Eilenberg–Zilber theorem

The Eilenberg–Zilber theorem gives a chain homotopy equivalence between the chain complex of a product of simplicial sets and the tensor product of their chain complexes.

Connect

Alexander–Whitney map: A chain map from the chains of a product of simplicial sets to the tensor product of their chain complexes. It gives one direction of the theorem’s chain homotopy equivalence.
Simplicial set: A combinatorial object consisting of simplices linked by face and degeneracy maps. The theorem begins with two simplicial sets and their product.
Samuel Eilenberg: A Polish-American mathematician who helped found modern algebraic topology and category theory. His work with Mac Lane established the theorem’s algebraic framework.
Künneth theorem: A theorem relating the homology of a product or tensor product to the homology of its factors. The Eilenberg–Zilber equivalence provides the chain-level comparison underlying Künneth calculations.
Eilenberg–Zilber shuffle map: A chain map from the tensor product of simplicial chain complexes to the chains of their product, built from shuffles. It supplies the reverse direction, whose composite with Alexander–Whitney is chain homotopic to the identity.
Tensor product of chain complexes: A chain complex formed from pairs of elements in two chain complexes, with a differential obeying the graded sign rule. It is the algebraic counterpart to chains on the product.
Joseph A. Zilber: A mathematician whose name is joined with Eilenberg’s in the Eilenberg–Zilber theorem. The theorem bears his name alongside Eilenberg’s.
Singular homology: A homology theory built from continuous maps of standard simplices into a topological space. The theorem’s simplicial chain comparison has a corresponding role for singular chains.
Chain homotopy: A relation between chain maps showing they induce the same map on homology. The theorem asserts equivalence up to chain homotopy, not equality of chain maps.
Chain complex: A sequence of abelian groups or modules connected by differentials whose successive composites are zero. The theorem compares two chain complexes through chain maps and homotopies.