Alexander-Whitney map

Definition

The Alexander-Whitney map is a natural transformation between the following two bifunctors on topological spaces: $(X,Y)\mapsto Sin{g}_{.}(X\times Y)$ and $Sin{g}_{.}(X)\otimes Sin{g}_{.}(Y)$ ($Sin{g}_{.}$ here denotes the singular chain complex functor).

The Eilenberg-Zilber map is a natural transformation in the reverse direction, and the composite both ways is naturally chain-homotopic to the identity transformations on the two functors.

Explicitly the Alexander-Whitney map is given as follows:

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