Talk:Poincaré duality

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Mathematics rating:	Start Class	High Priority	Field: Topology
Please update this rating as the article progresses, or if the rating is inaccurate. Click to show/hide comments. Please add to or update the comments to suggest improvements to the article. 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)

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.

[edit] New To Advanced Math

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

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

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)

[edit] Poincare duality of ring theory

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)