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I know that the Poincaré conjecture was first proved in dimension ≥ 5, then dimension 4, and finally 3. I'm just curious, does the Ricci flow approach by Perelman shed any light on the high dimension case? Or for some reason one cannot hope for a uniform proof because these dimensions are essentially different?

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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.) – Anton Petrunin Oct 26 '12 at 15:43
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. – Anton Petrunin Oct 26 '12 at 15:44
There's Brendle-Schoen:… You could also consider Ricci flow on cohomogeneity one metrics, which exist on Milnor spheres: – Ian Agol Oct 26 '12 at 15:45
This is in the opposite direction to the original question, but the Ricci approach also works in dimension 2: . (And in dimension 1 as well, I suppose, though this is a vacuous truth.) – Terry Tao Oct 26 '12 at 17:43
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. – Caramba Nov 21 '12 at 5:17

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