Journal of Noncommutative Geometry

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Volume 12, Issue 3, 2018, pp. 1133–1159
DOI: 10.4171/JNCG/299

Published online: 2018-10-30

Riemannian submersions and factorization of Dirac operators

Jens Kaad[1] and Walter D. van Suijlekom[2]

(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