You are currently browsing the tag archive for the ‘homotopy group’ tag.

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.