Texts and Readings in Mathematics

This monograph is a thorough introduction to the Atiyah-Singer index theorem for elliptic operators on compact manifolds without boundary. The main theme is only the classical index theorem and some of its applications, but not the subsequent developments and simplifications of the theory. The book is designed for a complete proof of the K-theoretic index theorem and its representation in terms of cohomological characteristic classes, with an effort to make the demands on the knowledge of background materials as modest as possible by supplying the proofs of all most every result. The applications include Hirzebruch signature theorem, Riemann-Roch-Hirzebruch theorem, Atiyah-Segal-Singer fixed point theorem, etc.

1. K-Theory
2. Fredholm Operators and Atiyah-Jänich Theorem
3. Bott Periodicity and Thom Isomorphism
4. Pseudo-Differential Operators
5. Characteristic Classes and Chern-Weil Construction
6. Spin Structure and Dirac Operator
7. Equivariant K-Theory
8. The Index Theorem
9. Cohomological Formulation of the Index Theorem
Biblography
Index