Talk:Poincaré duality

message: "This formulation (...)": what formulation?   — Preceding unsigned comment added by 89.164.90.215 (talk) 16:48, 12 June 2023 (UTC)
thread_title: Thom isomorphism subsection

message: 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)
thread_title:  Poincare's approach v.s. Modern  

message: 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)
thread_title:  Poincare duality of ring theory 

message: In this section, shouldn't the second homology group in each line be a cohomology group?? --Roentgenium111 (talk) 20:16, 29 June 2009 (UTC)
thread_title:  Bilinear pairings formulation 

message: 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)
thread_title:  Borel-Moore Homology 

message: Substituted at 02:29, 5 May 2016 (UTC)
thread_title: Assessment comment

message: 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)
thread_title:  Explanatory Examples 

message: 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)
thread_title:  Application to Euler Characteristics