A Cocycle Model for Topological and Lie Group Cohomology

Friedrich Wagemann (1) , Christoph Wockel (2)

1 LMJL - Laboratoire de Mathématiques Jean Leray
2 Department of Mathematics - University of Hamburg

We propose a unified framework in which the different constructions of cohomology groups for topological and Lie groups can all be treated on equal footings. In particular, we show that the cohomology of "locally continuous" cochains (respectively "locally smooth" in the case of Lie groups) fits into this framework, which provides an easily accessible cocycle model for topological and Lie group cohomology. We illustrate the use of this unified framework and the relation between the different models in various applications. This includes the construction of cohomology classes characterizing the string group and a direct connection to Lie algebra cohomology.

Mots clés: Lie group cohomology; cohomology of topoplogical groups; locally smooth cocycles; locally continuous cocycles; Buchsbaum argument; Lie algebra cohomology; string group; crossed module; abelian extension

Domaines: Géométrie différentielle [math.DG]; K-théorie et homologie [math.KT]; Topologie algébrique [math.AT]

Friedrich Wagemann, Christoph Wockel. A Cocycle Model for Topological and Lie Group Cohomology. Transactions of the American Mathematical Society, 2015, 367 (3), pp.1871-1909. ⟨hal-00874544⟩