Theorem

Eilenberg-Zilber Theorem

Topology

The Eilenberg-Zilber Theorem states that the singular chain complex of a product of two topological spaces is chain homotopy equivalent to the tensor product of the singular chain complexes of the two spaces separately, with explicit maps given by the Alexander-Whitney and shuffle constructions. Named for Samuel Eilenberg and Joseph Zilber, it is a foundational result of algebraic topology that underlies the Kunneth formula relating the homology of a product space to the homology of its factors.

Facts

Statement

There are chain maps F from C(X x Y) to C(X) tensor C(Y) and G in the other direction such that FG is the identity and GF is chain-homotopic to the identity. 1

Proof Year

1953 1

Classification

Statement Form

Existence Theorem 1

Statement Form

Identity or Equation 1

Connections

Has Statement Form

Equation

Existence Theorem

Identity

In Branch

Algebraic Topology

Source Eilenberg Zilber theorem (Wikipedia)

Sources

1. Eilenberg Zilber theorem (Wikipedia)

Lead section
The theorem first appeared in a 1953 paper in the American Journal of Mathematics by Samuel Eilenberg and Joseph A.
References
The theorem first appeared in a 1953 paper in the American Journal of Mathematics by Samuel Eilenberg and Joseph A. Zilber.
In Branch: Algebraic Topology, Lead sentence

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Eilenberg-Zilber Theorem

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In Branch