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Alexander-Whiteney duality

  1. Apr 15, 2009 #1
    What is Alexander-Whitney duality?
  2. jcsd
  3. Apr 15, 2009 #2
    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.
    Last edited by a moderator: May 4, 2017
  4. Apr 16, 2009 #3


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    At the risk of derailing the thread... how? :confused: 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
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