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

Frequently asked questions

Why are Perelman's papers cited as arXiv preprints rather than journal articles?

Perelman posted all three instalments (arXiv:math/0211159, arXiv:math/0303109, arXiv:math/0307245) directly to arXiv in 2002-2003 and never submitted them to a peer-reviewed journal; detailed book-length verifications filling in the technical details were subsequently published by Kleiner-Lott, Morgan-Tian and Cao-Zhu.

Source

Grigori Perelman, 2003

The entropy formula for the Ricci flow and its geometric applications

arXiv:math/0211159v1

Preprint

Condensed summary

Step 9 of 9: Conclusion: reassembling

M^3 \cong S^3

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In plain words

Running the whole movie in reverse from the moment of extinction, every piece that was ever discarded by surgery turns out to be one of a short list of simple building blocks; gluing them back together along the cuts reconstructs the original manifold as a connected sum of these blocks.

The assumption $\pi_1(M^3)=0$ rules out every building block except the round 3-sphere, so the reconstructed connected sum can only be a sum of spheres — which is again just a sphere — completing the proof.

M^3\cong\Big(\#_{i} S^3/\Gamma_i\Big)\,\#\,\Big(\#_{j}(S^2\times S^1)\Big)\ \ \xRightarrow{\pi_1(M^3)=0}\ \ M^3\cong S^3

The round sphere (shown in 2D analogy as

S^2

): after surgery and finite extinction, every piece of a simply connected closed

M^3

is a

3

-sphere summand

Detailed analysis

Working backwards from $T_{\mathrm{ext}}$ through the finitely many surgeries, every discarded high-curvature piece is diffeomorphic to a spherical space form $S^3/\Gamma_i$ , to $S^2\times S^1$ , or to $\mathbb{RP}^3\,\#\,\mathbb{RP}^3$ , and undoing the $2$ -sphere surgeries reassembles $M^3$ as a connected sum of such pieces.

When $\pi_1(M^3)=0$ , the van Kampen theorem forces every $\Gamma_i$ to be trivial (otherwise $\pi_1$ would contain a nontrivial subgroup) and rules out any $S^2\times S^1$ summand (which has infinite $\pi_1$ ), leaving a connected sum of copies of $S^3$ .

A connected sum of round 3-spheres is diffeomorphic — and hence homeomorphic — to $S^3$ , completing the proof that every simply connected closed 3-manifold is homeomorphic to $S^3$ : the Poincaré conjecture.

Terms in this step

Spherical space form ( $S^3/\Gamma$ ): A manifold obtained as the quotient of the round 3-sphere by a finite group $\Gamma$ acting freely by isometries; it is trivial (equal to $S^3$ itself) exactly when $\Gamma$ is the trivial group.
Connected sum ( $\#$ ): The operation of cutting a small ball out of each of two manifolds and gluing the resulting boundary spheres together; it is the topological operation that surgery keeps track of.

Knowledge used in this step

The fundamental group Smooth manifolds The Poincaré conjecture