Spectral sequences in smooth generalized cohomology

TL;DR

This paper develops spectral sequences for smooth generalized cohomology theories, enabling systematic analysis of torsion and cohomology operations, with explicit differentials identified for several key theories.

Contribution

It introduces a filtration-based spectral sequence framework for smooth generalized cohomology, including differential theories, and explicitly computes differentials for several important cases.

Findings

Explicit differentials in spectral sequences for smooth Deligne cohomology

Identification of differentials in differential topological K-theory

Introduction of a smooth extension of integral Morava K-theory

Abstract

We consider spectral sequences in smooth generalized cohomology theories, including differential generalized cohomology theories. The main differential spectral sequences will be of the Atiyah-Hirzebruch (AHSS) type, where we provide a filtration by the Cech resolution of smooth manifolds. This allows for systematic study of torsion in differential cohomology. We apply this in detail to smooth Deligne cohomology, differential topological complex K-theory, and to a smooth extension of integral Morava K-theory that we introduce. In each case we explicitly identify the differentials in the corresponding spectral sequences, which exhibit an interesting and systematic interplay between (refinement of) classical cohomology operations, operations involving differential forms, and operations on cohomology with U(1) coefficients.