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Worked solution: Perelman's Ricci flow with surgery (2002–2003), condensed

Step 7 of 9: Ricci flow with surgery through neck-pinches
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In plain words

Like snipping a narrow hourglass-shaped balloon at its waist and gluing a rounded cap onto each cut end, Perelman's surgery cuts the manifold along the thin neck that a singularity is about to pinch, replaces it with two smooth caps, and restarts the flow.

Because the canonical-neighbourhood theorem guarantees the pinching region really does look like a neck, this surgery can be performed with enough precision that all the earlier control — curvature pinching, no local collapsing — survives the operation and can be reused afterward.

S2×[−ϵ−1,ϵ−1] ⟶ B3⊔B3(M ⇝ M1 # M2)S^2\times[-\epsilon^{-1},\epsilon^{-1}]\ \longrightarrow\ B^3\sqcup B^3\qquad(M\ \leadsto\ M_1\,\#\,M_2)
A 2D surface analogy for an ϵ\epsilon-neck: a narrow cylindrical waist connecting two wider regions, where surgery cuts along a cross-sectional sphere and caps both ends
Three-dimensional hyperboloid of one sheet with a narrow circular waist flaring outward at top and bottom, illustrating a neck-pinch geometry where Ricci flow with surgery cuts the neck and caps the two sides.
Detailed analysis

In his second preprint (arXiv:math/0303109), Perelman constructs Ricci flow with surgery: at a singular time TT, connected components entirely covered by high-curvature canonical neighbourhoods are discarded (their topology is already known from the classification of the previous step), while each remaining "horn" ending at a singular neck is cut along a small cross-sectional sphere S2S^2 and capped with smooth positively curved balls B3B^3, and the flow restarts.

Choosing the surgery threshold δ(t)\delta(t) small enough preserves curvature pinching, κ\kappa-noncollapsing, and the canonical-neighbourhood bounds across each surgery; and because every surgery removes at least a fixed amount of volume (∼h3\sim h^3), only finitely many surgeries can occur in any finite time interval.

With surgeries controlled and finite in number, the flow can be continued all the way to any prescribed time — or until the manifold disappears entirely — which sets up the finite-extinction argument of the next step.

Terms in this step
Neck-pinch (ϵ\epsilon-neck)
A region shaped like a thin cylinder S2×IS^2\times I that is about to separate the manifold into two pieces as its cross-sectional radius shrinks toward zero.
Surgery
The topological operation of cutting a manifold along a small sphere and gluing in smooth caps, used to remove a developing singularity while continuing the flow on what remains.
Knowledge used in this step
Smooth manifoldsThe Poincaré conjecture
Common mistake. Topological 22-sphere surgery can split a manifold into connected-sum summands M≅M1 # M2M\cong M_1\,\#\,M_2, so one must check both that surgeries do not accumulate in finite time and that the topology of the original MM can be reconstructed from the discarded pieces.