nLab Alexander-Whitney map

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

(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). 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).

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, doi:10.2307/2372629)

• 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)

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:

• 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)