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

Schanuel's lemma

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 07:36:46. See the history of this page for a list of all contributions to it.