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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.
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;
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
defined on a pair of $n$-simplices $a \in A_p$ and $b \in B_q$ by
where (see here at products of simplices for the geometric interpretation):
the sum is over all $(p,q)$-shuffles
$sgn(\mu,\nu)$ is the signature of the corresponding permutation,
the maps $s_{\mu}$ and $s_\nu$ are iterated degeneracy maps:
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.
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.
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.
This Eilenberg-Zilber map (Def. ) co/restricts on the normalized chain complex inside the Moore complex, to a chain map of the form:
(cf. e.g. Kerodon, Exp. 2.5.7.12.)
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.
The EZ map (Def. ) is symmetric in that for all $A,B \in sAb$ the square
commutes, where $\sigma$ denotes the symmetry isomorphism in $sAb$ and $Ch_\bullet^+$.
(Eilenberg-Zilber/Alexander-Whitney deformation retraction)
Let
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.
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
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.
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:
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
Last revised on December 8, 2022 at 09:48:20. See the history of this page for a list of all contributions to it.