Journal of Noncommutative Geometry
Full-Text PDF (294 KB) | Metadata | Table of Contents | JNCG summary
Published online: 2018-10-30
Riemannian submersions and factorization of Dirac operatorsJens Kaad and Walter D. van Suijlekom (1) Syddansk Universitet, Odense, Denmark
(2) Radboud University Nijmegen, Netherlands
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.
Keywords: Unbounded KK-theory, Riemannian submersions, Dirac operators, spin-c structures, wrong way functoriality
Kaad Jens, van Suijlekom Walter: Riemannian submersions and factorization of Dirac operators. J. Noncommut. Geom. 12 (2018), 1133-1159. doi: 10.4171/JNCG/299