# Alexander-Whiteney duality

1. Apr 15, 2009

### wofsy

What is Alexander-Whitney duality?

2. Apr 15, 2009

### whybother

It's not a common term, but I assume you are really referring to the Alexander-Whitney map.

If $$X$$, $$Y$$ are topological spaces, then the AW map is the natural transformation between:

$$(X, Y) \mapsto Sing (X \times Y)$$ and $$Sing (X) \otimes Sing (Y)$$

Where $$Sing$$ is total singular chain complex for the specified top. space.

Since it's a http://en.wikipedia.org/wiki/Natural_transformation" [Broken], it could be labeled a duality.

Last edited by a moderator: May 4, 2017
3. Apr 16, 2009

### Hurkyl

Staff Emeritus
At the risk of derailing the thread... how? Dualities are typically expressible as contravariant functors -- the closest "natural transformation" gets to the notion of duality is there is typically a natural transformation (usually isomorphism) from an object to the dual of its dual.

Last edited by a moderator: May 4, 2017