Covering Groups of Nonconnected Topological Groups and 2-Groups

TL;DR

This paper explores the universal cover of nonconnected topological groups, revealing that it always exists as a topological 2-group and providing explicit formulas for the governing cohomological obstructions.

Contribution

It introduces explicit formulas for the Taylor and Sinh cocycles, clarifying their inverse relationship and the conditions for the existence and splitness of the universal cover as a 2-group.

Findings

Universal cover exists as a topological 2-group

Explicit formulas for Taylor and Sinh cocycles

Obstructions are inverses in cohomology

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.