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I'm trying to understand how the $E_{\infty}$-algebra structure on the singular cochain complex $C^{\bullet}(X)$ of a topological space $X$, in at least somewhat down-to-earth terms. (I'm coming at this from Lurie's notes on Steenrod-algebras, but maybe this is the wrong order of approaching things...)

What I already know is that the Eilenberg-Maclane spectrum is in fact a ring spectrum (in Switzer's sense) and I think that this should give (via Brown representability) the product structure on the cohomology ring of $X$.

I also know that it is possible to show this directly (Smirnov: On the cochain complex of topological spaces 1), but it doesn't seem like Smirnov is using the Eilenberg-Maclane spectrum anywhere (and it feels weird to just ignore it).

It feels like there should be a reference on it at some place, but so far, I couldn't find it.

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    $\begingroup$ Are you looking for what's in McClure-Smith? There is also Berger-Fresse and the more recent paper by Medina-Mardones. $\endgroup$
    – mme
    Commented Jul 5, 2021 at 20:48
  • $\begingroup$ I found the McClure-Smith article, but I'm not sure how to put the Eilenberg-Maclane spectrum in this context (I only skimmed it briefly though). $\endgroup$
    – Aaron Wild
    Commented Jul 5, 2021 at 20:55
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    $\begingroup$ You don't need E-ML spectra to construct the E-Infinity structure. It is something very combinatorial, like the cup product. You can find formulas in the aforementioned references. $\endgroup$
    – Fernando Muro
    Commented Jul 5, 2021 at 23:03

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