According to http://ocw.mit.edu/courses/mathematics/18-917-topics-in-algebraic-topology-the-sullivan-conjecture-fall-2007/lecture-notes/lecture2.pdf. , if $X$ is any topological space, then its cochain complex $C^* := C^*(X)$ has a (homotopy) symmetric multiplication

$$D_2(C^*) := (C^* \otimes C^* \otimes E(\mathbb{Z}/2))_{\mathbb{Z}/2} \rightarrow C^*$$

where $E(\mathbb{Z}/2)$ is a homological model for a contractible space with free $\mathbb{Z}/2$-action.

pre-Question: how can I describe concretely this symmetric multiplication? (For instance, if I pick a cellular model for $X$ then how do I write it down?)

If $X$ is a topological space with $G$-action, then we can form its homotopy quotient $X/G := (X \times EG)/G$, where $EG$ is a contractible space with free $G$-action. At the level of cochains, we have $C^*(X/G) = C^*(X)^{hG} := Hom(C_*(EG), X)^G$. So I should get a symmetric multiplication

$$ D_2(C^{hG}) \rightarrow C^{hG}$$

Question: more generally, if $C$ is any chain complex with symmetric multiplication $D_2(C) \rightarrow C$, is there an induced multiplication $D_2(C^{hG}) \rightarrow C^{hG}$? What is it?

negativedegrees, so the cochain complex of $X \times X \times E\mathbb{Z}/2$ wouldn't be what I call about $C \otimes C \otimes E(\mathbb{Z}/2)$). That's one of the reasons I am confused about this business! $\endgroup$