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How does one prove the following fact. I could not find anything in literature.

Let $\pi$ be a subgroup of the symmetric group $S_p$ and let $W$ be a free $\pi$-complex. Then for any space $X$ there is the isomorphism of homology groups in $\mathbb{F}_p$-coefficients

$H_*(W \otimes_{\pi} C_{\ast}(X)^{\otimes p}) \cong H_{\ast}(W \otimes_{\pi} H_{\ast}(X)^{\otimes p})$.

Thank you for your help.

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If $C_\ast$ is a chain complex of vector spaces over any field and $H_\ast$ is its homology, then, viewing $H_\ast$ as a chain complex, there is a chain map $r:C_\ast\to H_\ast$ inducing the obvious isomorphism in homology and there is a chain map $i:H_\ast\to C_\ast$ inducing its inverse. This results in an equivariant splitting $H_\ast^{\otimes n}\to C_\ast^{\otimes n}\to H_\ast^{\otimes n}$. The other factor contributes nothing to the homology of $W\otimes_{\pi}-$, because its homology is trivial and $W$ is free.

To see that the resulting isomorphism between $H_\ast(W\otimes_{\pi} C_\ast^{\otimes p})$ and $H_\ast(W\otimes_{\pi} H_\ast^{\otimes p})$ is independent of the chosen splitting $(r,i)$, first note that it depends only on $i$ and that its inverse (therefore also it) depends only on $r$. Now if $(r',i')$ is another such splitting then the splitting $(r',i)$ yields the same isomorphism as $(r,i)$ and also the same isomorphism as $(r',i')$.

By similar reasoning the isomorphism is natural (with respect to chain maps).

Note that the field did not have to be $\mathbb F_p$, and $p$ did not have to be prime, and the chain complex did not have to come from a space.

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This follows from the fact that $(C_\ast(X),\partial)$ and $(H_\ast(X),0)$ are weakly equivalent chain complexes when coefficients are taken in a field (which is surpressed from the notation). The reason for this is that the exact sequence $$ 0 \to \mathrm{Im}\partial \to \ker\partial \to H_\ast(X)\to 0$$ always splits at the level of vector spaces.

There is a related discussion in Section 2.4 of

André Haefliger, Points multiples d'une application et produit cyclique réduit, Amer. J. Math. 83 1961 57–70

where the author refers to an unpublished paper of Steenrod.

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