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Eilenberg-Zilber map

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Definition

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

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

Definition

For A,BsAbA,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

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

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

A,B:ab (μ,ν)sign(μ,ν)(s ν(a))(s μ(b))C p+q(AB)=A p+qB p+q, \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)(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)

and the corresponding degeneracy maps are

s μ=s μ ps μ 2s μ 1 s_{\mu} = s_{\mu_p} \circ \cdots s_{\mu_2} \circ s_{\mu_1}

and

s ν=s ν qs ν 2s ν 1. s_{\nu} = s_{\nu_q} \circ \cdots s_{\nu_2} \circ s_{\nu_1} \,.

The sign sign(μ,ν){1,1}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 Δ[p]×Δ[q]\Delta[p] \times \Delta[q]. See products of simplices for more on this.

Proposition

This map restricts to the normalized chains complex

A,B:N(A)N(B)N(AB). \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,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

  • 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 CC and NN 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 Δ A,B\Delta_{A,B}:

Id:NANB A,BN(AB)Δ A,BNANB. 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,YX,Y the EZ map X,Y\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 sAbsAb and Ch +Ch_\bullet^+ are in fact symmetric monoidal categories.

Proposition

The EZ map 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}}}\downarrow && \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^+.

In the context of filtered spaces X *,Y *X_*, Y_* and their associated fundamental crossed complex?es 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-Zilber map was introduced in (5.3) of

See also 29.7 of

  • Peter May, Simplicial objects in algebraic topology , Chicago Lectures in Mathematics, Chicago, (1967) (djvu)
  • A.P. Tonks, On the Eilenberg-Zilber Theorem for crossed complexes. J. Pure Appl. Algebra, 179~(1-2) (2003) 199–220,

and section 11.2 of

Revised on December 25, 2013 22:43:52 by Urs Schreiber (89.204.135.135)