Factorization of Dirac Operators on Almost-Regular Fibrations of Spinc Manifolds

Jens Kaad*, Walter D.van Suijlekom

*Corresponding author for this work

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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 ‘obstructing’ curvature term. We conclude by showing that the tensor sum factorization represents the interior Kasparov product in bivariant K-theory.

Original languageEnglish
JournalDocumenta Mathematica
Pages (from-to)2049-2084
Publication statusPublished - 2020


  • Dirac operators
  • half-closed chains
  • Kasparov product
  • KK-theory
  • proper Riemannian submersions
  • spinc-manifolds
  • Unbounded Kasparov modules
  • unbounded Kasparov product
  • unbounded KK-theory


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