The singular chain complex of a product is naturally chain-homotopy equivalent to the tensor product of the factors' chain complexes. The shuffle map and the front-face/back-face map implement the equivalence. It is the chain-level input to the Künneth theorem.
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The Eilenberg–Zilber theorem is a result in algebraic topology and homological algebra that provides a way to compute the singular homology of a product of two topological spaces. Specifically, the theorem addresses the relationship between the singular chains on the product of two spaces and the singular chains on the spaces themselves.