nLab
Eilenberg-Zilber theorem

Idea

The Dold-Kan correspondence relates simplicial groups to chain complexes. The Eilenberg-Zilber theorem says how in this context double complexes and their total complexes relate to bisimplicial groups and their diagonals/total simplicial sets.

Analogously there is also a version of the theorem for bi-cosimplicial abelian groups.

Statement

A version for simplicial abelian groups:

Let $A:{\Delta }^{\mathrm{op}}\times {\Delta }^{\mathrm{op}}\to \mathrm{Ab}$ be a bisimplicial abelian group. Write

• $C_{•}\mathrm{diag}A$ for the Moore complex of its diagonal simplicial group $\mathrm{diag}A:{\Delta }^{\mathrm{op}}\to {\Delta }^{\mathrm{op}}\times {\Delta }^{\mathrm{op}}\stackrel{A}{\to }\mathrm{Ab}$;

• $\mathrm{Tot}(CA)$ for the total complex of the double complex obtained by applying the Moore complex functor on both arguments of $A$.

Theorem

(Dold-Puppe generalization of Eilenberg-Zilber)

There is a quasi-isomorphism (even a chain-homotopy equivalence)

$$R:C_{•}\mathrm{diag}(A)\stackrel{\simeq }{\to }\mathrm{Tot}C(A).$$R : C_\bullet diag (A) \stackrel{\simeq}{\to} Tot C (A) \,.

Remark

Notice (see the discussion at bisimplicial set) that the diagonal simplicial set is isomorphic to the realization given by the coend

$$\mathrm{diag}{F}_{•,•}\simeq {\int }^{[n]\in \Delta }{\Delta }^n\times F_{n,•}.$$diag F_{\bullet,\bullet} \simeq \int^{[n] \in \Delta} \Delta^n \times F_{n,\bullet} \,.

Cosimplicial version

Let $A:\Delta \times \Delta \to \mathrm{Ab}$ be a bi-cosimplicial abelian group. And let $C:{\mathrm{Ab}}^{\Delta }\to {\mathrm{Ch}}^{•}$ the Moore cochain complex functor. Write $C(A)$ for the double complex obtained by applying $C$ to each of the two cosimplicial directions. Then we have natural isomorphisms in cohomology

Theorem

There is a natural isomorphism

$$H^{•}{C}^{•}\mathrm{diag}(A)\simeq H^{•}\mathrm{Tot}{C}^{•}(A)$$H^\bullet C^\bullet diag(A) \simeq H^\bullet Tot C^\bullet(A)

Crossed complex version

A version for crossed complexes is given by Andy Tonks. We give a summary:

First note that there is a tensor product for crossed complexes developed by Brown and Higgins. Letting $K$ and $L$ be simplicial sets.

• There is an Alexander-Whitney diagonal approximation defined as a natural transformation

$$a_{K,L}:\pi (K\times L)\to \pi K\otimes \pi L.$$a_{K,L}: \pi(K\times L)\to \pi K \otimes \pi L.

• Using shuffles, one defines an Eilenberg - Zilber map

$$b_{K,L}:\pi K\otimes \pi L\to \pi (K\times L),$$b_{K,L}:\pi K \otimes \pi L \to\pi(K\times L),

in a somewhat similar way to chain complexes.

• The composite

$$\pi (K\times L)\to \pi K\otimes \pi L\to \pi (K\times L),$$\pi(K\times L)\to \pi K \otimes \pi L\to\pi(K\times L),

is homotopic to the identity on $\pi (K\times L)$, whilst the other composite is the identity on $\pi K\otimes \pi L$, thus this is a strong deformation retract of $\pi (K\times L)$.

The Eilenberg - Zilber theorem for simplicial sets

Cegarra and REmedios have proved a version of the EIlenberg - Zilber theorem for simplicial sets. This is disccued under the entry n bisimplicial sets.

Applications

Homology groups of products of topological spaces

This is the original motivating application.

Let $X$ and $Y$ be two topological spaces. Their chain homology complexes $C_{•}(X)$ and $C_{•}(Y)$ are the Moore complexes of the simplicial abelian groups $\mathbb{Z}[\mathrm{Sing}X]$ and $\mathbb{Z}[\mathrm{Sing}Y]$. So from the Dold-Puppe quasi-isomorphism $R$ from above we have a quasi-isomorphism from the singular cohomology of their product topological space

$$\begin{array}{rl}{C}_{•}(X\times Y)& :=C_{•}(\mathbb{Z}[\mathrm{Sing}X\times \mathrm{Sing}Y])\\ & =C_{•}(\mathrm{diag}\mathbb{Z}[\mathrm{Sing}{X}_{•}]\otimes \mathbb{Z}[\mathrm{Sing}{Y}_{•}])\\ & \stackrel{R}{{\to }_{\simeq }}\mathrm{Tot}{C}_{•}(\mathbb{Z}[\mathrm{Sing}X])\otimes C_{•}(\mathbb{Z}[\mathrm{Sing}Y])\\ & =\mathrm{Tot}{C}_{•}(X)\otimes C_{•}(Y)\end{array}$$\begin{aligned} C_\bullet(X \times Y) & := C_\bullet( \mathbb{Z}[Sing X \times Sing Y] ) \\ &= C_\bullet( diag \mathbb{Z}[Sing X_\bullet] \otimes \mathbb{Z}[Sing Y_\bullet] ) \\ & \stackrel{R}{\to_\simeq} Tot C_\bullet(\mathbb{Z}[Sing X]) \otimes C_\bullet(\mathbb{Z}[Sing Y]) \\ & = Tot C_\bullet(X) \otimes C_\bullet(Y) \end{aligned}

and hence in particular an isomorphism in cohomology.

By following through these maps one can obtain an explicit description of the quasi isomorphism if needs be.

References

The original reference is

• Samuel Eilenberg, J. Zilber, On Products of Complexes , Amer. Jour. Math. 75 (1): 200–204, (1953) .

A weak version of the simplicial statement is in theorem 8.1.5 in

• Charles Weibel, An introduction to homological algebra

The stronger version as stated above is in chapter 4 of

• Paul Goerss, Rick Jardine, Simplicial homotopy theory (dvi)

The cosimplicial version of the theorem appears as theorem A.3 in

• L. Grunenfelder and M. Mastnak, Cohomology of abelian matched pairs and the Kac sequence (arXiv:math/0212124)

The crossed complex version is given in

• A.P. Tonks, On the Eilenberg-Zilber Theorem for crossed complexes. J. Pure Appl. Algebra, 179~(1-2) (2003) 199–220,

(for more detail see Tonks’ thesis),

and on page 360 of Nonabelian Algebraic Topology.