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#
INVARIANCE THEORY, THE HEAT EQUATION,

AND THE

ATIYAH-SINGER INDEX THEOREM

##
by Peter B. Gilkey

Electronic reprint © 1996 Peter B. Gilkey

Paper version originally published by Publish or Perish Inc., USA, 1984

Library of Congress Catalog Card Number 84-061166

ISBN 0-914098-20-9

*
The ELibM electronic version is a reprint of the first edition. A second,
revised edition was published in 1995 by CRC Press under the
ISBN 0-8493-7874-4 and is now out-of-print. Copies of this second
edition are still available directly from
Peter B. Gilkey at
US$ 60 per copy.
*

##

CONTENTS

Introduction

[DVI] [Postscript]
###
Chapter 1. Pseudo-Differential Operators

[DVI] [Postscript]

Introduction

1.1. Fourier Transform, Schwartz Class, and Sobolev Spaces

1.2. Pseudo-Differential Operators on **R**^m

1.3. Ellipticity and Pseudo-Differential Operators on Manifolds

1.4. Fredholm Operators and the Index of a Fredholm Operator

1.5 Elliptic Complexes, The Hodge Decomposition Theorem,
and Poincaré Duality

1.6. The Heat Equation

1.7. Local Formula for the Index of an Elliptic Operator

1.8. Lefschetz Fixed Point Theorems

1.9. Elliptic Boundary Value Problems

1.10. Eta and Zeta Functions
###
Chapter 2. Characteristic Classes

[DVI] [Postscript]

Introduction

2.1. Characteristic Classes of a Complex Bundle

2.2 Characteristic Classes of a Real Vector Bundle.
Pontrjagin and Euler Classes

2.3. Characteristic Classes of Complex Projective Space

2.4. The Gauss-Bonnet Theorem

2.5 Invariance Theory and the Pontrjagin Classes
of the Tangent Bundle

2.6 Invariance Theory and Mixed Characteristic Classes
of the Tangent Space and of a Coefficient Bundle
###
Chapter 3. The Index Theorem

[DVI] [Postscript]

Introduction

3.1. The Hirzebruch Signature Formula

3.2. Spinors and their Representations

3.3. Spin Structures on Vector Bundles

3.4. The Spin Complex

3.5. The Riemann-Roch Theorem for Almost Complex Manifolds

3.6. A Review of Kaehler Geometry

3.7 An Axiomatic Characterization of the Characteristic Forms
for Holomorphic Manifolds with Kaehler Metrics

3.8. The Chern Isomorphism and Bott Periodicity

3.9. The Atiyah-Singer Index Theorem
###
Chapter 4. Generalized Index Theorems and Special Topics

[DVI] [Postscript]

Introduction

4.1. The de Rham Complex for Manifolds with Boundary

4.2. The Gauss-Bonnet Theorem for Manifolds with Boundary

4.3. The Regularity at s = 0 of the Eta Invariant

4.4. The Eta Invariant with Coefficients in a Locally Flat Bundle

4.5. Lefschetz Fixed Point Formulas

4.6. The Eta Invariant and the K-Theory of Spherical Space Forms

4.7. Singer's Conjecture for the Euler Form

4.8. Local Formulas for the Invariants of the Heat Equation

4.9. Spectral Geometry
Bibliography

Index

[DVI] [Postscript]

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[DVI] (511,870 bytes)

[Postscript] (838,403 bytes)

*Electronic version published on ELibM 3 May 1999. Last modified 19 Apr 2001
© 1996, 1999 ELibM*