Riemannian submersions and factorization of Dirac operators

Abstract

We establish the factorization of Dirac operators on Riemannian submersions of compact spin${}^c$ manifolds in unbounded $KK$-theory. More precisely, we show that the Dirac operator on the total space of such a submersion is unitarily equivalent to the tensor sum of a family of Dirac operators with the Dirac operator on the base space, up to an explicit bounded curvature term. Thus, the latter is an obstruction to having a factorization in unbounded $KK$-theory. We show that our tensor sum represents the bounded KK-product of the corresponding $KK$-cycles and connect to the early work of Connes and Skandalis.