Alexander Duality & Cup Product: Commuting?

  • Context: Graduate 
  • Thread starter Thread starter wofsy
  • Start date Start date
  • Tags Tags
    Duality
Click For Summary

Discussion Overview

The discussion centers on the relationship between Alexander duality and the cup product in algebraic topology, specifically exploring whether Alexander duality commutes with the cup product. Participants examine the implications of this relationship in the context of cohomology groups and duality theories, including Poincaré duality.

Discussion Character

  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant questions whether Alexander duality commutes with the cup product.
  • Another participant raises concerns about the dimensionality of the cohomology groups involved, suggesting that the cup product of the images may belong to a different cohomology group than the image of the cup product.
  • A participant proposes that invoking Poincaré duality might resolve dimensionality issues and seeks clarification on the specific statement of Alexander duality being referenced.
  • A later reply elaborates on a special case of Alexander duality involving a circle embedded in S3, discussing the isomorphism of cohomology groups and the mapping of cup products between different groups.
  • The same participant conjectures about the existence of a natural homomorphism that relates the cohomology groups and suggests it may be linked to the degree of the linking map of the torus into the 2-sphere.

Areas of Agreement / Disagreement

Participants express differing views on the relationship between Alexander duality and the cup product, with some uncertainty regarding the dimensional aspects and the specific statements of duality being used. The discussion remains unresolved with multiple competing perspectives.

Contextual Notes

Participants note potential limitations related to the assumptions made about the cohomology groups and the specific cases of Alexander duality being considered, as well as the dependence on definitions of the involved concepts.

wofsy
Messages
724
Reaction score
0
does Alexander duality commute with cup product?
 
Physics news on Phys.org
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.
 

Similar threads

Graduate How Does Alexander-Whitney Duality Relate to Natural Transformations?
  • · Replies 2 ·
Replies
2
Views
3K
Graduate Seiberg Duality (or dualities)
  • · Replies 2 ·
Replies
2
Views
792
Undergrad About concept of wave-particle duality
  • · Replies 11 ·
Replies
11
Views
2K
High School Wave-particle duality of atoms and molecules
  • · Replies 38 ·
2
Replies
38
Views
6K
Undergrad "Wave-particle duality" and double-slit experiment
  • · Replies 36 ·
2
Replies
36
Views
9K
Graduate Hodge duality and some properties
  • · Replies 6 ·
Replies
6
Views
3K
Undergrad Diffraction of point source/s and duality
  • · Replies 2 ·
Replies
2
Views
2K
Fourier transform: duality property and convolution
  • · Replies 1 ·
Replies
1
Views
2K
Graduate Is wave-matter duality a proven theory?
  • · Replies 58 ·
2
Replies
58
Views
6K
Graduate Cubical cup poduct with a unit 1-cochain and the coboundary operator
  • · Replies 0 ·
Replies
0
Views
2K