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.