Spectral sequences in smooth generalized cohomology

Grady, Daniel; Sati, Hisham

doi:10.2140/agt.2017.17.2357

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Spectral sequences in smooth generalized cohomology

Daniel Grady and Hisham Sati

Algebraic & Geometric Topology 17 (2017) 2357–2412

DOI: 10.2140/agt.2017.17.2357

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 $U (1)$ coefficients.

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

References

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

Volume 17, issue 4 (2017)