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Eilenberg-Zilber theorem

Context

Homological algebra

homological algebra

and

nonabelian homological algebra

Context

Basic definitions

Stable homotopy theory notions

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diagram chasing

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Theorems

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

  • C diagAC_\bullet diag A for the Moore complex of its diagonal simplicial group diagA:Δ opΔ op×Δ opAAbdiag A : \Delta^{op} \to \Delta^{op} \times \Delta^{op} \stackrel{A}{\to} Ab;

  • Tot(CA)Tot (C A) for the total complex of the double complex obtained by applying the Moore complex functor on both arguments of AA.

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 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

Homology groups of products of topological spaces

This is the original motivating application.

Let XX and YY be two topological spaces. Their chain homology 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]. So from the Dold-Puppe quasi-isomorphism RR from above we have a quasi-isomorphism from the singular cohomology 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) & := 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

The stronger version as stated above is 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.

Revised on November 24, 2013 23:36:08 by Tim Porter (2.26.15.29)