Alexander Duality & Cup Product: Commuting?

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SUMMARY

The discussion centers on the relationship between Alexander duality and the cup product in algebraic topology, specifically regarding the case of circles embedded in S3. The key statement of Alexander duality discussed is H^*(S3-C) ≅ Hn-*(C). The participants explore how dimensions can be reconciled using Poincaré duality and the implications of cup products between embedded circles, leading to the conjecture of a natural homomorphism from H1(C1) x H1(C2) into Z, which relates to the degree of the linking map of the torus C1 x C2 into the 2-sphere.

PREREQUISITES
  • Understanding of Alexander duality in algebraic topology
  • Familiarity with cup products in cohomology
  • Knowledge of Poincaré duality
  • Basic concepts of homomorphisms in algebraic topology
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  • Study the implications of Alexander duality on cohomology groups
  • Research the properties and applications of cup products in algebraic topology
  • Explore Poincaré duality and its role in topological spaces
  • Investigate linking numbers and their significance in topology
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Mathematicians, particularly those specializing in algebraic topology, graduate students studying topology concepts, and researchers interested in the interplay between duality theories and cohomological operations.

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does Alexander duality commute with cup product?
 
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How could the dimensions add up? Unless I'm interpreting this the wrong way, taking the cup product of the images would be in a different cohomology group than the image of of the cup product.
 
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?
 
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?

I am really thinking about a special case of Alexander duality, the case of a circle embedded in S3,

H^*(S3-C) iso Hn-*(C)

If there are two embedded circles then cup product maps H^1(S3-C1)xH^1(S3-C2) -> H^2(S3-C1UC2).

The Alexander maps take these two groups into H1(C1)xH1(C2) and H0(C1UC2).
These are ZxZ and Z.

I was really wondering if there is a natural homomorphism from H1(C1)xH1(C2) into Z that completes the square. The conjecture is that it is the degree of the linking map of the torus C1xC2 into the 2 sphere.
 

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