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INDEX

A-roof (Â) genus, 99 Absolute boundary conditions for the de Rham complex, 243 Algebra of operators, 13 Allard, William, 45 Almost complex structure, 180 Arithmetic genus, 190, 200 Arzela-Ascoli Theorem, 8 Asymptotic expansion of symbol, 51 Asymptotics of elliptic boundary conditions, 72 of heat equation, 54 Atiyah-Bott Index Theorem, 247 Atiyah-Patodi-Singer Theorem, 268, 276 Atiyah-Singer Index Theorem, 233 Axiomatic characterization of Chern forms, 202 of Euler forms, 121 of Pontrjagin forms, 130 Bigrading exterior algebra, 107 Bott periodicity, 221 Bundle of Cauchy data, 70 Bundles of positive part of symbol, 76 Canonical bundle, 186 Cauchy data, 70 Čech Z_{2} cohomology, 166 Change of coordinates, 25 Characteristic forms, 91 Characteristic ring, 93, 98, 103 Chern character, 91, 174, 217 class of complex projective space, 111 form, 91 isomorphism in K-theory, 217 Christoffel symbols, 104 Clifford algebra, 148 matrices, 95 multiplication, 148 Clutching functions, 218 Cohomology of complex projective space, 109, 200 of elliptic complex, 37 of Kaehler manifolds, 195 Collared manifold, 253 Compact operators, 31 Complex characteristic ring, 93 projective space, 109 tangent space, 107 Connected sum, 154 Connection 1-form, 89 Continuity of operators, 12 Convolution, 2 Cotangent space, 25 Cup product, 151 Curvature of spin bundle, 172 2-form, 90 d-bar ( ¯∂), 107 Defect formulas, 286, 296 de Rham cohomology, 91, 197 complex, 39, 149, 246 de Rham, signature and spin bundles, 172 Dirac matrices, 95 Dirichlet boundary conditions, 71, 245 Dolbeault complex, 183 Dolbeault and de Rham cohomology, 197 Donnelly Theorem, 293 Double suspension, 220 Dual norm, 30 Elliptic boundary conditions, 72 complex, 37 operators, 23 Eta function, 81, 258, 263 invariant, 258, 270 Euler characteristic of elliptic complex, 37 form, 100, 121, 174, 308 Euler-Poincaré characteristic, 246 Exterior algebra, 148 multiplication, 39, 148 Flat K-theory, 298 Form valued polynomial, 129 Fourier transform, 3 Fredholm index, 32 operators, 32

348 Index

Fubini-Study metric, 111, 195, 200 Functional calculus, 48 Gärding’s inequality, 23 Gauss-Bonnet Theorem, 122, 179, 256 Geodesically convex, 166 Graded boundary conditions, 70 symbol, 71 vector bundle, 37 Greens operator, 44 Gunther and Schimming, 313 Half spin representations, 161 Heat equation, 47 local formulas, 326 Hirzebruch L-polynomial, 99, 115, 175 Signature Theorem, 154 Hodge Decomposition Theorem, 38, 246 manifolds, 196 ∗ operator, 246 Holomorphic connection, 108 manifold, 107 Holonomy of representation, 270 Homotopy groups of U(n), 221 Hyperplane bundle, 109 Hypoelliptic, 23 Index and heat equation, 47 form in K-theory, 299 formula (local), 58 in K-theory, 225 of representation, 271 R/Z valued, 270 Index Theorem for de Rham complex, 122 for Dolbeault complex, 189 for signature complex, 154 for spin complex, 179 Instanton bundles, 96 Integration along fibers, 235 Interior multiplication, 40, 148 Invariant form valued polynomial, 134 polynomials, 119 Invariants of heat equation, 326 Isospectral manifolds, 331 K-theory, 216, 225 of locally flat bundles, 298 of projective spaces, 302 of spheres, 221 of spherical space forms, 301 sphere bundle, 267 Kaehler manifolds, 195 2-form, 193 Kernel of heat equation, 54 of operator, 19 L-polynomial, 99, 115, 175 Lefschetz fixed point formulas, 63, 68, 286 number of elliptic complex, 62 Lens space, 276, 297 Levi-Cività connection, 104 Local formula for derivative of eta, 82 index formula, 58 Locally flat bundle, 270 Maurer-Cartan form, 275 Mellin transform, 78 Monopole bundles, 96 Neuman series, 29 Neumann boundary conditions, 71, 245 Nirenberg-Neulander Theorem, 182 Operators depending on a complex parameter, 50 Orientation form, 150 Parametrix, 29, 52 Peetre’s Inequality, 10 Plancherel Formula, 5 Poincaré duality, 41, 246 Pontrjagin form, 98, 174, 308 Projective space arithmetic genus, 200 cohomology, 109 signature, 152 Pseudo-differential operators, 11 Pseudolocal, 20 R/Z valued index, 270 Real characteristic ring, 99 Regularity of eta at 0, 263 Relative boundary conditions for the de Rham complex, 243 Rellich Lemma, 8 Residue of eta at s = 0, 84, 258, 266

Index 349

Resolvant of operator, 52 restriction of invariant, 120, 255 Riemann zeta function, 80 Riemann-Roch Theorem, 189, 202 Ring of invariant polynomials, 119 Schwartz class, 2 Secondary characteristic classes, 273 Serre duality, 199 Shape operator, 251 Shuffle formulas, 313 Signature complex, 150 tangential operator, 261 of projective space, 153 spin and de Rham bundles, 172 Theorem, 154 Simple covers, 166 Singer’s conjecture for the Euler form, 308 Snaith’s Theorem, 305 Sobolev Lemma, 7 spaces, 6 Spectral flow, 273 geometry, 331 Spectrum of compact operator, 42 of self-adjoint elliptic operator, 43 Sphere bundle K-theory, 267 Chern character, 96 Euler form, 101 K-theory, 221 Spherical space form, 295, 333 Spin and other elliptic complexes, 172 complex, 176 group, 159 structures, 165 on projective spaces, 170 SPIN_{c}, 186 Spinors, 159 Splitting principal, 190 Stable isomorphism of vector bundles, 216 Stationary phase, 64 Stieffel-Whitney classes, 167 Suspension of topological space, 215 of symbol, 224 Symbol de Rham complex, 40 Spin complex, 176 Dolbeault complex, 184 of operator on a manifold, 25 Tangential operator signature complex, 261 Tautological bundle of CP_{n}, 109, 195 Todd form, 97, 114, 236 Topological charge, 223 Trace heat operator, 47 Transgression, 92, 252 U(n), homotopy groups of, 221 Weyl spanning set, 121 Weyl’s theorem on invariants of orthogonal group, 132 Zeta function, 80