Chapter
CHAPTER IV: DIFFERENTIAL OPERATORS ON VECTOR BUNDLES
§3. Differential operators and their symbols
§4. Hermitian bundles and adjoint operators
§6. Some classical differential operators
§9. Connections and covariant derivatives
§10. Spin structures and Dirac operators
CHAPTER V: ANALYTICAL INDICES OF SOME CONCRETE OPERATORS
§1. Review of Hodge theory
§2. The Euler Characteristic
§3. The Hirzebruch signature theorem
§4. Odd-dimensional manifolds
CHAPTER VI: REVIEW OF FUNCTIONAL ANALYSIS
CHAPTER VII: FREDHDIM OPERATORS
CHAPTER VIII: CHAINS OF HTLBERTIAN SPACES
§2. Quadratic interpolation of pairs of hilbert spaces
§3. Quadratic interpolation of chains
§4. Scales and the chains (Z^n, V)
CHAPTER IX: THE DISCRETE SOBOLEV CHAIN OF A VECTOR BUNDLE
§2. The hilbert space H^0(ξ)
CHAPTER X: THE CONTINUOUS SOBOLEV CHAIN OF A VECTOR BUNDLE
§1. Continuous Sobolev chains
§2. The chains {H^k(T^n, V)}
§4. The Rellich, Sobolev, and restriction theorems
CHAPTER XI: THE SEELEY ALGEBRA
CHAPTER XII: HOMOTOPY INVARIANCE OF THE INDEX
CHAPTER XIII: WHITNEY SUMS
§1. Direct sums of chains of hilbertian spaces
§2. The Sobolev chain of a Whitney sum
§3. Behaviour of Smblk with respect to Whitney sums
§4. Behaviour of Intk and σ^k under Whitney sums
§5. Behaviour of the index under Whitney sums
CHAPTER XIV: TENSOR PRODUCTS
§1. Tensor products of chains of hilbertian spaces
§2. The Sobolev chain of a tensor product of bundles
§4. The property (S6) of the Seeley Algebra
§5. Multiplicativity of the index
CHAPTER XV: DEFINITION OF ia AND it ON K(M)
§1. Definition of the analytical index on K(B(M), S(M))
§2. Multiplicative properties of it
§4. Definition of it and ia on K(M)
§5. Summary of the properties of ia and it on K(M)
§6. Multiplicative properties of i on K(X )
§7. Direct check that ia = it in some special cases
CHAPTER XVI: CONSTRUCTION OF Intk
§1. The Fourier Transform
§2. Calderón-Zygnund operators
§3. Calderón-Zygmund operators for a compact manifold
§4. Calderón-Zygmund operators for vector bundles
§5. Definition and properties of Intr (ξ, η)
§6. An element of Into(S^1) with analytical index -1
§7. The topological index of the operator of §6
CHAPTER XVII: COBORDISM INVARIANCE OF THE ANALYTICAL INDEX
CHAPTER XVIII: BORDISM GROUPS OF BUNDLES
§2. Computation of Ωk(X) ⊗ Q
§3. The bordism ring of bundles
CHAPTER XIX: THE INDEX THEOREM: APPLICATIONS
§1. Proof of the index theorem
§2. An alternative formulation of the index theorem
§3. The non-orientable case of Theorem 2
§4. The Riemann-Roch-Hirzebruch theorem
§5. Generalities on integrality theorems
§6. The integrality theorems
APPENDIX I : THE INDEX THEOREM FOR MANIFOLDS WITH BOUNDARY
§1. Ellipticity for manifolds with boundary
§2. The difference element [σ(d, b)]
§3. Comments on the proof
APPENDIX II : NON STABLE CHARACTERISTIC CLASSES AND THE TOPOLOGICAL INDEX OF CLASSICAL ELLIPTIC OPERATORS
§1. Characteristic classes
§3. The character of classical elliptic operators
Seminar on Atiyah-Singer Index Theorem. (AM-57)
( Annals of Mathematics Studies )
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