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wofsy
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does Alexander duality commute with cup product?
zhentil said:On second thought, I think I can make the dimensions add up if you're also invoking Poincare duality. Can you tell me the statement of Alexander duality that you're using?
Alexander Duality is a mathematical theorem that relates the homology of a space to the cohomology of its complement. It is a fundamental tool in algebraic topology and has many applications in various fields of mathematics.
The Cup Product is a binary operation in cohomology that combines two cohomology classes to produce a third one. It is closely related to the intersection product in topology and has many important properties that make it a powerful tool in algebraic topology.
Alexander Duality and Cup Product commute when applied to certain pairs of spaces. This means that the order in which we apply these two operations does not matter, and we will get the same result regardless. This result is known as the Commutativity Theorem.
Alexander Duality and Cup Product have numerous applications in topology, geometry, and algebra. They are used to prove important theorems, such as the Poincaré Duality Theorem and the Lefschetz Fixed Point Theorem, and have applications in fields such as knot theory, algebraic geometry, and differential equations.
Despite their importance, there are still many open problems and conjectures related to Alexander Duality and Cup Product. Some of these include generalizing the Commutativity Theorem to more general situations, finding new applications in other fields of mathematics, and understanding the connections between these two operations and other mathematical concepts.