E-infinity structure on singular cochains Is there a transparent explanation of why the singular cochain complex of a topological space X is an $E_\infty$ algebra. There are combinatorial proofs using, say, the surjection operad, but is there a topological picture behind those? If we wanted to restrict,say, to the level of little (2-)discs, could we describe the structure using something like Eilenberg-Zilber contractions of $C^*(X^m)$ onto $C^*(X)^{\otimes m}$, various diagonal maps and so on?
 A: If you wrote $E(n)$ for the chain complex of natural transformations $C_*(-) \to C_*(-)^{\otimes n}$, the $E(n)$ collectively form an operad, parametrizing all natural "one-to-many" transformations on chains, called the Eilenberg-Zilber operad. The defining co-action on chains turns into a natural action on cochains, and so the Eilenberg-Zilber operad acts on the singular cochain functor $C^*(-)$. (I seem to recall that there is something to be careful with here regarding the order of composition and whether this is naturally an operad or the reverse of an operad, but the details escape me.)
One can show that the homology groups $H_* E(n)$ are just $\Bbb Z$, concentrated in degree zero, for all $n$, and that the composition operations make these homology groups into the commutative operad. The method of acyclic models is a nice way to prove this (but that seems to have fallen by the wayside as a standard part of the curriculum). It's not clear that Eilenberg-Zilber operad is an $E_\infty$ operad because the symmetric groups aren't guaranteed to act nicely enough, but this is enough to show you that it accepts a weak equivalence from an $E_\infty$ operad (e.g. a cofibrant replacement) without having to know a combinatorial construction.
It seems, from your question, like you would like a close connection to the little discs operads. Unfortunately, this point of view does not provide one at all, and I don't know of a "natural" way to show this that doesn't rely on either a specific combinatorial construction.
