@article{ae66c4055301482d9b82344ee13910c5,
title = "Factorization of Dirac Operators on Almost-Regular Fibrations of Spinc Manifolds",
abstract = "We establish the factorization of the Dirac operator on an almost-regular fibration of spinc 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 spinc 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 {\textquoteleft}obstructing{\textquoteright} curvature term. We conclude by showing that the tensor sum factorization represents the interior Kasparov product in bivariant K-theory.",
keywords = "Dirac operators, half-closed chains, Kasparov product, KK-theory, proper Riemannian submersions, spinc-manifolds, Unbounded Kasparov modules, unbounded Kasparov product, unbounded KK-theory",
author = "Jens Kaad and Suijlekom, \{Walter D.van\}",
note = "Funding Information: We would like to thank Claire Debord for pointing us to the thesis of St{\'e}phane Vassout. We would also like to thank Peter Hochs for a very useful suggestion on how to prove the compactness of the resolvent in the context of fiber bundles. We gratefully acknowledge the Southern Danish University and the Radboud University Nijmegen for their financial support in facilitating this collaboration. During the initial stages of this research project the first author was supported by the Radboud excellence fellowship. The first author was partially supported by the DFF-Research Project 2 “Automorphisms and Invariants of Operator Algebras”, no. 7014-00145B and by the Villum Foundation (grant 7423). The second author was partially supported by NWO under VIDI-grant 016.133.326. Publisher Copyright: {\textcopyright} 2020. Documenta Mathematica. All Rights Reserved. Copyright: Copyright 2021 Elsevier B.V., All rights reserved.",
year = "2020",
doi = "10.25537/dm.2020v25.2049-2084",
language = "English",
volume = "25",
pages = "2049--2084",
journal = "Documenta Mathematica",
issn = "1431-0635",
publisher = "European Mathematical Society Publishing House",
}