Topological index

Summary

The topological index of an elliptic operator on a closed manifold is an integer built only from topological data. These data are the K-theory class of the principal symbol and the characteristic classes of the manifold. Atiyah and Singer defined it by pushing the symbol class forward along an embedding of the manifold into a Euclidean space. Their index theorem states that it equals the analytic index. In cohomology it is an integral of the Chern character of the symbol times the Todd class.