nLab Eilenberg-Zilber map

Context

Homological algebra

homological algebra

and

nonabelian homological algebra

diagram chasing

Contents

Definition

Let $C : sAb \to Ch_\bullet^+$ be the chains/Moore complex functor of the Dold-Kan correspondence.

Let $(sAb, \otimes)$ be the standard monoidal category structure given degreewise by the tensor product on Ab and let $(Ch_\bullet^+, \otimes)$ be the standard monoidal structure on the category of chain complexes.

Definition

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

$\nabla_{A,B} : C(A) \otimes C(B) \to C(A \otimes B)$

defined on two $n$-simplices $a \in A_p$ and $b \in B_q$ by

$\nabla_{A,B} : a \otimes b \mapsto \sum_{(\mu,\nu)} sign(\mu,\nu) (s_\nu(a)) \otimes (s_\mu(b)) \;\; \in C_{p+q}(A \otimes B) = A_{p+q} \otimes B_{p+q} \,,$

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

$(\mu,\nu) = (\mu_1, \cdots, \mu_p, \nu_1, \cdots, \nu_q)$

and the corresponding degeneracy maps are

$s_{\mu} = s_{\mu_p - 1} \circ \cdots s_{\mu_2 - 1} \circ s_{\mu_1 - 1}$

and

$s_{\nu} = s_{\nu_q - 1} \circ \cdots s_{\nu_2 - 1} \circ s_{\nu_1 - 1} \,.$

(The shift in the indices is to be coherent with the convention that the shuffle $(\mu, \nu)$ is a permutation of $\{1, \dots, p+q\}$. In many references the shift disappears by making it a permutation of $\{0, \dots, p+q-1\}$ instead.) The sign $sign(\mu,\nu) \in \{-1,1\}$ is the signature of the corresponding permutation.

Remark

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

Proposition

This map restricts to the normalized chains complex

$\nabla_{A,B} : N(A) \otimes N(B) \to N(A \otimes B) \,.$

Properties

The specific maps introduced by Eilenberg-Mac Lane have stronger properties which for simplicial sets $K,L$ make $C(K) \otimes C(L)$ a strong deformation retract of $C(K \times L)$. 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 $\Delta_{A,B}$:

$Id : N A \otimes N B \stackrel{\nabla_{A,B}}{\to} N(A \otimes B) \stackrel{\Delta_{A,B}}{\to} N A \otimes N B \,.$
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 $sAb$ and $Ch_\bullet^+$ are in fact symmetric monoidal categories.

Proposition

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

$\array{ C A \otimes C B &\stackrel{\sigma}{\to}& C B \otimes C A \\ {}^{\mathllap{\nabla_{A,B}}}\downarrow && \downarrow^{\mathrlap{\nabla_{B,A}}} \\ C(A\otimes B) &\stackrel{C(\sigma)}{\to}& C(B \otimes A) }$

commutes, where $\sigma$ denotes the symmetry isomorphism in $sAb$ and $Ch_\bullet^+$.

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

$\eta: \Pi X_* \otimes \Pi Y_* \to \Pi (X_* \otimes Y_*)$

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