Superconnections and higher index theory

Abstract

Let M be a smooth closed spin manifold. The higher index theorem, as given for example in Proposition 6.3 of [CM], computes the pairing between the group cohomology of π 1 ( M ) and the Chern character of the “higher” index of a Dirac-type operator on M. Using superconnections, we give a heat equation proof of this theorem on the level of differential forms on a noncommutative base space. As a consequence, we obtain a new proof of the Novikov conjecture for hyperbolic groups.