nLab Eilenberg-Zilber theorem

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/Alexander-Whitney deformation retraction

Proposition

(Eilenberg-Zilber/Alexander-Whitney deformation retraction)

Let

• $A, B \,\in\, sAb =$ SimplicialAbelianGroups

and denote

• by $N(A), N(B) \,\in\, Ch^+_\bullet =$ ConnectiveChainComplexes their normalized chain complexes,

• by $A \otimes B \,\in\, sAb$ the degreewise tensor product of abelian groups,

• by $N(A) \otimes N(B)$ the tensor product of chain complexes.

Then there is a deformation retraction

where

• $\nabla_{A,B}$ is the Eilenberg-Zilber map;

• $\Delta_{A,B}$ is the Alexander-Whitney map.

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

See at transgression in group cohomology.

References

Original references:

• Samuel Eilenberg, Joseph Zilber, On Products of Complexes, Amer. Jour. Math. 75 (1): 200–204, (1953) (jstor:2372629)

• Samuel Eilenberg, Saunders MacLane, Section 2 of: On the Groups $H(\Pi,n)$, II: Methods of Computation, Annals of Mathematics, Second Series, Vol. 60, No. 1 (Jul., 1954), pp. 49-139 (jstor:1969702, doi:10.2307/2372629)

using the definition of the Eilenberg-Zilber map in:

• Samuel Eilenberg, Saunders MacLane, (5.3) of: On the groups $H(\Pi,n)$, I, Ann. of Math. (2) 58, (1953), 55–106. (jstor:1969820)

Review in:

• Albrecht Dold, Section VI 12.1 in Lectures on Algebraic Topology, Springer 1995 (doi:10.1007/978-3-642-67821-9, pdf)

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:

• Albrecht Dold, Dieter Puppe, Chapter 2 of: Homologie nicht-additiver Funktoren. Anwendungen, Ann. Inst. Fourier 11 (1961) pp. 201-312. (numdam:AIF_1961__11__201_0, pdf)

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

• Paul Goerss, J. F. Jardine, Simplicial homotopy theory

  (doi:10.1007/978-3-0346-0189-4, webpage)

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.