# nLab Eilenberg-Zilber map

### Context

#### Homological algebra

homological algebra

and

nonabelian homological algebra

diagram chasing

# Contents

## Definition

Let $C:\mathrm{sAb}\to {\mathrm{Ch}}_{•}^{+}$ be the chains/Moore complex functor of the Dold-Kan correspondence.

Let $\left(\mathrm{sAb},\otimes \right)$ be the standard monoidal category structure given degreewise by the tensor product on Ab and let $\left({\mathrm{Ch}}_{•}^{+},\otimes \right)$ be the standard monoidal structure on the category of chain complexes.

###### Definition

For $A,B\in \mathrm{sAb}$ two abelian simplicial groups, the Eilenberg-Zilber map or Eilenberg-MacLane map or shuffle map is the natural transformation on chain complexes

${\nabla }_{A,B}:C\left(A\right)\otimes C\left(B\right)\to C\left(A\otimes B\right)$\nabla_{A,B} : C(A) \otimes C(B) \to C(A \otimes B)

defined on two $n$-simplices $a\in {A}_{p}$ and $b\in {B}_{q}$ by

${\nabla }_{A,B}:a\otimes b↦\sum _{\left(\mu ,\nu \right)}\mathrm{sign}\left(\mu ,\nu \right)\left({s}_{\nu }\left(a\right)\right)\otimes \left({s}_{\mu }\left(b\right)\right)\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\in {C}_{p+q}\left(A\otimes B\right)={A}_{p+q}\otimes {B}_{p+q}\phantom{\rule{thinmathspace}{0ex}},$\nabla_{A,B} : a \otimes b \mapsto \sum_{(\mu,\nu)} sign(\mu,\nu) (s_\nu(a)) \otimes (s_\mu(b)) \;\; \in C_{p+q}(A \otimes B) = A_{p+q} \otimes B_{p+q} \,,

where the sum is over all $\left(p,q\right)$-shuffles

$\left(\mu ,\nu \right)=\left({\mu }_{1},\cdots ,{\mu }_{p},{\nu }_{1},\cdots ,{\nu }_{q}\right)$(\mu,\nu) = (\mu_1, \cdots, \mu_p, \nu_1, \cdots, \nu_q)

and the corresponding degeneracy maps are

${s}_{\mu }={s}_{{\mu }_{p}}\circ \cdots {s}_{{\mu }_{2}}\circ {s}_{{\mu }_{1}}$s_{\mu} = s_{\mu_p} \circ \cdots s_{\mu_2} \circ s_{\mu_1}

and

${s}_{\nu }={s}_{{\nu }_{q}}\circ \cdots {s}_{{\nu }_{2}}\circ {s}_{{\nu }_{1}}\phantom{\rule{thinmathspace}{0ex}}.$s_{\nu} = s_{\nu_q} \circ \cdots s_{\nu_2} \circ s_{\nu_1} \,.

The sign $\mathrm{sign}\left(\mu ,\nu \right)\in \left\{-1,1\right\}$ is the signature of the corresponding permutation.

###### Remark

The sum may be understood as being over all non-degenerate simplices in the product $\Delta \left[p\right]×\Delta \left[q\right]$. See products of simplices for more on this.

###### Proposition

This map restricts to the normalized chains complex

${\nabla }_{A,B}:N\left(A\right)\otimes N\left(B\right)\to N\left(A\otimes B\right)\phantom{\rule{thinmathspace}{0ex}}.$\nabla_{A,B} : N(A) \otimes N(B) \to N(A \otimes B) \,.

## Properties

The specific maps introduced by Eilenberg-Mac Lane have stronger properties which for simplicial sets $K,L$ make $C\left(K\right)\otimes C\left(L\right)$ a strong deformation retract of $C\left(K×L\right)$. This is exploited in

• R. Brown, The twisted Eilenberg-Zilber theorem. Simposio di Topologia (Messina, 1964), Edizioni Oderisi, Gubbio (1965), 33–37. pdf

which has been a foundation of the subject of Homological Perturbation Theory. The homotopies play a key role in the formulae and calculations.

###### Proposition

The Eilenberg-Zilber map is a lax monoidal transformation that makes $C$ and $N$ into lax monoidal functors.

See monoidal Dold-Kan correspondence for details.

###### Proposition

On normalized chain complexes the EZ map has a left inverse, given by the Alexander-Whitney map ${\Delta }_{A,B}$:

$\mathrm{Id}:NA\otimes NB\stackrel{{\nabla }_{A,B}}{\to }N\left(A\otimes B\right)\stackrel{{\Delta }_{A,B}}{\to }NA\otimes NB\phantom{\rule{thinmathspace}{0ex}}.$Id : N A \otimes N B \stackrel{\nabla_{A,B}}{\to} N(A \otimes B) \stackrel{\Delta_{A,B}}{\to} N A \otimes N B \,.
###### Proposition

For all $X,Y$ the EZ map ${\nabla }_{X,Y}$ is a quasi-isomorphism and in fact a chain homotopy equivalence.

This is in 29.10 of (May).

For the next statement notice that both $\mathrm{sAb}$ and ${\mathrm{Ch}}_{•}^{+}$ are in fact symmetric monoidal categories.

###### Proposition

The EZ map is symmetric in that for all $A,B\in \mathrm{sAb}$ the square

$\begin{array}{ccc}CA\otimes CB& \stackrel{\sigma }{\to }& CB\otimes CA\\ {}^{{\nabla }_{A,B}}↓& & {↓}^{{\nabla }_{B,A}}\\ C\left(A\otimes B\right)& \stackrel{C\left(\sigma \right)}{\to }& C\left(B\otimes A\right)\end{array}$\array{ C A \otimes C B &\stackrel{\sigma}{\to}& C B \otimes C A \\ {}^{\mathllap{\nabla_{A,B}}}\downarrow && \downarrow^{\mathrlap{\nabla_{B,A}}} \\ C(A\otimes B) &\stackrel{C(\sigma)}{\to}& C(B \otimes A) }

commutes, where $\sigma$ denotes the symmetry isomorphism in $\mathrm{sAb}$ and ${\mathrm{Ch}}_{•}^{+}$.

In the context of filtered spaces ${X}_{*},{Y}_{*}$ and their associated fundamental crossed complex?es $\Pi {X}_{*},\Pi {Y}_{*}$ there is a natural Eilenberg-Zilber morphism

$\eta :\Pi {X}_{*}\otimes \Pi {Y}_{*}\to \Pi \left({X}_{*}\otimes {Y}_{*}\right)$\eta: \Pi X_* \otimes \Pi Y_* \to \Pi (X_* \otimes Y_*)

which is difficult to define directly because of the complications of the tensor product of crossed complexes, but has a direct definition in terms of the associated cubical homotopy $\omega$–groupoids. This morphism is an isomorphism of free crossed complexes if ${X}_{*},{Y}_{*}$ are the skeletal filtrations of CW-complexes. For more on all this, see the book Nonabelian Algebraic Topology p. 533.

## References

The Eilenberg-Zilber map was introduced in (5.3) of