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wofsy
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What is Alexander-Whitney duality?
How Does Alexander-Whitney Duality Relate to Natural Transformations?
- Thread starter wofsy
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In summary, Alexander-Whitney duality is a natural transformation between two topological spaces, X and Y, and their singular chain complexes. It can be labeled as a duality because it can be expressed as a contravariant functor and involves a natural transformation between an object and its dual.
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whybother
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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" , 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" , it could be labeled a duality.
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- #3
Hurkyl
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At the risk of derailing the thread... how?whybother said:
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.
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FAQ: How Does Alexander-Whitney Duality Relate to Natural Transformations?
1. What is Alexander-Whitney duality?
Alexander-Whitney duality is a mathematical concept that relates the cohomology and homology groups of a topological space. It is named after mathematicians James Alexander and Hassler Whitney.
2. How does Alexander-Whitney duality work?
Alexander-Whitney duality states that for a topological space X, the cohomology and homology groups are dual to each other, meaning that they contain the same information but in different forms. This duality is expressed through the use of cup and cap products.
3. What is the significance of Alexander-Whitney duality?
Alexander-Whitney duality is significant because it allows for the study of a topological space using different mathematical tools, providing a deeper understanding of its structure. It also has applications in various fields such as algebraic topology, differential geometry, and algebraic geometry.
4. What are some examples of the application of Alexander-Whitney duality?
Alexander-Whitney duality is commonly used in the study of characteristic classes, fixed-point theory, and cohomology operations. It has also been applied to knot theory, representation theory, and algebraic K-theory.
5. Are there any limitations to Alexander-Whitney duality?
While Alexander-Whitney duality is a powerful tool, it has some limitations. It only applies to topological spaces and cannot be extended to more general structures such as manifolds or schemes. Additionally, it only applies to spaces that satisfy certain conditions, such as being locally compact and Hausdorff.