentication Unlicensed Licensed Download PDF	425