nLab Eilenberg-Zilber map

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

• $(sAb, \otimes)$ the monoidal category of simplicial abelian groups with tensor product given degreewise by the tensor product of abelian groups;

• $(Ch_\bullet^+, \otimes)$ the monoidal category of chain complexes with its tensor product of chain complexes.

• $C \colon sAb \to Ch_\bullet^+$ the chains/Moore complex functor of the Dold-Kan correspondence;

Definition

For $A,B \in sAb$ two simplicial abelian group, the Eilenberg-MacLane formula for the Eilenberg-Zilber map is the natural transformation of chain complexes

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

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

(1)$$\begin{aligned} \nabla_{A,B} \;\colon\; a \otimes b \;\mapsto\; & \sum_{(\mu,\nu) \in Sh(p,q)} sgn(\mu,\nu) \cdot \big(s_\nu(a)\big) \otimes \big(s_\mu(b)\big) \\ & \in C_{p+q}(A \otimes B) = A_{p+q} \otimes B_{p+q} \,, \end{aligned}$$

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

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

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

• $sgn(\mu,\nu)$ is the signature of the corresponding permutation,

• the maps $s_{\mu}$ and $s_\nu$ are iterated degeneracy maps:

  (2)$$s_{\mu} \coloneqq s_{\mu_p - 1} \circ \cdots \circ s_{\mu_2 - 1} \circ s_{\mu_1 - 1} \,, \phantom{----}\text{and}\phantom{----} s_{\nu} \coloneqq s_{\nu_q - 1} \circ \cdots \circ s_{\nu_2 - 1} \circ s_{\nu_1 - 1} \,.$$

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 $(\mu, \nu)$ is a permutation of $\{1, \dots, p+q\}$. In many references the shift disappears (here) by making it a permutation of $\{0, \dots, p+q-1\}$, instead.

Remark

The sum in (1) may be understood as being over all non-degenerate simplices in the Cartesian product $\Delta[p] \times \Delta[q]$ 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:

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

(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 $C$ and $N$ into lax monoidal functors.

See at monoidal Dold-Kan correspondence for details.

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

Proposition

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

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

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

Eilenberg-Zilber theorem

Proposition

(Eilenberg-Zilber/Alexander-Whitney deformation retraction)

Let

• $A, B \,\in\, sAb =$ SimplicialAbelianGroups

and denote

• by $N(A), N(B) \,\in\, Ch^+_\bullet$ (=ConnectiveChainComplexes) their normalized chain complexes,

• by $A \otimes B \,\in\, sAb$ the degreewise tensor product of abelian groups,

• by $N(A) \otimes N(B)$ the tensor product of chain complexes.

Then there is a deformation retraction

where

• $\nabla_{A,B}$ is the Eilenberg-Zilber map;

• $\Delta_{A,B}$ is the Alexander-Whitney map.

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 $X_*, Y_*$ and their associated fundamental crossed complexes $\Pi X_*, \Pi Y_*$ 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.

Related concepts

• Alexander-Whitney map

• Eilenberg-Zilber theorem

• product of simplices

• cap product, cup product

• transgression in group cohomology

References

The Eilenberg-MacLane formula was made explicit in:

• Samuel Eilenberg, Saunders MacLane, (5.3) of: On the groups $H(\Pi,n)$, I, Ann. of Math. 58 1 (1953) 55-106. [jstor:1969820]

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

• Samuel Eilenberg, Joseph A. Zilber, On Products of Complexes, American Journal of Mathematics 75 1 (1953) 200-204 [jstor:2372629, doi:10.2307/2372629]

Review and further discussion:

• Saunders MacLane, Thm. 8.8 in: Homology, Grundlehren der Mathematischen Wissenschaften 114, Springer (1963, 1974), reprinted as: Classics in Mathematics, Springer (1995) [pdf, doi:10.1007/978-3-642-62029-4]

• Peter May, Section 29.7 of Simplicial objects in algebraic topology , Chicago Lectures in Mathematics, University of Chicago Press 1967 (ISBN:9780226511818, djvu, pdf)

• Dan Quillen, part I, section 4 of: Rational homotopy theory, The Annals of Mathematics, Second Series 90 2 (1969) 205-295 [jstor:1970725]

• Jean-Louis Loday, Section 1.6 of: Cyclic Homology, Grund. Math. Wiss. 301 Springer (1992) [doi:10.1007/978-3-662-21739-9]

• Stefan Schwede, Brooke Shipley, §2.3 in: Equivalences of monoidal model categories, Algebr. Geom. Topol. 3 (2003) 287-334 [arXiv:math.AT/0209342, euclid:euclid.agt/1513882376]

• Kerodon, 2.5.7 The Shuffle Product (00RF)

  The Eilenberg-Zilber Homomorphism (00RS)

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