Eilenberg–Zilber theorem

In mathematics, specifically in algebraic topology, the Eilenberg–Zilber theorem is an important result in establishing the link between the homology groups of a product space $$X \times Y$$ and those of the spaces $$X$$ and $$Y$$. The theorem first appeared in a 1953 paper in the American Journal of Mathematics by Samuel Eilenberg and Joseph A. Zilber. One possible route to a proof is the acyclic model theorem.

Statement of the theorem
The theorem can be formulated as follows. Suppose $$X$$ and $$Y$$ are topological spaces, Then we have the three chain complexes $$C_*(X)$$, $$C_*(Y)$$, and $$C_*(X \times Y) $$. (The argument applies equally to the simplicial or singular chain complexes.) We also have the tensor product complex $$C_*(X) \otimes C_*(Y)$$, whose differential is, by definition,
 * $$\partial_{C_*(X) \otimes C_*(Y)}( \sigma \otimes \tau) = \partial_X \sigma \otimes \tau + (-1)^p \sigma \otimes \partial_Y\tau$$

for $$\sigma \in C_p(X)$$ and $$\partial_X$$, $$\partial_Y$$ the differentials on $$C_*(X)$$,$$C_*(Y)$$.

Then the theorem says that we have chain maps


 * $$F\colon C_*(X \times Y) \rightarrow C_*(X) \otimes C_*(Y), \quad G\colon C_*(X) \otimes C_*(Y) \rightarrow C_*(X \times Y)$$

such that $$FG$$ is the identity and $$GF$$ is chain-homotopic to the identity. Moreover, the maps are natural in $$X$$ and $$Y$$. Consequently the two complexes must have the same homology:


 * $$H_*(C_*(X \times Y)) \cong H_*(C_*(X) \otimes C_*(Y)).$$

Statement in terms of composite maps
The original theorem was proven in terms of acyclic models but more mileage was gotten in a phrasing by Eilenberg and Mac Lane using explicit maps. The standard map $$F$$ they produce is traditionally referred to as the Alexander–Whitney map and $$G$$ the Eilenberg–Zilber map. The maps are natural in both $$X$$ and $$Y$$ and inverse up to homotopy: one has


 * $$FG = \mathrm{id}_{C_*(X) \otimes C_*(Y)}, \qquad GF - \mathrm{id}_{C_*(X \times Y)} = \partial_{C_*(X) \otimes C_*(Y)}H+H\partial_{C_*(X) \otimes C_*(Y)}$$

for a homotopy $$H$$ natural in both $$X$$ and $$Y$$ such that further, each of $$HH$$, $$FH$$, and $$HG$$ is zero. This is what would come to be known as a contraction or a homotopy retract datum.

The coproduct
The diagonal map $$\Delta\colon X \to X \times X$$ induces a map of cochain complexes $$C_*(X) \to C_*(X \times X)$$ which, followed by the Alexander–Whitney $$F$$ yields a coproduct $$C_*(X) \to C_*(X) \otimes C_*(X)$$ inducing the standard coproduct on $$H_*(X)$$. With respect to these coproducts on $$X$$ and $$Y$$, the map


 * $$H_*(X) \otimes H_*(Y) \to H_*\big(C_*(X) \otimes C_*(Y)\big)\ \overset\sim\to\ H_*(X \times Y)$$,

also called the Eilenberg–Zilber map, becomes a map of differential graded coalgebras. The composite $$C_*(X) \to C_*(X) \otimes C_*(X)$$ itself is not a map of coalgebras.

Statement in cohomology
The Alexander–Whitney and Eilenberg–Zilber maps dualize (over any choice of commutative coefficient ring $$k$$ with unity) to a pair of maps


 * $$G^*\colon C^*(X \times Y) \rightarrow \big(C_*(X) \otimes C_*(Y)\big)^*, \quad F^*\colon \big(C_*(X) \otimes C_*(Y)\big)^*\rightarrow C^*(X \times Y)$$

which are also homotopy equivalences, as witnessed by the duals of the preceding equations, using the dual homotopy $$H^*$$. The coproduct does not dualize straightforwardly, because dualization does not distribute over tensor products of infinitely-generated modules, but there is a natural injection of differential graded algebras $$i\colon C^*(X) \otimes C^*(Y) \to \big(C_*(X) \otimes C_*(Y)\big)^*$$ given by $$\alpha \otimes \beta \mapsto (\sigma \otimes \tau \mapsto \alpha(\sigma)\beta(\tau))$$, the product being taken in the coefficient ring $$k$$. This $$i$$ induces an isomorphism in cohomology, so one does have the zig-zag of differential graded algebra maps


 * $$ C^*(X) \otimes C^*(X)\ \overset{i}{\to}\ \big(C_*(X) \otimes C_*(X)\big)^*\ \overset{G^*}{\leftarrow}\ C^*(X \times X) \overset{C^*(\Delta)}{\to} C^*(X)$$

inducing a product $$\smile\colon H^*(X) \otimes H^*(X) \to H^*(X)$$ in cohomology, known as the cup product, because $$H^*(i)$$ and $$H^*(G)$$ are isomorphisms. Replacing $$G^*$$ with $$F^*$$ so the maps all go the same way, one gets the standard cup product on cochains, given explicitly by


 * $$\alpha \otimes \beta \mapsto \Big(\sigma \mapsto (\alpha \otimes \beta)(F^*\Delta^*\sigma) =

\sum_{p=0}^{\dim \sigma} \alpha(\sigma|_{\Delta^{[0,p]}}) \cdot \beta(\sigma|_{\Delta^{[p,\dim \sigma]}})\Big)$$,

which, since cochain evaluation $$C^p(X) \otimes C_q(X) \to k$$ vanishes unless $$p=q$$, reduces to the more familiar expression.

Note that if this direct map $$C^*(X) \otimes C^*(X) \to C^*(X)$$ of cochain complexes were in fact a map of differential graded algebras, then the cup product would make $$C^*(X)$$ a commutative graded algebra, which it is not. This failure of the Alexander–Whitney map to be a coalgebra map is an example the unavailability of commutative cochain-level models for cohomology over fields of nonzero characteristic, and thus is in a way responsible for much of the subtlety and complication in stable homotopy theory.

Generalizations
An important generalisation to the non-abelian case using crossed complexes is given in the paper by Andrew Tonks below. This give full details of a result on the (simplicial) classifying space of a crossed complex stated but not proved in the paper by Ronald Brown and Philip J. Higgins on classifying spaces.

Consequences
The Eilenberg–Zilber theorem is a key ingredient in establishing the Künneth theorem, which expresses the homology groups $$H_*(X \times Y)$$ in terms of $$H_*(X)$$ and $$H_*(Y)$$. In light of the Eilenberg–Zilber theorem, the content of the Künneth theorem consists in analysing how the homology of the tensor product complex relates to the homologies of the factors.