Covering Groups of Nonconnected Topological Groups and 2-Groups

Abstract

We investigate the universal cover of a topological group that is not necessarily connected. Its existence as a topological group is governed by a Taylor cocycle, an obstruction in 3-cohomology. Alternatively, it always exists as a topological 2-group. The splitness of this 2-group is also governed by an obstruction in 3-cohomology, a Sinh cocycle. We give explicit formulas for both obstructions and show that they are inverse of each other.

9 pages. Version 2: historical review added. Version 3 (14 pages): historical review revised, minor corrections elsewhere, some of the results did not make it into the final version. Version 4: final journal version with a slightly different (to Version 3) approach