Homology cross product ($H_p(X)\otimes H_q(Y)\to H_{p+q}(X\times Y)$)

The singular-chain product induces this natural pairing on homology. Composing it with multiplication of a loop space gives the Pontryagin product on loop-space homology. This is different from the vector cross product.

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.