Mickelsson's twisted K-theory invariant and its generalizations

TL;DR

This paper reformulates Mickelsson's twisted K-theory invariant as a homomorphism related to cohomology and generalizes it to higher cohomological quotients, also constructing related characteristic classes.

Contribution

It introduces a new formulation of Mickelsson's invariant and extends it to higher cohomology, connecting to the Atiyah-Hirzebruch spectral sequence and constructing new characteristic classes.

Findings

Reformulation of Mickelsson's invariant as a homomorphism.

Generalization to quotients of fifth cohomology.

Construction of new characteristic classes for odd twisted K-theory.

Abstract

Mickelsson's invariant is an invariant of certain odd twisted K-classes of compact oriented three dimensional manifolds. We reformulate the invariant as a natural homomorphism taking values in a quotient of the third cohomology, and provide a generalization taking values in a quotient of the fifth cohomology. These homomorphisms are related to the Atiyah-Hirzebruch spectral sequence. We also construct some characteristic classes for odd twisted K-theory in a similar vein.