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Invariance Theory, the Heat Equation, and the Atiyah–Singer Index Theorem

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Invariance Theory, the Heat Equation, and the Atiyah–Singer Index Theorem INVARIANCE THEORY, THE HEAT EQUATION, AND THE ATIYAH-SINGER INDEX THEOREM by Peter B. Gilkey Electronic reprint, copyright 1996, Peter B. Gilkey Book originally published on paper by Publish or Perish Inc., USA, 1984 Library of Congress Catalog Card Number 84-061166 ISBN 0-914098-20-9 INTRODUCTION This book treats the Atiyah-Singer index theorem using heat equation methods. The heat equation gives a local formula for the index of any elliptic complex. We use invariance theory to identify the integrand of the index theorem for the four classical elliptic complexes with the invari- ants of the heat equation. Since the twisted signature complex provides a sufficiently rich family of examples,this approach yields a proof of the Atiyah-Singer theorem in complete generality. We also use heat equation methods to discuss Lefschetz fixed point formulas,the Gauss-Bonnet the- orem for a manifold with smooth boundary,and the twisted eta invariant. We shall not include a discussion of the signature theorem for manifolds with boundary. The first chapter reviews results from analysis. Sections 1.1 through 1.7 represent standard elliptic material. Sections 1.8 through 1.10 contain the material necessary to discuss Lefschetz fixed point formulas and other top- ics. Invariance theory and differential geometry provide the necessary link be- tween the analytic formulation of the index theorem given by heat equation methods and the topological formulation of the index theorem contained in the Atiyah-Singer theorem. Sections 2.1 through 2.3 are a review of char- acteristic classes from the point of view of differential forms. Section 2.4 gives an invariant-theoretic characterization of the Euler form which is used to give a heat equation proof of the Gauss-Bonnet theorem. Sections 2.5 and 2.6 discuss the Pontrjagin forms of the tangent bundle and the Chern forms of the coefficient bundle using invariance theory. The third chapter combines the results of the first two chapters to prove the Atiyah-Singer theorem for the four classical elliptic complexes. We first present a heat equation proof of the Hirzebruch signature theorem. The twisted spin complex provides a unified way of discussing the signature, Dolbeault,and de Rham complexes. In sections 3.2–3.4,we discuss the half-spin representations,the spin complex,and derive a formula for the Aˆ genus. We then discuss the Riemann-Roch formula for an almost complex manifold in section 3.5 using the SPINc complex. In sections 3.6–3.7 we give a second derivation of the Riemann-Roch formula for holomorphic Kaehler manifods using a more direct approach. In the final two sections we derive the Atiyah-Singer theorem in its full generality. vi Introduction The final chapter is devoted to more specialized topics. Sections 4.1–4.2 deal with elliptic boundary value problems and derive the Gauss-Bonnet theorem for manifolds with boundary. In sections 4.3–4.4 we discuss the twisted eta invariant on a manifold without boundary and we derive the Atiyah-Patodi-Singer twisted index formula. Section 4.5 gives a brief dis- cussion of Lefschetz fixed point formulas using heat equation methods. In section 4.6 we use the eta invariant to calculate the K-theory of spherical space forms. In section 4.7,we discuss Singer’s conjecture for the Euler form and related questions. In section 4.8,we discuss the local formulas for the invariants of the heat equation which have been derived by several authors,and in section 4.9 we apply these results to questions of spectral geometry. The bibliography at the end of this book is not intended to be exhaustive but rather to provide the reader with a list of a few of the basic papers which have appeared. We refer the reader to the bibliography of Berger and Berard for a more complete list of works on spectral geometry. This book is organized into four chapters. Each chapter is divided into a number of sections. Each Lemma or Theorem is indexed according to this subdivision. Thus,for example,Lemma 1.2.3 is the third Lemma of section 2 of Chapter 1. CONTENTS Introduction ................................ v Chapter 1. Pseudo-Differential Operators Introduction ................................ 1 1.1. Fourier Transform,S chwartz Class,and Sobolev Spaces ..... 2 1.2. Pseudo-Differential Operators on Rm ...............11 1.3. Ellipticity and Pseudo-Differential Operators on Manifolds . 23 1.4. Fredholm Operators and the Index of a Fredholm Operator . 31 1.5 Elliptic Complexes,The Hodge Decomposition Theorem, and Poincar´e Duality ......................37 1.6. The Heat Equation .........................42 1.7. Local Formula for the Index of an Elliptic Operator .......50 1.8. Lefschetz Fixed Point Theorems ..................62 1.9. Elliptic Boundary Value Problems .................70 1.10. Eta and Zeta Functions .......................78 Chapter 2. Characteristic Classes Introduction ................................87 2.1. Characteristic Classes of a Complex Bundle ............89 2.2 Characteristic Classes of a Real Vector Bundle. Pontrjagin and Euler Classes ..................98 2.3. Characteristic Classes of Complex Projective Space ...... 104 2.4. The Gauss-Bonnet Theorem ................... 117 2.5 Invariance Theory and the Pontrjagin Classes of the Tangent Bundle .................... 123 2.6 Invariance Theory and Mixed Characteristic Classes of the Tangent Space and of a Coefficient Bundle ...... 141 viii Contents Chapter 3. The Index Theorem Introduction ............................... 147 3.1. The Hirzebruch Signature Formula ............... 148 3.2. Spinors and their Representations ................ 159 3.3. Spin Structures on Vector Bundles ................ 165 3.4. The Spin Complex ........................ 176 3.5. The Riemann-Roch Theorem for Almost Complex Manifolds . 180 3.6. A Review of Kaehler Geometry ................. 193 3.7 An Axiomatic Characterization of the Characteristic Forms for Holomorphic Manifolds with Kaehler Metrics ...... 204 3.8. The Chern Isomorphism and Bott Periodicity .......... 215 3.9. The Atiyah-Singer Index Theorem ................ 224 Chapter 4. Generalized Index Theorems and Special Topics Introduction ............................... 241 4.1. The de Rham Complex for Manifolds with Boundary ..... 243 4.2. The Gauss-Bonnet Theorem for Manifolds with Boundary . 250 4.3. The Regularity at s = 0 of the Eta Invariant .......... 258 4.4. The Eta Invariant with Coefficients in a Locally Flat Bundle . 270 4.5. Lefschetz Fixed Point Formulas ................. 284 4.6. The Eta Invariant and the K-Theory of Spherical Space Forms 295 4.7. Singer’s Conjecture for the Euler Form ............. 307 4.8. Local Formulas for the Invariants of the Heat Equation .... 314 4.9. Spectral Geometry ........................ 331 Bibliography .............................. 339 Index .................................. 347 CHAPTER 1 PSEUDO-DIFFERENTIAL OPERATORS Introduction In the first chapter,we develop the analysis needed to define the index of an elliptic operator and to compute the index using heat equation meth- ods. Sections 1.1 and 1.2 are brief reviews of Sobolev spaces and pseudo- differential operators on Euclidean spaces. In section 1.3,we transfer these notions to compact Riemannian manifolds using partition of unity argu- ments. In section 1.4 we review the facts concerning Fredholm operators needed in section 1.5 to prove the Hodge decomposition theorem and to discuss the spectral theory of self-adjoint elliptic operators. In section 1.6 we introduce the heat equation and in section 1.7 we derive the local for- mula for the index of an elliptic operator using heat equation methods. Section 1.8 generalizes the results of section 1.7 to find a local formula for the Lefschetz number of an elliptic complex. In section 1.9,we dis- cuss the index of an elliptic operator on a manifold with boundary and in section 1.10,we discuss the zeta and eta invariants. Sections 1.1 and 1.4 review basic facts we need,whereas sections 1.8 through 1.10 treat advanced topics which may be omitted from a first reading. We have attempted to keep this chapter self-contained and to assume nothing beyond a first course in analysis. An exception is the de Rham theorem in section 1.5 which is used as an example. A number of people have contributed to the mathematical ideas which are contained in the first chapter. We were introduced to the analysis of sections 1.1 through 1.7 by a course taught by L. Nirenberg. Much of the organization in these sections is modeled on his course. The idea of using the heat equation or the zeta function to compute the index of an elliptic operator seems to be due to R. Bott. The functional calculus used in the study of the heat equation contained in section 1.7 is due to R. Seeley as are the analytic facts on the zeta and eta functions of section 1.10. The approach to Lefschetz fixed point theorems contained in section 1.8 is due to T. Kotake for the case of isolated fixed points and to S. C. Lee and the author in the general case. The analytic facts for boundary value problems discussed in section 1.9 are due to P. Greiner and R. Seeley. 1.1. Fourier Transform, Schwartz Class, And Sobolev Spaces. The Sobolev spaces and Fourier transform provide the basic tools we shall need in our study of elliptic partial differential operators. Let x = m m (x1, ...,xm) ∈ R . If x, y ∈ R ,we define: 1/2 x · y = x1y1 + ···+ xmym and |x| =(x · x) as the Euclicean dot product and length. Let α =(α1, ...,αm) be a multi- index. The αj are non-negative integers. We define: α α1 αm |α| = α1 + ···+ αm ,α!=α1!...αm! ,x= x1 ...xm . Finally,we define: α1 αm α ∂ ∂ α |α| α dx = ... and Dx =(−i) dx ∂x1 ∂xm as a convenient notation for multiple partial differentiation.
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