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 Connections
Has Statement Form
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Entity-backed identity for the statement-form enum value this theorem already carries, resolved to a mathematics concept by an explicit value-to-entity map (phase 3 bucket conversion, docs\design_entity_backed_browse_buckets_20260928.md). The statement-form fact itself stays on the theorem unchanged.
Entity-backed identity for the statement-form enum value this theorem already carries, resolved to a mathematics concept by an explicit value-to-entity map (phase 3 bucket conversion, docs\design_entity_backed_browse_buckets_20260928.md). The statement-form fact itself stays on the theorem unchanged.
In Branch
Source Eilenberg Zilber theorem (Wikipedia)
Sources
1. Eilenberg Zilber theorem (Wikipedia)
Statement of the theorem
Then the theorem says that we have chain maps F : C ∗ ( X × Y ) → C ∗ ( X ) ⊗ C ∗ ( Y ) , G : C ∗ ( X ) ⊗ C ∗ ( Y ) → C ∗ ( X × Y ) such that F G is the identity and G F is chain-homotopic to the identity.
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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