Timeline for Wilking's connectivity theorem
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10 events
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| Sep 14, 2012 at 6:49 | comment | added | Mark Grant | In the manifold case the proof uses Morse-Bott theory on the space of broken geodesics in $X$ starting and ending in $Y$. Maybe there is something similar in the Alexandrov case? (I'm afraid I don't know Alexandrov geometry.) Another thought: in the non-manifold case Poincaré duality may fail, so my argument suggests to try and prove the result about vanishing of compactly supported cohomology of $X-Y$ instead. | |
| Sep 14, 2012 at 6:01 | comment | added | jiangsaiyin | My English is poor,Sorry that I have not made me clear.The conjecture for Alexandrov space is "has homology only up to 2k-2",which is far from the manifold case "n-2k+1 connected".I wonder whether we can get another conjecture for Alexandrov space case.But by now I have no idea. | |
| Sep 13, 2012 at 9:37 | comment | added | Mark Grant | I'm not sure what you mean by "put forward a result nearer to the manifold case". As far as I understand it (from briefly skimming the review on MathSciNet(!)) Wilking's theorem is stated only for $X$ and $Y$ compact, and probably fails otherwise. So if by "manifold" you mean "possibly non-compact manifold" then there may not be any result to put forward. | |
| Sep 13, 2012 at 8:18 | comment | added | jiangsaiyin | Oh,different dimensions!Thank you for your answer.I wonder whether we can put forward a result nearer to the manifold case. | |
| Sep 12, 2012 at 18:56 | history | edited | Mark Grant | CC BY-SA 3.0 |
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| Sep 12, 2012 at 15:25 | comment | added | Mark Grant | By the way, in your comment you seem to be composing the iso on homology with Poincaré duality to get an iso on cohomology with compact supports. But note that this introduces a degree shift, since $X$ and $Y$ have different dimensions. So you may have to use the UCT, or something similar. | |
| Sep 12, 2012 at 14:27 | comment | added | Mark Grant | It looks like I am claiming that since the inclusion map $f\colon Y\to X$ is $n-2k+1$ connected, the map $f^\ast\colon H^i_c(X)\to H^i_c(Y)$ is iso for $i<n-2k+1$ and mono for $i=n-2k+1$. This is certainly true if $X$ and $Y$ are compact, by the naturality of the universal coefficient sequence and the five lemma. I think it should be true more generally, but may require an argument involving direct limits. | |
| Sep 12, 2012 at 13:46 | comment | added | jiangsaiyin | My algebraic topology is poor.Y is n−2k+1 connected in X,then homology H_k(Y) iso to H_k(X) when k<n-2k+1 and surjective when k=n-2k+1.So I think compact cohomology of X iso to Y when K>2k-1,surjective(?)when k=2k-1.Then by the long exact sequence,compact cohomology of X-Y =0 when k>=2k-2.So H_k(X-Y)=0 when k<=n-2k+2.Why I am wrong? | |
| Sep 12, 2012 at 10:59 | history | edited | Mark Grant | CC BY-SA 3.0 |
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| Sep 12, 2012 at 10:52 | history | answered | Mark Grant | CC BY-SA 3.0 |