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The problem: Let $X$ be a product of equidimensional spheres of arbitrary dimension, say $k$, and $G$ a finite group acting freely on $X$. Assume that the induced $G$-action on the $Z_2$-cohomology ring of $X$ is trivial. We then have a Serre spectral sequence with the $E_2$-page given by the expression $H^*(G; H^*(X; Z_2))$ and converging to $H^*(X/G; Z_2)$.

The first non-zero differential is certainly $d^{k+1}$. Let $M$ be the image of the transgression (hence $M$ is a subgroup of $H^{k+1}(G; Z_2))$. Now suppose that $M = 0$.

How do I deduce that $E_{\infty} = E_{k+2} = E_2$?

Background: I've been trying to learn some spectral sequences these past few days. After a while I had decided I kind of have a grasp of what is going on and have been trying to understand some research-level examples. The one described above comes from the paper Free compact group actions on products of spheres by R. Oliver. In fact, he takes G to be the alternating group on four letters, but I don't think this has anything to do with my question. (My guess is that it somehow follows from the multiplicative structure of the spectral sequence.)

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