Alexander-Whitney map: Difference between revisions

Revision as of 22:39, 1 December 2007

Definition

The Alexander-Whitney map is a natural transformation between the following two bifunctors on topological spaces: $(X, Y) \mapsto S i n g_{.} (X \times Y)$ and $S i n g_{.} (X) \otimes S i n g_{.} (Y)$ ( $S i n g_{.}$ here denotes the singular chain complex functor).

The Eilenberg-Zilber map is a natural transformation in the reverse direction, and the composite both ways is naturally chain-homotopic to the identity transformations on the two functors.

Explicitly the Alexander-Whitney map is given as follows:

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 ==Definition==
-The '''Alexander-Whitney map''' is a natural transformation between the following two bifunctors on topological spaces: <math>(X,Y) \mapsto Sing_.(X \times Y)</math> and <math>Sing_.(X) \otimes Sing_.(Y)</math>.
+The '''Alexander-Whitney map''' is a natural transformation between the following two bifunctors on topological spaces: <math>(X,Y) \mapsto Sing_.(X \times Y)</math> and <math>Sing_.(X) \otimes Sing_.(Y)</math> (<math>Sing_.</math> here denotes the [[singular chain complex]] functor).
 The [[Eilenberg-Zilber map]] is a natural transformation in the reverse direction, and the composite both ways is naturally chain-homotopic to the identity transformations on the two functors.