nLab
Alexander-Whitney map

Contents

Context

Homological algebra

homological algebra

(also nonabelian homological algebra)

Introduction

Context

Basic definitions

Stable homotopy theory notions

Constructions

Lemmas

diagram chasing

Homology theories

Theorems

Contents

Definition

Let C:sAbCh +C : sAb \to Ch_\bullet^+ be the chains/Moore complex functor of the Dold-Kan correspondence.

Let (sAb,)(sAb, \otimes) be the standard monoidal category structure given degreewise by the tensor product on Ab and let (Ch +,)(Ch_\bullet^+, \otimes) be the standard monoidal structure on the category of chain complexes.

Definition

For A,BsAbA,B \in sAb two abelian simplicial groups, the Alexander-Whitney map is the natural transformation on chain complexes

Δ A,B:C(AB)C(A)C(B) \Delta_{A,B} : C(A \otimes B) \to C(A) \otimes C(B)

defined on two nn-simplices aA na \in A_n and bB nb \in B_n by

Δ A,B:ab p+q=n(d˜ pa)(d 0 qb), \Delta_{A,B} : a \otimes b \mapsto \oplus_{p + q = n} (\tilde d^p a) \otimes (d^q_0 b) \,,

where the front face map d˜ p\tilde d^p is that induced by

[p][p+q]:ii [p] \to [p+q] : i \mapsto i

and the back face d 0 qd^q_0 map is that induced by

[q][p+q]:ii+p. [q] \to [p+q] : i \mapsto i+p \,.
Definition

This AW map restricts to the normalized chains complex

Δ A,B:N(AB)N(A)N(B). \Delta_{A,B} : N(A \otimes B) \to N(A) \otimes N(B) \,.

Properties

Proposition

The Alexander-Whitney map is an oplax monoidal transformation that makes CC and NN into oplax monoidal functors.

Beware that the AW map is not symmetric. For details see monoidal Dold-Kan correspondence.

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). 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).

References

The Eilenberg-Zilber theorem is due to

using the definition of the Eilenberg-Zilber map in:

Review:

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

Last revised on September 13, 2021 at 03:36:46. See the history of this page for a list of all contributions to it.