Let $\pi\subseteq\Sigma_r$ and $V$ be a right $\pi$space. We may assume that $V$ is free, if necessary. Consider the morphism of singular chain complexes (over a fixed commutative ring) $$f:C_*(V) \otimes C_*(X)^{\otimes r} \to C_*(V\times X^r) \to C_*(V\times_\pi X^r) $$ where the first arrow is the EilenbergZilber map and the second is induced by the projection. I think that it is clear that $f$ factors as $$\overline{f} : C_*(V) \otimes_\pi C_*(X)^{\otimes r} \to C_*(V\times_\pi X^r) $$ Now my question is: Is $\overline{f}$ a quasiisomorphism?
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1$\begingroup$ The map $\bar f$ appears to be induced from the map $C^*(V)⊗C^*(X)^{⊗r}→C^*(V×X^r)$ by taking πcoinvariants. If the action of π is free, then πcoinvariants are also homotopy πcoinvariants, and taking homotopy coinvariants of a weak equivalence again produces a weak equivalence. $\endgroup$ – Dmitri Pavlov Dec 23 '19 at 23:17

$\begingroup$ This looks like a nice argument! Do you have a reference for your last sentence about homotopy coinvariants of weak equivalences? $\endgroup$ – FKranhold Dec 24 '19 at 11:50

$\begingroup$ Or maybe one can see more easily that a quasiisomorphism $f:C\to C'$ of free chain complexes of left $R\pi$modules induce quasiisomorphisms $f_\pi:C_\pi\to C'_\pi$ on coinvariants. $\endgroup$ – FKranhold Dec 24 '19 at 12:04

$\begingroup$ The claim about homotopy colimits (such as homotopy coinvariants) preserving weak equivalences follows immediately from the definition of homotopy colimits, which are defined as left derived functors of the colimit functor, see, for example, Section 5.1 in Riehl's Categorical Homotopy Theory. Left derived functors preserve weak equivalences by construction. $\endgroup$ – Dmitri Pavlov Dec 24 '19 at 19:46