Talk:Alexander duality

Untitled

this had two errors of substance; if you're going to (partially) revert then be aware of them. First, it is patently false that removing a point from the complement of a compact set in the sphere or euclidean space does not alter its homotopy type or homology/cohomology groups. Second (because of that) the statement was incorrect. An example is furnished by the 1 dimensional line and the point. According to the previous version, we have the reduced homology of the point (0) equal to the reduced cohomology of two intervals (which is the ground ring). —Preceding unsigned comment added by 71.190.4.212 (talk) 01:51, 14 August 2009 (UTC)[reply]

I shall certainly check the statement in Spanier's book. The article as it stands has no reference of any sort, which is unsatisfactory. Removing the reference to Spanier isn't therefore best practice. Charles Matthews (talk) 07:13, 14 August 2009 (UTC)[reply]

Duality without local contracibility

In the "mordern treatment" section, it is written "Note that we can drop "local contractibility" as part of the hypothesis, if we use Čech cohomology, which is designed to deal with local pathologies."

My question is what homology theory do you use in this case- would it be strong homology?

143.210.42.231 (talk) 14:40, 3 October 2011 (UTC)[reply]

Erroneous statement

One sentence reads as follows, supposedly as an example of Alexander-Whitehead duality:

"For example the Clifford torus construction in the 3-sphere shows that the complement of a solid torus is another solid torus; which will be open if the other is closed, but this doesn't affect its homology."

This construction shows no such thing, because the claim that "the complement of a solid torus [in the 3-sphere is another solid torus" is false! (For example, consider the complement of a knotted torus.)

Rather. what the duality theorem shows is that the complement of a solid torus in the 3-sphere has the same reduced homology as the solid torus itself. This should be fixed, since it's a reasonable example of the application of the theorem.

Also, together with the Hurewicz theorem, this shows that if T is a (closed) solid torus embedded in S³ in any manner, and Y denotes S³ - T, then the fundamental group G of Y made abelian — i.e., factored out by its commutator subgroup [G, G] — must be isomorphic to the integers Z.

For, by the Hurewicz theorem, G / [G, G] = H₁(Y) = H₁(S³ - T). And by Alexander duality, H₁(S³ - T) is isomorphic to H_(3-1-1)(T) = H₁(T), which is known to equal Z.Daqu (talk) 05:09, 8 February 2013 (UTC)[reply]

Poorly thought out

The section "Alexander's 1915 result" is very poorly written.

All the stuff about flipping the Betti numbers and shifting them "not working out" is incomprehensible to most readers because it gives no reason to do what is done to the Betti numbers, and completely fails to state just what it is that "doesn't work out".

This should be written instead by someone who is able to understand the reader's point of view.50.205.142.35 (talk) 04:40, 23 December 2019 (UTC)[reply]

I agree. All those rows and rows of Betti numbers and permutations thereof is neither encyclopedic, nor illuminating. Too cute by half. I think that whole section needs to be trimmed down. Turgidson (talk) 12:17, 23 December 2019 (UTC)[reply]