# nLab Eilenberg-Zilber theorem

Contents

### Context

#### Homological algebra

homological algebra

Introduction

diagram chasing

# Contents

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

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

• $Tot (C A)$ 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)

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

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

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

### Cosimplicial version

Let $A : \Delta \times \Delta \to Ab$ be a bi-cosimplicial abelian group. And let $C : Ab^\Delta \to Ch^\bullet$ 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

$H^\bullet C^\bullet diag(A) \simeq H^\bullet Tot C^\bullet(A)$

### 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
$a_{K,L}: \pi(K\times L)\to \pi K \otimes \pi L.$
• Using shuffles, one defines an Eilenberg - Zilber map
$b_{K,L}:\pi K \otimes \pi L \to\pi(K\times L),$

in a somewhat similar way to chain complexes.

• The composite
$\pi(K\times L)\to \pi K \otimes \pi L\to\pi(K\times L),$

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 discussed under the entry on 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 $C_\bullet(X)$ and $C_\bullet(Y)$ are the Moore complexes of the simplicial abelian groups $\mathbb{Z}[Sing X]$ and $\mathbb{Z}[Sing Y]$. 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{aligned} C_\bullet(X \times Y) & \coloneqq C_\bullet( \mathbb{Z}[Sing X \times Sing Y] ) \\ &= C_\bullet( diag \mathbb{Z}[Sing X_\bullet] \otimes \mathbb{Z}[Sing Y_\bullet] ) \\ & \underoverset{\simeq}{R}{\longrightarrow} 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.

The original reference is

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

The stronger version as stated above, published by Dold and Puppe, in chapter 2 of

where is is ascribed to Pierre Cartier. This result is discussed in chapter 4 of

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.