nLab
Eilenberg-Zilber map

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

Definition

Denote by

Definition

For A,BsAbA,B \in sAb two simplicial abelian group, the Eilenberg-Zilber map (or Eilenberg-MacLane map or shuffle map) is the natural transformation of chain complexes

A,B:C(A)C(B)C(AB) \nabla_{A,B} \;\colon\; C(A) \otimes C(B) \longrightarrow C(A \otimes B)

defined on two n-simplices aA pa \in A_p and bB qb \in B_q by

A,B:ab (μ,ν)Sh(p,q)sgn(μ,ν)(s ν(a))(s μ(b))C p+q(AB)=A p+qB p+q, \nabla_{A,B} \;\colon\; a \otimes b \;\mapsto\; \sum_{(\mu,\nu) \in Sh(p,q)} sgn(\mu,\nu) \cdot \big(s_\nu(a)\big) \otimes \big(s_\mu(b)\big) \;\; \in C_{p+q}(A \otimes B) = A_{p+q} \otimes B_{p+q} \,,

where:

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

    (μ,ν)=(μ 1,,μ p,ν 1,,ν q), (\mu,\nu) = (\mu_1, \cdots, \mu_p, \nu_1, \cdots, \nu_q) \,,
  • sgn(μ,ν)sgn(\mu,\nu) is the signature of the corresponding permutation,

  • the maps s μs_{\mu} and s νs_\nu are defined by:

    s μs μ p1s μ 21s μ 11 s_{\mu} \coloneqq s_{\mu_p - 1} \circ \cdots \circ s_{\mu_2 - 1} \circ s_{\mu_1 - 1}

    and

    s νs ν q1s ν 21s ν 11. s_{\nu} \coloneqq s_{\nu_q - 1} \circ \cdots \circ s_{\nu_2 - 1} \circ s_{\nu_1 - 1} \,.

Remark

The shift in the indices in Def. is to be coherent with the convention that the shuffle (μ,ν)(\mu, \nu) is a permutation of {1,,p+q}\{1, \dots, p+q\}. In many references the shift disappears by making it a permutation of {0,,p+q1}\{0, \dots, p+q-1\}, instead.

Remark

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

Proposition

This Eilenberg-Zilber map (Def. ) co/restricts to the normalized chain complex inside the Moore complex, to a chain map of the form:

A,B:N(A)N(B)N(AB). \nabla_{A,B} \;\colon\; N(A) \otimes N(B) \longrightarrow N(A \otimes B) \,.

Properties

Monoidal properties

Proposition

The Eilenberg-Zilber map (Def. ) is a lax monoidal transformation that makes CC and NN into lax monoidal functors.

See at monoidal Dold-Kan correspondence for details.

For the next statement notice that both sAbsAb and Ch +Ch_\bullet^+ are in fact symmetric monoidal categories.

Proposition

The EZ map (Def. ) is symmetric in that for all A,BsAbA,B \in sAb the square

CACB σ CBCA A,B B,A C(AB) C(σ) C(BA) \array{ C A \otimes C B &\stackrel{\sigma}{\to}& C B \otimes C A \\ {}^{\mathllap{\nabla_{A,B}}} \big\downarrow && \big\downarrow^{\mathrlap{\nabla_{B,A}}} \\ C(A\otimes B) &\stackrel{C(\sigma)}{\to}& C(B \otimes A) }

commutes, where σ\sigma denotes the symmetry isomorphism in sAbsAb and Ch +Ch_\bullet^+.

Eilenberg-Zilber theorem

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.

Applications

References

The Eilenberg-Zilber map was introduced in:

following the Eilenberg-Zilber theorem of

Review and further discussion:

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