Twisted differential generalized cohomology theories and their Atiyah-Hirzebruch spectral sequence

TL;DR

This paper develops a spectral sequence for twisted differential generalized cohomology theories, extending classical and prior differential cohomology frameworks, with detailed focus on twisted differential K-theory and explicit computational methods.

Contribution

It introduces a new Atiyah-Hirzebruch spectral sequence for twisted differential cohomology theories, including a detailed analysis of twisted differential K-theory and its differentials.

Findings

Constructed the AHSS for twisted differential cohomology theories.

Established that twisted differential spectra are bundles of spectra with flat connections.

Provided explicit computations and examples for twisted differential K-theory.

Abstract

We construct the Atiyah-Hirzebruch spectral sequence (AHSS) for twisted differential generalized cohomology theories. This generalizes to the twisted setting the authors' corresponding earlier construction for differential cohomology theories, as well as to the differential setting the AHSS for twisted generalized cohomology theories, including that of twisted K-theory by Rosenberg and Atiyah-Segal. In describing twisted differential spectra we build on the work of Bunke-Nikolaus, but we find it useful for our purposes to take an approach that highlights direct analogies with classical bundles and that is at the same time amenable for calculations. We will, in particular, establish that twisted differential spectra are bundles of spectra equipped with a flat connection. Our prominent case will be twisted differential K-theory, for which we work out the differentials in detail. This involves differential refinements of primary and secondary cohomology operations the authors developed in earlier papers. We illustrate our constructions and computational tools with examples.