Culture and History of Mathematics

The Atiyah-Singer index theorem, proved in 1963, is one of the highlights of twentieth century mathematics. It subsumed many classical results that connected topological properties of manifolds with their differential-geometric properties, and led to a lot of new research in topology, geometry, analysis and physics.

This book published in 1965, was one of the first expositions of this theorem. It is a document of historical interest and much can be learned from it even now. 1: Statement of the Theorem, Outline of the Proof- A. Borel.
2: Review of K-Theory-R. Solovay.
3: The Topological Index of an Operator Associated To A G-Structure-R. Solovay.
4: Differential Operators on Vector Bundles-R.S. Palais.
5: Analytical Indices of Some Concrete Operators-M. Solovay.
6: Review of Functional Analysis-R.S. Palais.
7: Fredholm Operators- R. S. Palais.
8: Chains of Hilbertian Spaces-R. S. Palais.
9: The Discrete Sobolev Chain of a Vector Bundle- R. S. Palais.
10: The Continuous Sobolev Chain of a Vector Bundle-R. S. Palais.
11: The Seeley Algebra-R. S. Palais.
12: Homotopy Invariance of the Index- R. S. Palais.
13: Whitney Sums- R. S. Palais.
14: Tensor Products- R. S. Palais.
15: Definition of 1a and 1t on K(M)-R. M. Solovay.
16: Construction of Intk-R. S. Palais and R. T. Seeley.
17: Cobordism Invariance of the Analytical Index-R. S. Palais and R. T. Seeley.
18: Hordism Groups of Bundles-E. E. Floyd.
19: The Index Theorem: Applications-R.M. Solovay.
Appendix I: The Index Theorem for Manifolds with Boundary-M.F. Atiyah
Appendix II: Non Stable Characteristic Classes and the Topological Index of Classical Elliptic Operators-W. Shih