Spectral sequences in smooth generalized cohomology

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 Čech 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 (refinements of) classical cohomology operations, operations involving differential forms and operations on cohomology with $U\left(1\right)$ coefficients.