nLab
Eilenberg-Zilber theorem

Contents

Context

Homological algebra

homological algebra

(also nonabelian homological algebra)

Introduction

Context

Basic definitions

Stable homotopy theory notions

Constructions

Lemmas

diagram chasing

Homology theories

Theorems

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:Δ op×Δ opAbA \colon \Delta^{op} \times \Delta^{op} \to Ab be a bisimplicial abelian group. Write

Theorem

(Dold-Puppe generalization of Eilenberg-Zilber)

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

R:C diag(A)TotC(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

diagF , [n]ΔΔ n×F n,. diag F_{\bullet,\bullet} \simeq \int^{[n] \in \Delta} \Delta^n \times F_{n,\bullet} \,.

Cosimplicial version

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

Theorem

There is a natural isomorphism

H C diag(A)H TotC (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 KK and LL be simplicial sets.

  • There is an Alexander-Whitney diagonal approximation defined as a natural transformation
a K,L:π(K×L)πKπL.a_{K,L}: \pi(K\times L)\to \pi K \otimes \pi L.
  • Using shuffles, one defines an Eilenberg - Zilber map
b K,L:πKπLπ(K×L),b_{K,L}:\pi K \otimes \pi L \to\pi(K\times L),

in a somewhat similar way to chain complexes.

  • The composite
π(K×L)πKπLπ(K×L),\pi(K\times L)\to \pi K \otimes \pi L\to\pi(K\times L),

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

(Eilenberg-Zilber/Alexander-Whitney deformation retraction)

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

So from the Dold-Puppe quasi-isomorphism RR from above we have a quasi-isomorphism of chain complexes of their product topological space:

C (X×Y) C ([SingX×SingY]) =C (diag[SingX ][SingY ]) RTotC ([SingX])C ([SingY]) =TotC (X)C (Y) \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

See at transgression in group cohomology.

References

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.

Last revised on July 13, 2021 at 06:34:32. See the history of this page for a list of all contributions to it.