Central Extensions and Cohomology

TL;DR

This paper explores the relationship between central extensions of topological groups and their cohomology classes, establishing conditions for injectivity, bijectivity, and liftings via cohomology pullbacks.

Contribution

It proves the injectivity and often bijectivity of the cohomology class association for central extensions of CW-complex groups.

Findings

The association between central extensions and cohomology classes is injective.

In many cases, this association is bijective.

A homomorphism lifts iff the pullback of the cohomology class vanishes.

Abstract

Let G be a group which is topologically a CW-complex, BG a classifying space for G, and A a discrete abelian group. To a central extension of G by A, one can associate a cohomology class in $H^2(BG,A)$. We show this association is injective, and bijective in many cases. A homomorphism to G lifts to the extension iff the pullback of the associated cohomology class vanishes.