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arXiv · 1710.03182

Factorization of Dirac operators on almost-regular fibrations of spin$^c$ manifolds

Jens Kaad·Walter D. van Suijlekom
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Abstract

We establish the factorization of the Dirac operator on an almost-regular fibration of spin$^c$ manifolds in unbounded KK-theory. As a first intermediate result we establish that any vertically elliptic and symmetric first-order differential operator on a proper submersion defines an unbounded Kasparov module, and thus represents a class in KK-theory. Then, we generalize our previous results on factorizations of Dirac operators to proper Riemannian submersions of spin$^c$ manifolds. This allows us to show that the Dirac operator on the total space of an almost-regular fibration can be written as the tensor sum of a vertically elliptic family of Dirac operators with the horizontal Dirac operator, up to an explicit `obstructing' curvature term. We conclude by showing that the tensor sum factorization represents the interior Kasparov product in bivariant K-theory.

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Jens Kaad, Walter D. van Suijlekom. 2017-10-09. Factorization of Dirac operators on almost-regular fibrations of spin$^c$ manifolds. https://arxiv.org/abs/1710.03182

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