nLab Eilenberg-Zilber/Alexander-Whitney deformation retraction

Idea

The Dold-Kan correspondence between connective chain complexes and simplicial abelian groups is not quite compatible with the tensor product-structure on both sides, but it is so up to homotopy (see at monoidal Dold-Kan correspondence). One aspect of this is the Eilenberg-Zilber theorem, saying that the Eilenberg-Zilber map from the tensor product of chain complexes of normalized chain complexes of two abelian groups to the normalized chain complex of their degree-wise tensor product of abelian groups is not quite an isomorphism, but is a homotopy equivalence, in fact a deformation retraction, with homotopy-inverse the Alexander-Whitney map.

Statement

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), review in MacLane 1975, VIII 8, Dold 1995, VI 12.1. The above deformation retraction for normalized chain complexes is Eilenberg & MacLane 1954, Thm. 2.1a. Both are reviewed in May 1967, Cor. 29.10, the latter at least mentioned in MacLane 1975, VIII Cor. 8.9. Explicit description of the homotopy operator is given in Gonzalez-Diaz & Real 1999, p. 7.

Related concepts

• Alexander-Whitney map

• Eilenberg-Zilber map

References

The Eilenberg-Zilber theorem is due to

• Samuel Eilenberg, Joseph Zilber, On Products of Complexes, Amer. Jour. Math. 75 (1): 200–204, (1953) (jstor:2372629)

• Samuel Eilenberg, Saunders MacLane, Section 2 of: On the Groups $H(\Pi,n)$, II: Methods of Computation, Annals of Mathematics, Second Series, Vol. 60, No. 1 (Jul., 1954), pp. 49-139 (jstor:1969702, doi:10.2307/2372629)

using the definition of the Eilenberg-Zilber map in:

• Samuel Eilenberg, Saunders MacLane, (5.3) of: On the groups $H(\Pi,n)$, I, Ann. of Math. (2) 58, (1953), 55–106. (jstor:1969820)

Review and further discussion:

• Peter May, Section 29 of: Simplicial objects in algebraic topology , Chicago Lectures in Mathematics, University of Chicago Press 1967 (ISBN:9780226511818, djvu, pdf)

• Saunders Mac Lane, Homology (1975) reprinted as Classics in Mathematics. Springer-Verlag, Berlin, 1995. x+422 pp. ISBN 3-540-58662-8 (doi:10.1007/978-3-642-62029-4)

• Albrecht Dold, Section VI 12.1 in Lectures on Algebraic Topology, Springer 1995 (doi:10.1007/978-3-642-67821-9, pdf)

• Rocio Gonzalez-Diaz, Pedro Real, A Combinatorial Method for Computing Steenrod Squares, Journal of Pure and Applied Algebra 139 (1999) 89-108 (arXiv:math/0110308)