Abstract

Index Theory and Non-Commutative Geometry I. Higher families index theory

We prove an index theorem for foliated manifolds. We do so by constructing a push forward map in cohomology for a K-oriented map from an arbitrary manifold to the space of leaves of an oriented foliation, and by constructing a Chern-Connes character from the K-theory of the compactly supported smooth functions on the holonomy groupoid of the foliation to the Haefliger cohomology of the foliation. Combining these with the Connes-Skandalis topological index map and the classical Chern character gives a commutative diagram from which the index theorem follows immediately.