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Eilenberg-Zilber/Alexander-Whitney deformation retraction

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Context

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homological algebra

(also nonabelian homological algebra)

Introduction

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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

and denote

Then there is a deformation retraction

where

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.

References

The Eilenberg-Zilber theorem is due to

using the definition of the Eilenberg-Zilber map in:

Review and further discussion:

Last revised on July 13, 2021 at 15:36:47. See the history of this page for a list of all contributions to it.