nLab Eilenberg-Zilber map



Homological algebra

homological algebra

(also nonabelian homological algebra)



Basic definitions

Stable homotopy theory notions



diagram chasing

Schanuel's lemma

Homology theories




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


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} \,.


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.


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.


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.


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.


Monoidal properties


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.


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


(Eilenberg-Zilber/Alexander-Whitney deformation retraction)


and denote

Then there is a deformation retraction


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 *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.


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.