Invariance Theory, the Heat Equation, and the AtiyahSinger Index Theorem


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1 PETER B. GILKEY Department of Mathematics, University of Oregon Invariance Theory, the Heat Equation, and the AtiyahSinger Index Theorem Second Edition CRC PRESS Boca Raton Ann Arbor London Tokyo
2 Contents Pseudodifferential operators Introduction Fourier transform and Sobolev spaces Convolution product Fourier transform Sobolev spaces Duality and interpolation PseudoDifferential Operators on R m Continuity properties Equivalence of symbols A wider class of symbols Adjoints and compositions Operators denned by kernels Pseudolocality Completeness Pseudodifferential operators on manifolds Ellipticity Change of coordinates Operators on manifolds Sobolev spaces on manifolds Extension to vector bundles Index of Fredholm Operators Compact operators Fredhplm operators Compositions and adjoints Index of Fredholm operators Properties of the index Elliptic pseudodifferential operators Elliptic complexes Hodge decomposition theorem de Rham complex Spectral theory Selfadjoint compact operators Selfadjoint elliptic operators 51 iii
3 iv Contents Bounding the spectrum from below Heat equation Trace and kernel 55 1:7.0 Heat equation and index theory The heat equation Dependence on a complex parameter Spectral theory Heat equation Local index formula Asymptotic expansions Index theory Variational formulas Generalized heat equation asymptotics Properties of the trace Conformal geometry Lefschetz fixed point theorems Generalized Lefschetz number Equivariant asymptotics Isolated fixed points Heat asymptotics and Lefschetz number Elliptic boundary value problems Notational conventions Operators of Dirac and Laplace type Spectral theory Heat equation Index theory of operators of Dirac type Nonlocal boundary conditions The Zeta function Notational conventions Zeta function and heat equation Zeta function of powers Positive semidefinite operators Eigenvalue growth estimates The Eta function Eta invariant and spectral asymmetry Characteristic classes Introduction Characteristic classes of complex bundles Notational conventions ChernWeil homomorphism Functorial constructions Chern classes Chern character 132
4 Contents v 2.2 Characteristic classes of real bundles Generating functions Euler class Directional covariant derivative Complex projective space Holomorphic manifolds Fiber metrics and connections Complex projective space Characteristic classes of complex projective space Dual basis to the characteristic forms Todd class and Hirzebruch L polynomial Invariance theory Notational conventions Dimensional analysis Invariants of the orthogonal group Diffeomorphism invariance ; Diagonalization Lemma The GaussBonnet theorem The restriction map The proof of the GaussBonnet theorem Next term in the heat equation Shuffle formulas Invariance theory and Pontrjagin classes GaussBonnet for manifolds with boundary Boundary conditions Associated boundary conditions de RhamHodge theorem Heat'equation asymptotics Invariance theory 196 """ The GaussBonnet theorem for manifolds with boundary Doubling the manifold Boundary characteristic classes Singer's question Invariance theory Singer's question Form valued invariants The index theorem Introduction Clifford modules.">, Notational conventions Homotopy groups of the orthogonal group Clifford modules 220
5 vi Contents Clifford modules on manifolds Decomposing compatible connections Hirzebruch signature formula The LeviCivita connection on differential forms Twisted signature complex Product formulas Invariants of the heat equation Hirzebruch signature formula Applications of the signature formula Generalized signature formula Spinors Two dimensional spinors Stiefel Whitney classes Spin bundle The spin and exterior bundles Characteristic classes The spin complex Twisted spin complex Product manifolds Invariants of the heat equation Spin, de Rham, and signature complexes Index theorem for spin complex Twisted de Rham complex YangMills complex Geometrical index theorem The Riemann Roch theorem Almost complex manifolds The arithmetic genus Holomorphic manifolds Relations with holomorphic and Kaehler geometry The spin c complex : Ktheory Ktheory Chern isomorphism Classifying spaces Bott periodicity Suspension and clutching data Orientations External tensor product Integration along the fibers The Atiyah Singer index theorem Extending the index to Ktheory Even dimensional manifolds 275
6 Contents vu Cohomology and Ktheory extensions Odd dimensional manifolds The real Todd genus The~ regularity at s=0 of the eta function Analytic facts Interpretation in Ktheory Odd dimensional manifolds ' Even dimensional manifolds Lefschetz fixed point formulas Isolated fixed points de Rham complex Equivariant invariants Index theorem for manifolds with boundary The induced structures on the boundary Nonlocal boundary conditions Index theorem for product structures Asymptotic expansions for non product structures The transgression Spinors Compatible elliptic complexes of Dirac type Twisted signature complex The eta invariant of locally flat bundles Flat structures on bundles Relative eta invariant Secondary characteristic classes Index theorem on trivial bundles Relative eta invariant...' 323 Spectral geometry Introduction Operators of Laplace type Spectrum of flat tori Local geometry of operators of Laplace type Vanishing theorems ; Formulas for the heat equation asymptotics The form valued Laplacian A recursion relation Leading terms in the asymptotics Variational formulas: Table Isospectral manifolds Geometry "of the spectrum Isospectral non isometric manifolds Compactness results Spherical space forms 349
7 viii Contents ^ Isospectral non isometric metacyclic spherical space forms Spherical harmonics ~7 "Isospectral non isometric lens spaces Non minimal operators Operators of Dirac type Local formulas Reconstruction of the divergence terms Non vanishing of the invariants Manifolds with boundary Boundary conditions Dirichlet and Neumann boundary conditions Mixed boundary conditions Absolute boundary conditions Other asymptotic formulas Asymptotics of operators of Dirac type Non minimal operators with absolute boundary conditions Heat asymptotics on small geodesic balls Operators of Laplace type Heat content asymptotics of nonminimal operators The eta invariant of spherical space forms Properties of the eta function The Hurwicz zeta function The square root of the normalized spherical Laplacian The eta invariant on real projective space Equivariant zeta function Equivariant eta function Eta invariant of spherical space forms Ktheory of spherical space forms Metrics of positive scalar curvature Bibliographic information Acknowledgement Introduction Historical summary The formation of index theory The general AtiyahSinger index theorem The heat equation method Index theory on open manifolds Index theory on singular spaces ifhomology and operator Ktheory Index theory and physics Other topics 433
8 Contents ix 5.3 List of references 435 Notation 509 Index 511
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