Seminar on Atiyah-Singer Index Theorem. (AM-57), Volume 57 (Annals of Mathematics Studies)

The description for this book, Seminar on Atiyah-Singer Index Theorem. (AM-57), Volume 57, will be forthcoming. CONTENTS PREFACE CHAPTER I: STATEMENT OF THE THEOREM OUTLINE OF THE PROOF §1. The index theorem §2. The topological index §3. The analytical index §4. Appendix CHAPTER II : REVIEW OF K-THEORY §1. K(X) a finite CW-complex §2. The Chern character §3. The difference construction §4. L-theory §5. Products in L-theory CHAPTER III : THE TOPOLOGICAL INDEX OF AN OPERATOR ASSOCIATED TO A G-STRUCTURE CHAPTER IV: DIFFERENTIAL OPERATORS ON VECTOR BUNDLES §1. Notation §2. Jet bundles §3. Differential operators and their symbols §4. Hermitian bundles and adjoint operators §5. Green's forms §6. Some classical differential operators §7. Whitney sums §8. Tensor products §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 §1. Chains §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 §1. The spaces C^k(ξ) §2. The hilbert space H^0(ξ) §3. The spaces H^k(ξ) CHAPTER X: THE CONTINUOUS SOBOLEV CHAIN OF A VECTOR BUNDLE §1. Continuous Sobolev chains §2. The chains {H^k(T^n, V)} §3. An extension theorem §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 §3. The # operation §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 §3. Proof of Lemma 1 §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 §8. Sign conventions CHAPTER XVII: COBORDISM INVARIANCE OF THE ANALYTICAL INDEX CHAPTER XVIII: BORDISM GROUPS OF BUNDLES §1. Introductory remarks §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 §2. τ-homomorphism §3. The character of classical elliptic operators This book consists mainly of slightly revised notes of a seminar held at the Institute for Advanced Study in 1963-64 upon the initiative of A. Borel. Aside from going through the details of the proof of the Index Theorem, the major emphasis of the seminar was placed on developing the topological and analytical machinery associated with integro-differential operators. On the topological side the agreement was to assume a reasonable degree of sophistication. Thus, the basic facts concerning K-theory and characteristic classes are reviewed rather than proved and the emphasis is on showing how, with these tools, elliptic operators give rise to cohomology classes and on studying the properties of these classes. On the analytical side, it was decided to start more or less ab initio. The reason for this somewhat unbalanced exposition is in part due to the predilections of the organizers of the seminar, but also in part it is due to the fact that while most of the algebraic topology involved is covered in complete detail in easily accessible published papers, much of the analysis is quite recent, and the published versions often refer explicitly only to the case of trivial bundles over domains in Euclidean space A classic treatment of the Atiyah-Singer index theorem from the acclaimed Annals of Mathematics Studies series Princeton University Press is proud to have published the Annals of Mathematics Studies since 1940. One of the oldest and most respected series in science publishing, it has included many of the most important and influential mathematical works of the twentieth century. The series continues this tradition as Princeton University Press publishes the major works of the twenty-first century. 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