Central Extensions of Topological Groups and the Cohomology of Classifying Spaces

Abstract

For a group $G$ and an abelian group $A$, the theory of group cohomology gives an isomorphism $\mathbb{E}(G,A)\to H^2(BG,A)$, where $\mathbb{E}(G,A)$ is the group of central extensions of $G$ by $A$. We generalize this construction to the case where $G$ and $A$ are (sufficiently nice) topological groups by producing a map $\alpha:\mathbb{E}(G,A)\to H^2(BG,A)$. Here, $\mathbb{E}(G,A)$ consists of central extensions which are also principal $A$-bundles, and $H^2(BG,A)$ is defined using the $\Omega$-spectrum $A, BA, B^2A,\cdots$. The study of $\ker\alpha$ naturally leads us to define certain maps $\alpha^n:H_{\text{c}}^n(G,A)\to H^n(BG,A)$, where $H_{\text{c}}^{\ast}(G,A)$ is the homology of the chain complex of continuous inhomogeneous cochains. When $G$ and $A$ are discrete, $\alpha^n$ agrees with the classical isomorphism between group cohomology $H_{\text{gp}}^n(G,A)$ and $H^n(BG,A)$. Contingent on a conjecture regarding the cohomology of the Milgram--Steenrod filtration (equivalently, Milnor's filtration) of $BG$, we obtain the following satisfactory characterization of $\ker\alpha^n$: a cohomology class lies in $\ker\alpha^n$ if and only if the algebraic information it contains can be killed by homotopy, loosely speaking. The special case $n=2$ gives a similar characterization of the extensions contained in $\ker \alpha$. We demonstrate several examples where $\ker\alpha^n$ and $\ker\alpha$ can be characterized independent of the conjecture. The study of $\alpha^n$ is of independent interest, since it generalizes the homotopy-theoretic approach to classical group cohomology. Furthermore, it complements the analytic and categorical lenses employed in existing literature on continuous group cohomology.