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Summary
This book treats the Atiyah-Singer index theorem using the heat equation, which gives a local formula for the index of any elliptic complex. Heat equation methods are also used to discuss Lefschetz fixed point formulas, the Gauss-Bonnet theorem for a manifold with smooth boundary, and the geometrical theorem for a manifold with smooth boundary. The author uses invariance theory to identify the integrand of the index theorem for classical elliptic complexes with the invariants of the heat equation.
Table of Contents
| Pseudo-Differential Operators | |
| Introduction | |
| Fourier Transform and Sobolev Spaces | |
| Pseudo-Differential Operators on Rm | |
| Pseudo-Differential Operators on Manifolds | |
| Index of Fredholm Operators | |
| Elliptic Complexes | |
| Spectral Theory | |
| The Heat Equation | |
| Local Index Formula | |
| Variational Formulas | |
| Lefschetz Fixed Point Theorems | |
| The Zeta Function | |
| The Eta Function | |
| Characteristic Classes | |
| Introduction | |
| Characteristic Classes of Complex Bundles | |
| Characteristic Classes of Real Bundles | |
| Complex Projective Space | |
| Invariance Theory | |
| The Gauss-Bonnet Theorem | |
| Invariance Theory and Pontrjagin Classes | |
| Gauss-Bonnet for Manifolds with Boundary | |
| Boundary Characteristic Classes | |
| Singer's Question | |
| The Index Theorem | |
| Introduction | |
| Clifford Modules | |
| Hirzebruch Signature Formula | |
| Spinors | |
| The Spin Complex | |
| The Riemann-Roch Theorem | |
| K-Theory | |
| The Atiyah-Singer Index Theorem | |
| The Regularity at s = 0 of the Eta Function | |
| Lefschetz Fixed Point Formulas | |
| Index Theorem for Manifolds with Boundary | |
| The Eta Invariant of Locally Flat Bundles | |
| Spectral Geometry | |
| Introduction | |
| Operators of Laplace Type | |
| Isospectral Manifolds | |
| Non-Minimal Operators | |
| Operators of Dirac Type | |
| Manifolds with Boundary | |
| Other Asymptotic Formulas | |
| The Eta Invariant of Spherical Space Forms | |
| A Guide to the Literature | |
| Acknowledgment | |
| Introduction | |
| Bibliography | |
| Notation |
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