Timeline for homology with compact supports
Current License: CC BY-SA 2.5
7 events
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| Feb 25, 2010 at 3:40 | history | edited | Paul | CC BY-SA 2.5 |
removed wrong assertion.
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| Feb 25, 2010 at 3:39 | comment | added | Paul |
OK, I see my mistake. $H^1_c(X)$' is not dual to $H_1(X)$', but it is dual to `$H_{1,BM}(X)$. I'll get rid of the second paragraph of my answer, but then the first paragraph becomes irrelevant to the question.
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| Feb 25, 2010 at 3:31 | comment | added | Paul |
$ H^{n−k}_{comp}(X)=H_k(X)$ and $H^{n−k}(X)=H_{k,BM}(X)$ seems OK to me: If you think of a regular cell structure and its dual cell structure, the Poincare dual to a finite chain is a compactly supported cochain and the dual to a locally finite infinite chain is a regular cochain. But I agree your assertion contradicts what I said in the second paragraph. What if $X$ is an infinite genus 2-manifold? Then (say rational coeffs) $H^1_{comp}(X)=H_1(X)$ is a direct sum of infinitely many $Q$s, so $(H_1(X))^*$ is a infinite product of $Q$s. I'm confused.
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| Feb 25, 2010 at 2:16 | comment | added | Emerton | I'm not sure that I understand the claims here. For example, one cannot pair an ordinary cocycle with a locally finite chain (since the resulting sum may still be infinite), but only with a finite chain. Also, it is true that $H^k_c$ and Borel--Moore homology in degree $k$ (which is what I think you mean by $H_{k,inf}(X)$) are canonically dual (with field coefficients). (This is not a statement of Poincare duality, but is easier, and is analogous to the canonical duality of cohomology and usual homology with field coefficients given by one of the universal coefficient-type formulas.) | |
| Feb 25, 2010 at 0:48 | history | edited | Paul | CC BY-SA 2.5 |
added 6 characters in body
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| Feb 25, 2010 at 0:33 | history | edited | Paul | CC BY-SA 2.5 |
added 20 characters in body; added 2 characters in body
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| Feb 25, 2010 at 0:24 | history | answered | Paul | CC BY-SA 2.5 |