In the previous lecture, we saw that Ricci flow with surgery ensures that the second homotopy group $\pi_2(M)$ became extinct in finite time (assuming, as stated in the above erratum, that there is no embedded $\Bbb{RP}^2$ with trivial normal bundle). It turns out that the same assertion is true for the third homotopy group, at least in the simply connected case:

Theorem 1. (Finite time extinction of $\pi_3(M)$) Let $t \mapsto (M(t),g(t))$ be a Ricci flow with surgery on compact 3-manifolds with $t \in [0,+\infty)$, with M(0) simply connected. Then for all sufficiently large t, $\pi_3(M(t))$ is trivial (or more precisely, every connected component of M(t) has trivial $\pi_3$).

[Aside: it seems to me that this theorem should also be true if one merely assumes that M(0) contains no embedded copy of $\Bbb{RP}^2$ with trivial bundle, as opposed to M(0) being simply connected, but I will be conservative and only state Theorem 1 with this stronger hypothesis, as this is all that is necessary for proving the Poincaré conjecture.]

Suppose we apply Ricci flow with surgery to a compact simply connected Riemannian 3-manifold (M,g) (which, by Lemma 1 from Lecture 2, has no embedded $\Bbb {RP}^2$ with trivial normal bundle). From the above theorem, as well as Theorem 1 from the previous lecture, we know that all components of M(t) eventually have trivial $\pi_2$ and $\pi_3$ for all sufficiently large t. Also, since M is initially simply connected, we see from Exercise 2 of Lecture 2, as well as Theorem 2.1 of Lecture 2, that all components of M(t) also have trivial $\pi_1$. The finite time extinction result (Theorem 4 from Lecture 2) then follows immediately from Theorem 1 and the following topological result, combined with the following topological observation:

Lemma 1. Let M be a compact non-empty connected 3-manifold. Then it is not possible for $\pi_1(M)$, $\pi_2(M)$, and $\pi_3(M)$ to simultaneously be trivial.

This lemma follows immediately from the Hurewicz theorem, but for sake of self-containedness we give a proof of it here.

There are two known approaches to establishing Theorem 1; one due to Colding and Minicozzi, and one due to Perelman. The former is conceptually simpler, but requires a certain technical concentration-compactness type property for a min-max functional which has only been established recently. This approach will be the focus of this lecture, while the latter approach of Perelman, which has also been rigorously shown to imply finite time extinction, will be the focus of the next lecture.

— A little algebraic topology —

[Aside: My algebraic topology is rather rusty; I haven’t studied it since taking an undergraduate class on the topic at Flinders 17 years ago! There may be some technical subtleties involving, e.g. the difference between simplicial complexes and CW complexes, or between free homotopy and homotopy with fixed base point, which I may not quite have rendered correctly here, though I believe that the broad details of what I wrote below are right, at least. As always, I of course welcome any and all corrections.]

We begin by proving Lemma 1. We need to recall some (very basic) singular homology theory (over the integers ${\Bbb Z}$). Fix a compact manifold M, and let k be a non-negative integer. Recall that a singular k-chain (or k-chain for short) is a formal integer-linear combination of k-dimensional singular simplices $\sigma(\Delta_k)$ in M, where $\Delta_k$ is the standard k-simplex and $\sigma: \Delta_k \to M$ is a continuous map. There is a boundary map $\partial$ taking k-chains to (k-1)-chains, defined by mapping ${}\sigma(\Delta_k)$ to an alternating sum of restrictions of $\sigma$ to the (k-1)-dimensional boundary simplices of $\Delta_k$, and then extending by linearity. A k-chain is said to be a k-cycle if its boundary vanishes, and is a k-boundary if it is the boundary of a (k+1)-chain. One easily verifies that $\partial^2 = 0$, and so every k-boundary is a k-cycle. We say that M has trivial $k^{th}$ homology group $H_k(M)$ if the converse is true, i.e. every k-cycle is a k-boundary.

Our main tool for proving Lemma 1 is

Proposition 1 (Baby Hurewicz theorem). Let M be a triangulated connected compact manifold such that the fundamental groups $\pi_1(M),\ldots,\pi_k(M)$ all vanish for some $k \geq 1$. Then M has trivial $j^{th}$ homology group $H_j(M)$ for every $1 \leq j \leq k$.

Proof. Because of all the vanishing fundamental groups, one can show by induction on j that for any integer $1 \leq j \leq k$, any singular complex in M involving singular simplices of dimension at most j can be continuously deformed to a point (while preserving all boundary relationships between the singular simplices in that complex). As a consequence, every j-cycle, being the combination of singular simplices in a singular complex involving singular simplices of dimension at most j, can be expressed as the boundary of a (j+1)-chain, and the claim follows. $\Box$

Remark 1. The full Hurewicz theorem asserts some further relationships between homotopy groups and homology groups, in particular that under the assumptions of Proposition 1, that the Hurewicz homomorphism from $\pi_{k+1}(M)$ to $H_{k+1}(M)$ is in fact an isomorphism (and, in the k=0 case, that $H_1(M)$ is canonically isomorphic to the abelianisation of $\pi_1(M)$). However, we do not need this slightly more advanced result here. $\diamond$

Now we can quickly prove Lemma 1.

Proof of Lemma 1. Suppose for contradiction that we have a non-empty connected compact 3-manifold M with $\pi_1(M),\pi_2(M),\pi_3(M)$ all trivial. Since M is simply connected, it is orientable (as all loops are contractible, there can be no obstruction to extending an orientation at one point to the rest of the manifold). Also, it is a classical result of Moise that every 3-manifold can be triangulated. Using a consistent orientation on M, we may therefore build a 3-cycle on M consisting of the sum of oriented 3-simplices with disjoint interiors that cover M (i.e. a fundamental class), thus the net multiplicity of this cycle at any point in M is odd. On the other hand, the net multiplicity of any 3-boundary at any point can be seen to necessarily be even. Thus we have found a 3-cycle which is not a 3-boundary, which contradicts Proposition 1. [Aside: one can avoid the use of Moise’s theorem here by working in the category of smooth manifolds, or by using more of the basic theory of singular homology. See comments. Also, the use of orientability can be avoided by working with homologies over ${\Bbb Z}/2{\Bbb Z}$ rather than over ${\Bbb Z}$.] $\Box$

Remark 2. Using (slightly) more advanced tools from algebraic topology, one can in fact say a lot more about the homology and homotopy groups of connected and simply connected compact 3-manifolds M. Firstly, since $\pi_1(M)$ is trivial, one sees from the full Hurewicz theorem that $H_1(M)$ is also trivial. Also, as M is connected, $H_0(M) \equiv {\Bbb Z}$. From orientability (which comes from simple connectedness) and triangularisability we have Poincaré duality, which implies that the cohomology group $H^2(M)$ is trivial and $H^3(M) \equiv {\Bbb Z}$, which by the universal coefficient theorem for cohomology implies that $H_2(M)$ is trivial and $H_3(M) \equiv {\Bbb Z}$. Of course, being 3-dimensional, all higher homology groups vanish, and so M is a homology sphere. On the other hand, by orientability, we can find a map from M to $S^3$ that takes a fundamental class of M to a fundamental class of $S^3$, by taking a small ball in M and contracting everything else to a point; this map is thus an isomorphism on each homology group. Using the relative Hurewicz theorem (and the simply connected nature of M and $S^3$) we conclude that this map is also an isomorphism on each homotopy group, and thus by Whitehead’s theorem, the map is a homotopy equivalence, thus M is a homotopy sphere. Thus, to complete the proof of the Poincaré conjecture, it suffices to show that every compact 3-manifold which is a homotopy sphere is also homeomorphic to a sphere. Unfortunately this observation does not seem to significantly simplify the proof of that conjecture, although it does allow one at least to get the extinction of $\pi_2$ from the previous lecture “for free” in the simply connected case. (Note also that there are homology 3-spheres that are not homeomorphic to the 3-sphere, such as the Poincaré homology sphere; thus homology theory is not sufficient by itself to resolve this conjecture.) $\diamond$

— The Colding-Minicozzi approach to $\pi_3$ extinction —

We now sketch the Colding-Minicozzi approach towards proving Theorem 1. Our discussion here will not be fully rigorous; further details can be found in the original paper of Colding and Minicozzi.

In the previous lecture, we obtained the differential inequality

$\frac{d}{dt} \int_{f(S^2)}\ d\mu \leq - 4\pi - \frac{1}{2} R_{\min} \int_{f(S^2)}\ d\mu$; (1)

for any minimal immersed 2-sphere $f(S^2)$ in a Ricci flow $t \mapsto (M(t),g(t))$. The inequality also holds for the slightly larger class of minimal 2-spheres that are branched immersions rather than just immersions; furthermore, an inspection of the proof reveals that the surface does not actually have to be a local area minimiser, but merely needs to have zero mean curvature (i.e. to be a critical point for the area functional, rather than a local minimum). The inequality (1) was a key ingredient in the proof of finite time extinction of the second homotopy group $\pi_2(M(t))$.

The Colding-Minicozzi approach seeks to exploit the same inequality (1) to also prove finite time extinction of $\pi_3(M(t))$. It is not immediately obvious how to do this, since $\pi_3()$ involves immersed 3-spheres $f(S^3)$ in M, whereas (1) involves immersed 2-spheres $f(S^2)$. However, one can view the 3-sphere as a loop of 2-spheres with fixed base point; indeed if one starts with the cylinder ${}[0,1] \times S^2$ and identifies $\{0,1\} \times S^2 \cup [0,1] \times \{N\}$ to a single point, where N is a single point in $S^2$, one obtains a (topological) 3-sphere. Because of this, any immersed 3-sphere $f(S^3)$ is swept out by a loop $s \mapsto f_s$ of immersed 2-spheres $f_s(S^2)$ for $0 < s < 1$ with fixed base point $f_s(N)=p$, with $f_s$ varying continuously in t for $0 \leq s \leq 1$, and $f_0=f_1\equiv p$ being the trivial map.

Suppose that we have a Ricci flow in which M is connected and $\pi_3(M)$ is non-trivial; then we have at least one immersed 3-sphere $f(S^3)$ which is not contractible to a point. We then define the functional $W_3(t)$ by the min-max formula

$W_3(t) := \inf_f \sup_{0 \leq s \leq 1} \int_{f_s(S^2)}\ d\mu$ (2)

where f ranges over all incontractible immersed 3-spheres, and $\mu$ is the volume element of $f_s(S^2)$ with respect to the restriction of the ambient metric g(t).

It can be shown (see e.g. page 125 of Jost’s book) that $W_3(t)$ is strictly positive; in other words, if the area of each 2-sphere in a loop of immersed 2-spheres is sufficiently small, then the whole loop is contractible to a point.

Suppose for the moment that the infimum in (2) was actually attained, thus there exists an incontractible immersed 3-sphere f whose maximum value of $\int_{f_s(S^2)}\ d\mu$ is exactly $W_3(t)$. Applying mean curvature flow for a short time to reduce the area of any sphere which does not already have vanishing mean curvature, we may assume without loss of generality that the maximum value is only attained when $f_s(S^2)$ has zero mean curvature. (To make this rigorous, one either has to prove a local well-posedness result for , or else to use a cruder version of this flow, for instance deforming f a small amount along a vector field which points in the same direction as the mean curvature. We omit the details.) If we then use (1), we thus (formally, at least) obtain the differential inequality

$\frac{d}{dt} W_3(t) \leq - 4\pi - \frac{1}{2} R_{\min} W_3(t)$ (3)

much as in the previous lecture. Arguing as in that lecture, we obtain a contradiction if the Ricci flow persists without developing singularities for a sufficiently long time.

It should be possible that a similar analysis can also be performed when the infimum in (2) is not actually attained, in which case one has a minimising sequence of loops of 2-spheres whose width approaches $W_3(t)$. This sequence can be analysed by Sacks-Uhlenbeck theory (together with some later analysis of bubbling due to Siu and Yau) and a finite number of minimal 2-spheres extracted as a certain “limit” of the above sequence, although as in the previous lecture, these 2-spheres need only branched immersions rather than immersions. From this one can establish (3) (in a suitably weak sense) in the general case in which the infimum in (2) is not necessarily attained), assuming that one can show that all the 2-spheres with area close to $W_3(t)$ that appear in a loop in the minimising sequence are close to the union of the limiting minimal 2-spheres in a certain technical sense; see the paper of Colding and Minicozzi (and the references therein) for details. This property (which is a sort of concentration-compactness type property for the min-max functional (2), which is a partial substitute for the failure of the Palais-Smale condition for this functional) was recently established, again by Colding and Minicozzi, using the theory of harmonic maps.

There is also the issue of how to deal with surgery. This follows the same lines that were briefly (and incompletely) sketched out in the previous lecture. Namely, one first observes that after finitely many surgeries, all remaining surgeries are along 2-spheres that are homotopically trivial (i.e. contractible to a point). Because of this, one can show that any incontractible 3-sphere will, after surgery, lead to at least one incontractible 3-sphere on one of the components of the post-surgery manifold. Furthermore, it turns out that there is a distance-decreasing property of surgery which can be used to show that $W_3(t)$ does not increase at any surgery time. We will discuss these sorts of issues in a bit more detail in the next lecture, when we turn to the Perelman approach to $\pi_3$ extinction.

[Update, April 16: Simplicial homology replaced with singular homology.]