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
StatementThere 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 Classification
Statement Form Statement Form Eilenberg-Zilber Theorem
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Source Eilenberg Zilber theorem (Wikipedia)
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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.
Principal Result Of: Samuel Eilenberg, Lead sentence of the article on the result
The theorem first appeared in a 1953 paper in the American Journal of Mathematics by Samuel Eilenberg and Joseph A.
- In Branch: Algebraic Topology, Lead sentence
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