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

TL;DR

This paper proves that the Dirac operator on almost-regular fibrations of spin$^c$ manifolds can be factorized into a tensor sum of vertical and horizontal components within unbounded KK-theory, extending previous results.

Contribution

It generalizes the factorization of Dirac operators to proper Riemannian submersions of spin$^c$ manifolds using unbounded KK-theory, including explicit curvature terms.

Findings

Dirac operator factorization as tensor sum

Representation of the Dirac operator in KK-theory

Extension of previous results to Riemannian submersions

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.