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$R$ is a ring. Applying the Serre spectral sequence to a fibration $F\to E\to B$, to avoid local coefficients, we need to require that $\pi_1(B)$ acts on $H^*(F, R)$ trivially. Consider a fibration of classifying spaces $BH\to BG \to BM$. Here $H,G,M$ are nice (e.g. compact) topological groups s.t. $1\to H\to G \to M\to 1$.

What condition, as general as possible, can ensure that $\pi_1(BM)$ acts on $H^*(BH, R)$ trivially? I could verify the triviality/non-triviality in the problems I have on hand, but I'm curious about a general characterization.

I'm particularly interested in the case that $H$ is path-connected and $M$ is a finite group. For example, $BSO(n)\to BO(n) \to K(\mathbb{Z}/2,1)$. (In this example, the action is always trivial when $R=\mathbb{Z}/2$, but depends on the parity of $n$ when $R=\mathbb{Z}$.)

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  • $\begingroup$ I think you could investigate the principal $\pi_0(M)$-bundle over $BG$ that is the change of structure group from the $M$-bundle $BH \to BG$ using $M\to \pi_0(M)$. This reduces the more general case to be closer to what you are interested in. $\endgroup$
    – David Roberts ♦
    Jun 14, 2022 at 11:39
  • $\begingroup$ @DavidRoberts Thanks! Before going to the more general case, I would be satisfied with an answer that only applies to the special case. Then, the best condition I can find is, there exists another path-connected group $X$ s.t. $G\subseteq X$ and $H$ is a normal subgroup in $X$. But I'm not satisfied with this condition... $\endgroup$
    – Leo
    Jun 14, 2022 at 19:26
  • $\begingroup$ I'd be tempted to try the case R is a p-group, and compare to p-torsion in M, which wlog I think can be taken to be discrete. $\endgroup$
    – David Roberts ♦
    Jun 15, 2022 at 5:30

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