# nLab Eilenberg-Zilber theorem

Contents

### Context

#### Homological algebra

homological algebra

Introduction

diagram chasing

# Contents

## Idea

The Dold-Kan correspondence relates simplicial abelian groups to connective 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 $A \colon \Delta^{op} \times \Delta^{op} \to Ab$ be a bisimplicial abelian group. Write

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

• $Tot (C A)$ 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 cochain 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

### Eilenberg-Zilber/Alexaner-Whitney deformation retraction

###### Proposition

Let

and denote

Then there is a deformation retraction

where

For unnormalized chain complexes, where we have a homotopy equivalence, this is the original Eilenberg-Zilber theorem (Eilenberg & Zilber 1953, Eilenberg & MacLane 1954, Thm. 2.1). The above deformation retraction for normalized chain complexes is Eilenberg & MacLane 1954, Thm. 2.1a. Both are reviewed in May 1967, Cor. 29.10. Explicit description of the homotopy operator is given in Gonzalez-Diaz & Real 1999.

### Homology of products of topological spaces

The following is the special case of the Eilenberg-Zilber/Alexander-Whitney deformation retraction (Prop. ) for singular simplicial sets, observing that forming free simplicial abelian groups preserves monoidal products. This is the original motivating application in Eilenberg &Zilber 1953

Let $X$ and $Y$ be two topological spaces. Their singular homology chain complexes $C_\bullet(X)$ and $C_\bullet(Y)$ are the Moore complexes of the simplicial abelian groups $\mathbb{Z}[Sing X]$ and $\mathbb{Z}[Sing Y]$ free abelian group on their singular simplicial complexes.

So from the Dold-Puppe quasi-isomorphism $R$ from above we have a quasi-isomorphism of chain complexes 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 singular cohomology.

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

### Transgression in group cohomology

Original references:

using the definition of the Eilenberg-Zilber map in:

Review in:

A weak version of the simplicial statement:

• Charles Weibel, Theorem 8.1.5 in: An introduction to homological algebra

The stronger version as stated above:

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

The cosimplicial version of the theorem appears as:

• L. Grunenfelder and M. Mastnak, Theorem A.3 in: 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.