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Spectral sequences in smooth generalized cohomology

Daniel Grady and Hisham Sati

Algebraic & Geometric Topology 17 (2017) 2357–2412

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.

differential cohomology, smooth cohomology, generalized cohomology, Atiyah-Hirzebruch spectral sequence, cohomology operations
Mathematical Subject Classification 2010
Primary: 55N15, 55T10, 55T25
Secondary: 53C05, 55S05, 55S35
Received: 3 June 2016
Revised: 11 October 2016
Accepted: 3 January 2017
Published: 3 August 2017
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