#### Volume 17, issue 4 (2017)

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Spectral sequences in smooth generalized cohomology

### Daniel Grady and Hisham Sati

Algebraic & Geometric Topology 17 (2017) 2357–2412
##### 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.

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##### Keywords
differential cohomology, smooth cohomology, generalized cohomology, Atiyah-Hirzebruch spectral sequence, cohomology operations
##### Mathematical Subject Classification 2010
Primary: 55N15, 55T10, 55T25
Secondary: 53C05, 55S05, 55S35
##### Publication
Received: 3 June 2016
Revised: 11 October 2016
Accepted: 3 January 2017
Published: 3 August 2017
##### Authors
 Daniel Grady Department of Mathematics New York University, Abu Dhabi Abu Dhabi United Arab Emirates Hisham Sati Department of Mathematics University of Pittsburgh Pittsburgh, PA United States