Talk:Poincaré duality

A fact from Poincaré duality appeared on Wikipedia's Main Page in the Did you know column on 21 March 2004. The text of the entry was as follows:

Did you know... that Henri Poincaré first stated the Poincaré duality in terms of Betti numbers?

A record of the entry may be seen at Wikipedia:Recent additions/2004/March.

Wikipedia

Thom isomorphism subsection[edit]

"This formulation (...)": what formulation? — Preceding unsigned comment added by 89.164.90.215 (talk) 16:48, 12 June 2023 (UTC)[reply]

Untitled[edit]

I have a problem with the line, ``kth homology group H^k(M) to the (n−k)-th cohomology group H_{n − k}(M).`` It is my understanding that homology groups are represented with a subscript, and cohomology with a superscript; reverse from what was written. I have changed this. If I am in error, please feel free to revert the edit and make a comment on the discussion page. 00:08 CST, 29 December 2004.

New To Advanced Math[edit]

Hi; I'm trying desperately to understand many of these advanced principals of mathematics, such as Poincaré duality, but no matter how many times I review the material, it doesn't sink in. Could someone please provide examples, problems to solve (with their solutions) and/or ways to visualize this? beno 26 Jan 2006

Poincare's approach v.s. Modern[edit]

I really enjoyed this entry, thanks a lot! I think that perhaps it would be a small improvement to it if the following is taken into consideration:

 You started by given a historical introduction, with Poincare's understanding of duality as concerned about Betti numbers.  (I really enjoyed learning this).  But them you stated the modern perpective, which is close to the first attempt, but it does not imply the "duality" between the Betti numbers right away.  I think that it would be nice to reconcile the modern paragraph with the historical one by stating that by using the universal coeficients theorem the modern approach implies Poincare's approach.

See the `bilinear pairings' section -- it answers your question, I think. Rybu —Preceding comment was added at 18:31, 28 October 2007 (UTC)[reply]

Poincare duality of ring theory[edit]

Cohen-Macaulay Rings by Bruns and Herzog, pp. 123~126, mentions Poincare duality in somewhat different context from this article. The main theorem (Avramov-Golod theorem) seems to be that Noetherian local ring R is Gorenstein iff. H.(R) is a Poincare algebra iff. k-linear map H_n-1 (R) -> Hom_k (H_1 (R),H_n (R)) induced by the multiplication on H.(R) is a monomorphism. I can't why this is called Poincare duality, as I can't see how this is related to the fact in this article that H_(n-k) (M) is isomorphic to H^k (M). Can someone provide an explanation? --Acepectif 06:04, 28 October 2007 (UTC)[reply]

I know some mathematicians that call the isomorphism between a vector space and its dual (given by an inner product), Poincare duality. The further away you get from topology the more vaguely and inaccurately people use the phrase 'Poincare duality'. In that sense, there's many non-standard and borderline useless usages of the phrase Poincare duality. I tend to just ignore them unless the author can make a compelling case -- most often I feel like that answer is no. Rybu (talk) 04:56, 5 July 2009 (UTC)[reply]

Bilinear pairings formulation[edit]

In this section, shouldn't the second homology group in each line be a cohomology group?? --Roentgenium111 (talk) 20:16, 29 June 2009 (UTC)[reply]

Nope. The whole point of this section is to *not* mention cohomology in the statement of Poincare duality. Rybu (talk) 04:50, 5 July 2009 (UTC)[reply]

I've expanded the bilinear pairings formulation in a sense, with the new Thom Isomorphism Formulation, this allows us to mention what Poincare duality means for general homology theories. Rybu (talk) 00:38, 14 November 2009 (UTC)[reply]

Borel-Moore Homology[edit]

It is explained that one can generalise to non-compact manifolds if one changes to cohomology with compact supports. It seems like quite a large omission, at least to me, to have no mention of Borel-Moore homology. In many regards, this is a more "natural" generalisation, in that it is the adjustment needed to obtain a non-compact fundamental class for the manifold, and the duality is induced by taking the cap-product with this fundamental class, just as for the compact case. 143.210.42.231 (talk) 15:35, 10 March 2014 (UTC)[reply]

Assessment comment[edit]

The comment(s) below were originally left at Talk:Poincaré duality/Comments, and are posted here for posterity. Following several discussions in past years, these subpages are now deprecated. The comments may be irrelevant or outdated; if so, please feel free to remove this section.

Comment(s)

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One of the very important results in algebraic topology, and one with major influence elsewhere too. Should aim to get this to high level of completeness. Things to add:

Duality in cohomology in terms of cup product and evaluation against the fundamental class; the de Rham picture with differential forms and integration is particularly good as an introduction;
More details the cap-product-based pairing currently discussed;
Discussion on relaxing the conditions of the "easy" case: orientability (use orientation bundle or Z/2Z coefficients), compactness (use cohomology with supports or homology/cohomology pairing) and smoothness (intersection cohomology);
Brief introduction to Verdier duality and how it generalises the previous models;
Discuss restrictions PD implies for, e.g., Betti numbers of smooth compact manifolds;
Add applications in differential geometry / topology, elsewhere;
Give pointers to Poincaré-type duality theorems in other fields (Serre duality, the more general coherent duality in algebraic / analytic geometry, duality in étale cohomology)

Stca74 13:22, 15 May 2007 (UTC)[reply]

Last edited at 19:26, 16 May 2007 (UTC). Substituted at 02:29, 5 May 2016 (UTC)

Explanatory Examples[edit]

This page should have an explanatory example showing how poincare duality and cap products work. This could be done with a torus, but more non-trivial examples should be discussed as well. Also, it should discuss the duality of cup products and intersections of chains. — Preceding unsigned comment added by 50.246.213.170 (talk) 18:26, 12 August 2017 (UTC)[reply]

Application to Euler Characteristics[edit]

It is unclear to me what "manifold that bounds" means. I am guessing it's something like that the manifold is the boundary of a lower dimensional manifold, but am not sure, and for someone more unfamiliar with the subject I imagine it would be even less clear. Could someone put more details here? 35.20.161.132 (talk) 18:26, 20 September 2023 (UTC)[reply]