User:Chan-Ho Suh/todo/PC proof
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[rough proof off the top of my head...no refs, very rough! Understandable to perhaps no one who doesn't already know the proof!]
The Poincare conjecture states that any homotopy 3-sphere is homeomorphic to the 3-sphere. So consider a homotopy 3-sphere M. Start with any metric on it. Now run Hamilton's Ricci flow, which is given by a geometric evolution equation relying on curvature. The heuristic behind the flow is that it will evolve the metric in such a way that curvature gets distributed as evenly as possible. Unfortunately, things can get complicated as singularities develop. Singularities occur when there are regions of high curvature; imagine a round sphere of shrinking radius - as it collapses to a point the curvature is getting unbounded.
Perelman showed that singularities are basically neck-pinches, which are localized in only part of the manifold, or the whole manifold can collapse to a point. The neck pinch pinches off a part of M by collapsing a 2-sphere to a point. The picture here is that of a long thin tube. Now the modification of Ricci flow, called Ricci flow with surgery, will continue the flow through these pinches (a pinch is called a "surgery" by topologists). Once there is more than one piece evolving through Ricci flow, at a singularity time we may have entire pieces crunching to a point and also neck pinches pinching off more pieces.
Note that after a surgery, if the pieces are all 3-spheres, then undoing the surgery (reattaching, which is reeally a connected sum) shows the manifold before the surgery had all pieces being 3-spheres. So if we can show that at some time all the pieces are 3-spheres, we can deduce the original manifold (at time t=0) was a 3-sphere.
Two important things need to be shown. That only finitely many singularity times exist in any time interval, so that we need only do finitely many surgeries. And also that eventually the flow becomes extinct, which is a fancy way of saying that all the pieces have collapsed to points and there is nothing left to flow on.
Perelman showed both these things. Prior work by Hamilton demonstrates that such pieces that collapse to points are actually becoming rounder and rounder, i.e. spherical space-forms. The fact that we start out with a homotopy sphere means that these pieces are become metrically closer to a sphere and so are certainly topologically spheres. From our prior observation, this proves the conjecture.
