Homology, Homotopy and Applications

Volume 21 (2019)

Number 1

Higher-twisted periodic smooth Deligne cohomology

Pages: 129 – 159

DOI: https://dx.doi.org/10.4310/HHA.2019.v21.n1.a7


Daniel Grady (Department of Mathematics, New York University Abu Dhabi, United Arab Emirates)

Hisham Sati (Department of Mathematics, New York University Abu Dhabi, United Arab Emirates)


Generalizing degree one twisting of Deligne cohomology, as a differential refinement of integral cohomology, from previous work, here we consider higher degree twists. The Rham complex, hence de Rham cohomology, admits twists of any odd degree. However, in order to consider twists of integral cohomology we need a periodic version. Combining the periodic versions of both ingredients leads us to introduce a periodic form of Deligne cohomology. We demonstrate that this theory indeed admits a twist by a gerbe of any odd degree. We present the main properties of the new theory and illustrate its use with examples and computations, mainly via a corresponding twisted differential Atiyah–Hirzebruch spectral sequence.


Deligne cohomology, differential cohomology, gerbe, twisted cohomology, stack, Atiyah–Hirzebruch spectral sequence

2010 Mathematics Subject Classification

14D23, 14F43, 18G40, 19L50, 53C08, 55R20

Received 28 March 2018

Received revised 21 June 2018

Accepted 2 July 2018

Published 5 September 2018