Eilenberg–Zilber theorem
# 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

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

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

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

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 $i\colon C^{*}(X)\otimes C^{*}(Y)\to {\bigC_{*}(X)\otimes C_{*}(Y){\big )}^{*}$ 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

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

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.

