nLab Eilenberg-Zilber map

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Contents

Context

Homological algebra

homological algebra

(also nonabelian homological algebra)

Introduction

Context

Basic definitions

Stable homotopy theory notions

Constructions

Lemmas

diagram chasing

Schanuel's lemma

Homology theories

Theorems

Contents

Idea

The Eilenberg-Zilber map is a natural transformation intertwining the tensor products of chain complexes with that of their corresponding simplicial abelian groups, which is part of the monoidal Dold-Kan correspondence.

Its explicit relation by the Eilenberg-MacLane formula expresses it in terms of sums of non-degenerate simplices inside a product of simplices.

Definition

Denote by

Definition

For A,BsAbA,B \in sAb two simplicial abelian group, the Eilenberg-MacLane formula for the Eilenberg-Zilber 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 a pair of n n -simplices aA pa \in A_p and bB qb \in B_q by

(1) 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, \begin{aligned} \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} \,, \end{aligned}

where (see here at products of simplices for the geometric interpretation):

  • 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 iterated degeneracy maps:

    (2)s μs μ p1s μ 21s μ 11,ands νs ν q1s ν 21s ν 11. s_{\mu} \coloneqq s_{\mu_p - 1} \circ \cdots \circ s_{\mu_2 - 1} \circ s_{\mu_1 - 1} \,, \phantom{----}\text{and}\phantom{----} s_{\nu} \coloneqq s_{\nu_q - 1} \circ \cdots \circ s_{\nu_2 - 1} \circ s_{\nu_1 - 1} \,.

Remark

The explicit formula (1) is due to Eilenberg & MacLane (1953), eq. (5.3), there called the “\nabla-product”; review includes MacLane (1963), eq. (8.9); May (1967), p. 133; Quillen (1969), eq. (4.2); Loday (1992), Def. 1.6.11; Gonzalez-Diaz & Real (1999), p. 7.

The map that is expressed by this formula was previously shown to exist, more abstractly, by Eilenberg & Zilber (1953); cf. also Kerodon, Rem. 2.5.7.16.

Remark

The shift in the indices in (2) is to be consistent 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 (here) by making it a permutation of {0,,p+q1}\{0, \dots, p+q-1\}, instead.

Remark

The sum in (1) 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 (here) for more on this.

Proposition

This Eilenberg-Zilber map (Def. ) co/restricts on 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) \,.

(cf. e.g. Kerodon, Exp. 2.5.7.12.)

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

  • The Eilenberg-Zilber map induces a functor from simplicial Lie algebras to dg-Lie algebras (see here).

  • The Eilenberg-Zilber map controls the formula for transgression in group cohomology, see there fore more.

  • In the context of filtered spaces X *,Y *X_*, Y_* and their associated fundamental crossed complexes ΠX *,ΠY *\Pi X_*, \Pi Y_* there is a natural Eilenberg-Zilber morphism

    η:ΠX *ΠY *Π(X *Y *)\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 ω\omega–groupoids. This morphism is an isomorphism of free crossed complexes if X *,Y *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-MacLane formula was made explicit in:

realizing a transformation that was shown more indirectly to exist in the proof of the Eilenberg-Zilber theorem:

Review and further discussion:

See also:

  • A.P. Tonks, On the Eilenberg-Zilber Theorem for crossed complexes. J. Pure Appl. Algebra, 179 (1-2) (2003) 199-220

  • Tim Porter, Section 11.2 of: Crossed Menagerie,

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

Last revised on December 8, 2022 at 09:48:20. See the history of this page for a list of all contributions to it.