Ricci flow with surgery

Plain Ricci flow has a fatal flaw if you want to use it to classify shapes: it crashes. Necks pinch off, curvature blows up, and at the singular time the flow simply stops existing. The fix, due to Hamilton and made to work in full by Perelman, is brutally direct: when a region is about to become singular, pause the flow, cut out the bad piece with surgical scissors, sew in standard smooth caps, throw away any tiny leftover pieces, and restart the flow. Do this again and again, and you get a process that runs for all time, punctuated by discrete surgeries, that you can actually use to read off topology.

Here is the procedure in plain steps. (1) Run Ricci flow until just before a singular time T. (2) By Perelman's canonical-neighborhood theorem, every region whose curvature has grown beyond a chosen huge threshold looks, after rescaling, like one of a short list of models — most importantly a thin epsilon-neck (close to a round cylinder S^2 x R). (3) Cut the manifold along the middle spheres of those necks, removing the high-curvature parts. (4) Glue a standard round cap (a smooth hemisphere-like metric) onto each cut boundary, lowering the maximum curvature back to a controlled level. (5) Discard any closed components that have become recognizable standard pieces (round spheres, spherical space forms, S^2 x S^1, etc.) — these are understood and removed from further study. (6) Restart Ricci flow from the post-surgery metric and repeat. The deep content is quantitative control: Perelman proved one can choose the surgery parameters so that only finitely many surgeries happen on any finite time interval and the process does not run away.

This is the actual engine of the proofs of the Poincaré and geometrization conjectures: you track which topological pieces are removed at each surgery, and in the limit the original manifold is reconstructed by gluing back the standard pieces, which is exactly the geometric decomposition the conjectures predict. The crucial honesty point — and a very common misconception — is that Perelman did not prove the conjectures with smooth Ricci flow alone. The flow by itself develops singularities and dies; the entire achievement is that surgery can be carried out with enough control to extract the topology, and that the surgeries do not accumulate. Saying 'Ricci flow proves Poincaré' without the word surgery skips the hardest part of the argument.