# nLab Alexander-Whitney map

Contents

### Context

#### Homological algebra

homological algebra

Introduction

diagram chasing

# Contents

## Definition

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

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

###### Definition

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

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

defined on two $n$-simplices $a \in A_n$ and $b \in B_n$ by

$\Delta_{A,B} : a \otimes b \mapsto \oplus_{p + q = n} (\tilde d^p a) \otimes (d^q_0 b) \,,$

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

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

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

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

This AW map restricts to the normalized chains complex

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

## Properties

###### Proposition

The Alexander-Whitney map is an oplax monoidal transformation that makes $C$ and $N$ into oplax monoidal functors.

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

###### Proposition

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.