The aim of this course is to introduce topological K -theory, formulate the index theorem, sketch the K -theoretic proof and exhibit its power by deriving celebrated results such the Gauss-Bonnet theorem and Hirzebruch’s signature theorem.
An outline of the course is as follows. We begin by defining vector bundles and generalising standard constructions such as direct sum, tensor product and exterior powers, from vector spaces to vector bundles. We explain how to define K-theory by considering formal differences of vector bundles. After establishing some formal properties of K-theory, we turn to the theory of characteristic classes and Chern-Weil theory. As a first application we present an easy proof, due to Adams and Atiyah, of the Hopf invariant one theorem using certain operations on K-theory. One easy consequence of this theorem is that there are no real division algebras in dimensions other than 1, 2, 4 and 8.
Next we introduce elliptic differential operators (or more generally, elliptic complexes) on compact manifolds and explain how their symbol determines a class in compactly supported K-theory. We define the notion of analytic index and topological index for elliptic operators. The Atiyah-Singer index theorem is the statement that these two index maps are equal. After explaining the basic idea behind the proof and deriving a cohomological formula, we consider several applications of the index theorem.
A. Hatcher, Algebraic Topology.
A. Hatcher, Vector Bundles and K-theory.
P. Shanahan, The Atiyah-Singer Index Theorem, Lecture Notes in Mathematics, 638, Springer, Berlin, 1978.
H. B. Lawson and M.-L. Michelson, Spin Geometry, Princeton Mathematical Series 38, Princeton University Press, Princeton, 1989.