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What is Alexander-Whitney duality?

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- Thread starter wofsy
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- #1

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What is Alexander-Whitney duality?

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

If [tex]X[/tex], [tex]Y[/tex] are topological spaces, then the AW map is the natural transformation between:

[tex] (X, Y) \mapsto Sing (X \times Y) [/tex] and [tex] Sing (X) \otimes Sing (Y) [/tex]

Where [tex]Sing[/tex] 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.

If [tex]X[/tex], [tex]Y[/tex] are topological spaces, then the AW map is the natural transformation between:

[tex] (X, Y) \mapsto Sing (X \times Y) [/tex] and [tex] Sing (X) \otimes Sing (Y) [/tex]

Where [tex]Sing[/tex] 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.

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- #3

Hurkyl

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

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