Factorization of Dirac Operators on Almost-Regular Fibrations of $\mathrm{Spin}^c$ Manifolds

Kaad, Jens ; van Suijlekom, Walter D.

Documenta mathematica, Tome 25 (2020), pp. 2049-2084.

Résumé

We establish the factorization of the Dirac operator on an almost-regular fibration of $\mathrm{spin}^c$ 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 $\mathrm{spin}^c$ 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.

Détail

Classification : 19K35, 53C27, 58B34
Keywords: unbounded Kasparov modules, half-closed chains, Dirac operators, \(\mathrm{spin}^c\)-manifolds, proper Riemannian submersions, KK-theory, unbounded KK-theory, Kasparov product, unbounded Kasparov product