Reading Off the Topology

The flow with surgery of Book 12B runs for all time, and at every surgery it records what it did: it cut along a sphere, or discarded a component it recognised. This chapter shows how to read the topology of the original manifold off that record. The answer is cleanest when the flow becomes extinct in finite time, so that nothing is left. Then the manifold is a connected sum of spherical space forms and copies of $S^2\times S^1$, and if it is simply connected, it is $S^3$. That reduces the Poincaré conjecture to one analytic statement, finite extinction for simply connected manifolds, which is 12C.2 Finite Extinction.

By the end of this chapter you will be able to:

• list the pieces that a flow with surgery can discard, and say why each is on the list;
• prove that a manifold whose flow with surgery becomes extinct is a connected sum of spherical space forms and copies of $S^2\times S^1$;
• deduce that a simply connected manifold whose flow becomes extinct is $S^3$;
• state Perelman's three-way classification of closed 3-manifolds by the behaviour of the flow, and the elliptization conjecture as a consequence.

1.1A ledger

1.2What the flow can discard

Collect the components that a Ricci flow with surgery on a closed oriented 3-manifold can remove (12B.4 Surgery):

1. Components covered by necks and caps at a singular time, when $\Omega$ is empty or at a surgery: $S^3$, $\mathbb{RP}^3$, $S^2\times S^1$ and $\mathbb{RP}^3\#\mathbb{RP}^3$ (12B.4 Surgery, rehearsal exercise).
2. Nearly round components, $\varepsilon$-close to a quotient of the round sphere, which are declared extinct: spherical space forms $S^3/\Gamma$.
3. Closed positively curved components that become extinct at a singular time; their canonical neighbourhoods are of type (c) or (d) of 12B.2 The Structure of κ-Solutions, so they are $S^3$, $\mathbb{RP}^3$ or quotients of the round sphere.
4. Components with no point of low curvature at a surgery time (capped horns and double horns). These are removed, but they are not closed manifolds; their effect on the topology is accounted for by the caps glued in and the connected sum formula below.

Every closed piece on the list is a spherical space form ($S^3$ and $\mathbb{RP}^3$ included), $S^2\times S^1$, or a connected sum of these ($\mathbb{RP}^3\#\mathbb{RP}^3$). Perelman's §3 computation makes this precise for one singular time: $M$ is diffeomorphic to the connected sum of the surviving components, with their horns closed off, together with finitely many copies of $S^2\times S^1$ and of $\mathbb{RP}^3$ (12B.4 Surgery).

1.3Extinction gives a connected sum

Conversely, every such connected sum carries a metric of positive scalar curvature (Gromov–Lawson and Schoen–Yau, 9B.6 Scalar Curvature and Topology), and for such a metric the flow becomes extinct (12B.5 Ricci Flow with Surgery for All Time). So the manifolds of the theorem are exactly those whose flow with surgery can become extinct, and exactly those with a metric of positive scalar curvature. Figure 1.1 sets out the chain of reasoning that leads to the Poincaré conjecture.

1.4The simply connected case

So the Poincaré conjecture follows from one statement: on a closed simply connected 3-manifold, the flow with surgery becomes extinct in finite time. Perelman proved more in his third preprint: the flow becomes extinct, for every initial metric, whenever the prime decomposition of $M$ has no aspherical factors (no factor whose universal cover is contractible). 12C.2 Finite Extinction gives the proof, in the version of Colding and Minicozzi.

1.5The three types

Perelman's second preprint ends with a classification of closed 3-manifolds by what the flow with surgery does (II, §8.2), expressed through $\lambda$, the bottom eigenvalue of $-4\Delta + R$ (12A.2 Ricci Flow as a Gradient Flow):

type	flow with surgery	topology
$M$ has a metric with $\lambda > 0$	becomes extinct (for a suitable cutoff)	connected sum of spherical space forms and copies of $S^2\times S^1$
$\sup\lambda V^{2/3} = 0$ over all metrics	—	a graph manifold
$\sup\lambda V^{2/3} < 0$	runs forever, with a hyperbolic thick part after splitting off summands of the first type, a hyperbolic piece embedded with graph-manifold complement (II, §8.2(c))

The second and third rows are the subject of 12C.4 Geometrization. Perelman remarks that these results are exact analogues, at least in the nonpositive case, of conjectures of Michael Anderson about the sigma constant. The first row is the theorem above, together with the fact that $\lambda > 0$ forces extinction.

Elliptization. A closed 3-manifold with finite fundamental group has no aspherical prime factors: the fundamental group of each prime factor injects into that of the connected sum, and an aspherical closed 3-manifold has infinite fundamental group. So by Perelman's finite extinction theorem its flow becomes extinct, and by Theorem 1.1 and a short argument with free products, it is a spherical space form. Perelman notes that this gives the elliptization conjecture, with the long-time analysis of the flow not needed at all (Exercise 1.6). The Poincaré conjecture is the case of the trivial group.

1.6History

Perelman's second preprint (2003) described the topology at a singular time in §3 and classified manifolds into three types in §8.2. His third preprint (July 2003) began by recalling that the first type was characterised by admitting metrics whose flow with surgery becomes extinct, and asked whether extinction happens for every metric on such a manifold; it proved that it does. Perelman remarked there that this, with §§1–5 of the second preprint, gives the elliptization conjecture directly, and that for a homotopy sphere one can even avoid the Kneser finiteness theorem by following homotopy equivalences through each surgery.

1.7Exercises

Solution

$\pi_1(S^2\times S^1) = \pi_1(S^2)\times\pi_1(S^1) = \mathbb{Z}$. $S^3 \to S^3/\Gamma$ is a covering with deck group $\Gamma$ and $S^3$ is simply connected, so $\pi_1(S^3/\Gamma) \cong \Gamma \neq 1$. $\pi_1(\mathbb{RP}^3\#\mathbb{RP}^3) = \mathbb{Z}/2*\mathbb{Z}/2$, the infinite dihedral group: the product $ab$ of the two generators has infinite order.

Solution

The homomorphism is surjective, so $G$ is a quotient of $G * H = 1$, hence trivial; similarly $H$.

Solution

$S^3$ minus an open ball is a closed ball, by Alexander's theorem (10A.3 The Prime Decomposition) or directly for the round sphere. So $S^3\#N$ is $N$ minus a ball with a ball glued back along the boundary sphere, which is $N$.

Solution

(a) As in the theorem, each $P_i$ is a spherical space form or $S^2\times S^1$. (b) $\pi_1(M)$ is the free product of the $\pi_1(P_i)$. A free product of two nontrivial groups is infinite (it contains elements $ab$ of infinite order), and $\mathbb{Z}$ is infinite, so at most one factor is nontrivial and it is finite. (c) The other summands are $S^3$, which can be dropped (Exercise 1.5), so $M \cong S^3/\Gamma$.

Solution

(a), (b), (d): yes; their prime factors are spherical space forms and $S^2\times S^1$, none aspherical. (c) No: $T^3$ is aspherical (its universal cover is $\mathbb{R}^3$); a flat metric is static and never becomes extinct. (e) No: aspherical, and the flow of a hyperbolic metric expands forever. (f) It is $T^3$ (Exercise 1.5), so no.