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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
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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
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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
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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?
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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?
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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
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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?
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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) -:
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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
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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.
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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.
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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;
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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.
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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)
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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.
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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:
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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:
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I hope someone familiar with this subject can add something about
Poincaré duality for topological manifolds.
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The article says little or nothing about what kind of manifolds the manifolds discussed in the article are.
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Nope. The whole point of this section is to *not* mention cohomology in the statement of
Poincare duality.
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feel free to revert the edit and make a comment on the discussion page. 00:08 CST, 29 December 2004.
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Brief introduction to Verdier duality and how it generalises the previous models;
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See the `bilinear pairings' section -- it answers your question, I think.
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I've expanded the bilinear pairings formulation in a sense, with the new
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In this section, shouldn't the second homology group in each line be a
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Smooth? Piecewise linear? Homeomorphic to a simplicial complex?
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Add applications in differential geometry / topology, elsewhere;
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More details the cap-product-based pairing currently discussed;
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