Algebraic topology Homology and cohomology Theorem

Eilenberg–Zilber theorem

Statement

Let $X$ and $Y$ be topological spaces with singular chain complexes $C_\bullet(X)$ and $C_\bullet(Y)$ . Let $C_\bullet(X) \otimes C_\bullet(Y)$ be the tensor product chain complex, with $(C_\bullet(X) \otimes C_\bullet(Y))_n = \bigoplus_{i + j = n} C_i(X) \otimes_{\Z} C_j(Y)$ and $\partial(a \otimes b) = \partial a \otimes b + (-1)^i a \otimes \partial b$ for $a \in C_i(X)$ . Then there are chain maps $C_\bullet(X \times Y) \rightleftarrows C_\bullet(X) \otimes C_\bullet(Y),$ natural in $X$ and $Y$ , that are inverse to each other up to chain homotopy. In particular, $H_n(X \times Y) \cong H_n\big(C_\bullet(X) \otimes C_\bullet(Y)\big)$ for every $n$ .

Remarks

A standard choice of maps is the Alexander–Whitney map from $C_\bullet(X \times Y)$ and the shuffle map into it. The existence and uniqueness up to homotopy of such natural maps follows from the method of acyclic models. Both functors are free on the models $(\Delta^p, \Delta^q)$ and acyclic on them, because $\Delta^p \times \Delta^q$ is contractible.

Example: with the algebraic Künneth theorem, the theorem gives the topological Künneth formula. Over a field $F$ , $H_n(X \times Y; F) \cong \bigoplus_{i + j = n} H_i(X; F) \otimes_F H_j(Y; F)$ . For the torus $S^1 \times S^1$ this gives $H_1 \cong F^2$ and $H_2 \cong F$ .

Going deeper

For a student

The theorem is the bridge between the homology of a product and the algebra of tensor products of chain complexes. Combined with the Künneth theorem of homological algebra, it computes the homology of products. The Alexander–Whitney map, composed with the diagonal $X \to X \times X$ , also gives the cup product on singular cohomology.

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Singular homology concept
Eilenberg–Zilber theorem statement

Tag	`V3G5YQTQ00`
URI	`m2://V3G5YQTQ00`
Kind	statement, theorem
Writing	W0 stub
MSC	55U25, 55N10
Status	This copy of the site has no build data. Run `m2 status V3G5YQTQ00`.
About	Singular homology
Uses	Singular homology
Same as	Wikidata Q1349282 Wikipedia Eilenberg–Zilber theorem
Changed	2 November 2026