Eilenberg–Zilber theorem

The set of Eilenberg - Zilber, named after S. Eilenberg and JA Zilber, is a set from the mathematical branch of algebraic topology. It provides a connection between the singular homology groups of a Cartesian product of two topological spaces and homology groups fro the rooms themselves.

Tensor products of chain complexes

Are and two chain complexes, so the tensor product of chain complex is with

This is explained on producers, and the bill

Shows that actually is present again a chain complex.

If the boundary operators and should not be noted as one writes simply, this is especially true for singular chain complex of topological spaces, where the boundary operators are given.

Wording of the sentence

Are and topological spaces, the singular chain complex of the product space is chain- homotopy equivalent to the tensor.

Importance

Because of the homotopy equivalence and have the same homology groups. The calculation of the singular homology groups of a product space is therefore reduced to a problem of homological algebra, namely the calculation of the homology of a tensor product of chain complexes. This algebraic problem is solved by the set of Kiinneth.