TITLE: References for Eilenberg-Zilber shuffle product
QUESTION [17 upvotes]: Most of the treatments I can find in the literature for the Eilenberg-Zilber shuffle product approach it from the point of view of simplicial sets (including the original Eilenberg-MacLane paper). I was wondering if anyone knows of a reference written directly in the language of singular or PL chains. By this I mean that one should show that the (p,q)-shuffles each produce a distinct embedded p+q simplex in $\Delta^p\times \Delta^q$ and that, collectively, this produces a triangulation of $\Delta^p\times \Delta^q$. Furthermore, this should be a chain map, in the sense that the boundary of the triangulation is compatible with the same triangulation process applied to the boundaries (or in other words, the map $S_p(X)\otimes S_q(X)\to S_{p+q}(X\times Y)$ obtained by applying this process is a chain map). Dold constructs this map explicitly in his book, but it's in an exercise and he doesn't provide any proofs of properties (he does helpfully indicate with an asterisk that it is a challenging exercise to do so!)
The reason I ask is that I'm working on some lecture notes (for publication) for which I need to have an explicit construction of the homology cross product at the chain level, but the intended audience is not expected to know about simplicial sets. 
If what I'm looking for does not exist, can anyone recommend the simplicial set treatment that provides the best description of the connection to the explicit geometric triangulation side of things?

REPLY [2 votes]: This is very late, I know, but I just stumbled across this question as I happen to be thinking about these things myself. There is a nice discussion of the simplicial cross product in Hatcher's "Algebraic Topology" text, on pp. 277-278. There Hatcher refers to Eilenberg and Steenrod's "Foundations of Algebraic Topology" textbook, p. 68, for a proof of precisely the fact you mention. I have not yet waded through their proof, but this would seem to be the canonical reference.
In fact they seem to be proving something more general, that the shuffles induce a natural $\Delta$-complex structure on a product of (finite) $\Delta$-complexes.