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We know that the Borel group cohomology (group cohomology of measurable functions) of a group $G$, ${\cal H}_B^d(G,Z)$, is given by the cohomology of the classifying space: ${\cal H}_B^d(G,Z)=H^d(BG,Z)$. However, ${\cal H}_B^d(G,R/Z)\neq H^d(BG,R/Z)$, since for example: $H^d(BU(1),R/Z) = R/Z$ for even $d$ and $H^d(BU(1),R/Z) = 0$ for odd $d$; while ${\cal H}_B^d(BU(1),R/Z)=0$ for even $d$ and ${\cal H}_B^d(BU(1),R/Z)=Z$ for odd $d$. Instead, we have ${\cal H}^d(G,R/Z) = H^{d+1}(BG,Z)$.

My question is that do we have any relations between ${\cal H}_B^*(G,M)$ and $H^*(BG,M')$ where $G$ can be continuous, $M$ can be $Z_n$, and $M'$ can be different from $M$? In particular, I would like to know how ${\cal H}_B^*(G,Z_n)$ is related to $H^*(BG,M')$ when $G$ is continuous.

A related question group cohomology and cohomology of classifying space is closed. I hope it can be reopen.

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Yes, there is a relation between these cohomology groups. The results you are referring to are contained for instance in Austin-Moore "Continuity properties of measurable group cohomology" [http://arxiv.org/abs/1004.4937] and in parts already in Wigner "Algebraic cohomology of topological groups" [http://projecteuclid.org/euclid.bams/1183532101].

To understand the relation in general one can consider topological group cohomology and different models of it. This has the draw-back that it restricts the groups to which the results apply (for instance to finite-dimensional Lie groups), but has the advantage that the different models for it provide many useful relations to other concepts from algebraic topology, where classifying space cohomology is one instance of.

To make it short, the measurable cohomology (which I think you refer to with "Borel group cohomology") of a compact group $G$ with coefficients in a finite-dimensional connected abelian Lie group $A$ is the cohomology of the classifying space with a degree shift: $H^n_{B}(G,A)\cong H^{n+1}(BG,\pi_1(A))$. This explains the phenomenon you describe. More on this can be found in [http://arxiv.org/abs/1110.3304] (see in particular Remark 4.13 for the relation to measurable cohomology).

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  • $\begingroup$ Christoph Wockel: Thank you very much. Do your results apply to Borel group cohomology $H^d_B(G,Z_n)$ when $G$ is continuous? $\endgroup$ – Xiao-Gang Wen Nov 7 '14 at 4:33
  • $\begingroup$ Our results work also for finite-dimensional locally compact groups. The problems with $Z_n$-coefficients would be to find a short exact sequence $Z_n \to E \to A$ where you see that $H^n_B(G,E)$ vanishes. (To this sequence you would like to apply the long exact sequence to in oder to identify $H^n_B(G,Z_n)$ with $H^n_B(G,A)$ in case $H^n_B(G,E)$ vanishes. For $Z$ you can take $Z \to R \to U(1)$.) $\endgroup$ – Christoph Wockel Nov 7 '14 at 7:51
  • $\begingroup$ Christoph: Do you have an example of a short exact sequence $Z_n\to E\to A$ such that that $H^n_B(G,E)$ vanishes? $\endgroup$ – Xiao-Gang Wen Nov 8 '14 at 12:57
  • $\begingroup$ @Xiao-Gang: Yes, this runs under the name "dimension shifting". Take $E Z_n$ the group of (equivalence classes of) measurable $Z_n$-valued functions on the unit interval. With the distance in measure this should be a contractible polish group. Then take $C(G,E Z_n)$, which is still contractible and polish. Moreover, it is soft, so the topological group cohomology (and under the above assumption on $G$) of it vanishes (see the appendix in our paper). Since $Z_n$ is finite, it acts properly discontinuously on $C(G,E Z_n)$ and so $E:=C(G,E Z_n)\to C(G,E Z_n)/Z_n=:A$ has a continuous section. $\endgroup$ – Christoph Wockel Nov 11 '14 at 8:45
  • $\begingroup$ Christoph: thank you very much for explaining. Using your above construction, now we have $H^n_B(G,A)\simeq H^{n+1}_B(G,Z_n)$ for measurable group cohomology, or $H^n(BG,A)\simeq H^{n+1}(BG,Z_n)$ for topological cohomology, which is very very interesting. My question here is can we use a similar construction to find the relation between group cohomology $H^*_B(G,Z_n)$ and topological cohomology $H^*(BG,M)$ with some properly chosen $M$ (where $G$ can be continuous). $\endgroup$ – Xiao-Gang Wen Nov 11 '14 at 11:53
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For a modern and general treatment of group cohomology and classifying space, see Flach's paper.

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