Timeline for Does the Ricci flow approach shed light on Poincare conjecture in higher dimension?
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| Nov 21, 2012 at 5:17 | comment | added | Caramba | This is somehow hopeless (at this moment) if you want to get general topological conclusions. Ricci flow encounters singularities, and we expect rescaled limit of singularities are Ricci solitons. The problem is we don't have a classification of solitons - in dimension 4 or higher we don't "even" have a classification of Einstein manifolds. In dimension 3, Einstein manifolds are of constant curvature and compact solitons are just Einstein, and one knows a lot on noncompact solitons (Hamilton, Ivey, Perelman, Brendle, ...) That is a necessary condition that Ricci flow is fruitful in 3d. | |
| Oct 26, 2012 at 17:43 | comment | added | Terry Tao | This is in the opposite direction to the original question, but the Ricci approach also works in dimension 2: arxiv.org/abs/math/0505163 . (And in dimension 1 as well, I suppose, though this is a vacuous truth.) | |
| Oct 26, 2012 at 15:45 | comment | added | Ian Agol | There's Brendle-Schoen: ams.org/journals/bull/2011-48-01/S0273-0979-2010-01312-4/… You could also consider Ricci flow on cohomogeneity one metrics, which exist on Milnor spheres: arxiv.org/abs/math/0601765 | |
| Oct 26, 2012 at 15:44 | comment | added | Anton Petrunin | An other reason, is that in dimensions 5 and up, the neck surgeries do not lead to a simpler manifold. At least in principle, they can cancel each other. | |
| Oct 26, 2012 at 15:43 | comment | added | Anton Petrunin | One may think of Ricci flow with surgeries as an algorithm. Since there is no algorithm to say that more-than-three-dimensional manifold is simply connected, you do not have much chance. (It will not help even if an old man told you that manifold is simply connected and you trust him.) | |
| Oct 26, 2012 at 13:51 | comment | added | Tim Perutz | Is there any connection, in high dimensions, between Ricci curvature and the property of being homeomorphic (not necessarily diffeomorphic) to a sphere? | |
| Oct 26, 2012 at 13:47 | comment | added | YangMills | Note: if the Ricci flow on a homotopy sphere did something like you said, it almost certainly would give you that your manifold was diffeomorphic to the sphere, not just homeomorphic. Morally, this is because (perhaps after surgeries) the flow should give you a positive constant curvature metric on the manifold, which is then diffeomorphic to the sphere. But this statement is still open in dimension $4$, and plain false in general dimensions bigger than or equal to $7$. | |
| Oct 26, 2012 at 6:39 | history | asked | 36min | CC BY-SA 3.0 |