The structure of solutions at the first singular time

Consider a smooth solution $g_{ij}(t)$ to the Ricci flow on $M\times [0,T),$ where $M$ is a closed oriented 3-manifold, $T<\infty.$ Assume that curvature of $g_{ij}(t)$ does not stay bounded as $t\to T.$ Recall that we have a pinching estimate $Rm\ge -\phi(R)R$ for some function $\phi$ decreasing to zero at infinity [H 4,§4], and that the solution is $\kappa$-noncollapsed on the scales $\le r$ for some $\kappa>0, r>0$ [I, §4].Then by Theorem I.12.1 and the conclusions of 1.5 we can find $r=r(\epsilon )>0,$ such that each point $(x,t)$ with $R(x,t)\ge r^{-2}$ satisfies the estimates (1.3) and has a neighborhood, which is either an $\epsilon$-neck, or an $\epsilon$-cap, or a closed positively curved manifold. In the latter case the solution becomes extinct at time $T,$ so we don't need to consider it any more.

If this case does not occur, then let $\Omega$ denote the set of all points in $M,$ where curvature stays bounded as $t\to T.$ The estimates (1.3) imply that $\Omega$ is open and that $R(x,t)\to \infty$ as $t\to T$ for each $x\in M\backslash\Omega .$ If $\Omega$ is empty, then the solution becomes extinct at time $T$ and it is entirely covered by $\epsilon$-necks and caps shortly before that time, so it is easy to see that $M$ is diffeomorphic to either $\mathbb{S}^3$, or $\mathbb{RP}^3$, or $\mathbb{S}^2\times\mathbb{S}^1$, or $\mathbb{RP}^3\ \sharp\ \mathbb{RP}^3.$

Otherwise, if $\Omega$ is not empty, we may (using the local derivative estimates due to W.-X.Shi, see [H 2,§ 13]) consider a smooth metric $\bar{g}_{ij}$ on $\Omega ,$ which is the limit of $g_{ij}(t)$ as $t\to T.$ Let $\Omega _\rho$ for some $\rho<r$ denotes the set of points $x\in\Omega ,$ where the scalar curvature $\bar{R}(x)\le \rho^{-2}.$ We claim that $\Omega _\rho$ is compact. Indeed, if $\bar{R}(x)\le \rho^{-2},$ then we can estimate the scalar curvature $R(x,t)$ on $[T-\eta^{-1}\rho^2,T)$ using (1.3), and for earlier times by compactness, so $x$ is contained in $\Omega$ with a ball of definite size, depending on $\rho.$

Now take any $\epsilon$-neck in $(\Omega ,\bar{g}_{ij})$ and consider a point $x$ on one of its boundary components. If $x\in \Omega \backslash\Omega _{\rho} ,$ then there is either an $\epsilon$-cap or an $\epsilon$-neck, adjacent to the initial $\epsilon$-neck. In the latter case we can take a point on the boundary of the second $\epsilon$-neck and continue. This procedure can either terminate when we reach a point in $\Omega _{\rho}$ or an $\epsilon$-cap, or go on indefinitely, producing an $\epsilon$-horn. The same procedure can be repeated for the other boundary component of the initial $\epsilon$-neck. Therefore, taking into account that $\Omega$ has no compact components, we conclude that each $\epsilon$-neck of $(\Omega ,\bar{g}_{ij})$ is contained in a subset of $\Omega$ of one of the following types:

(a) An $\epsilon$-tube with boundary components in $\Omega _{\rho} ,$ or

(b) An $\epsilon$-cap with boundary in $\Omega _{\rho} ,$ or

(c) An $\epsilon$-horn with boundary in $\Omega _{\rho} ,$ or

(d) A capped $\epsilon$-horn, or

(e) A double $\epsilon$-horn.

Clearly, each $\epsilon$-cap, disjoint from $\Omega _{\rho} ,$ is also contained in one of the subsets above. It is also clear that there is a definite lower bound (depending on $\rho$) for the volume of subsets of types (a),(b),(c), so there can be only finite number of them. Thus we can conclude that there is only a finite number of components of $\Omega ,$ containing points of $\Omega _{\rho} ,$ and every such component has a finite number of ends, each being an $\epsilon$-horn. On the other hand, every component of $\Omega ,$ containing no points of $\Omega _{\rho} ,$ is either a capped $\epsilon$-horn, or a double $\epsilon$-horn.

Now, by looking at our solution for times $t$ just before $T,$ it is easy to see that the topology of $M$ can be reconstructed as follows: take the components $\Omega _j, 1\le j\le i$ of $\Omega$ which contain points of $\Omega _{\rho} ,$ truncate their $\epsilon$-horns, and glue to the boundary components of truncated $\Omega _j$ a collection of tubes $\mathbb{S}^2\times\mathbb{I}$ and caps $\mathbb{B}^3$ or $\mathbb{RP}^3\backslash\mathbb{B}^3.$ Thus, $M$ is diffeomorphic to a connected sum of $\bar{\Omega }_j , 1\le j\le i,$ with a finite number of $\mathbb{S}^2\times\mathbb{S}^1$ (which correspond to gluing a tube to two boundary components of the same $\Omega _j$), and a finite number of $\mathbb{RP}^3;$ here $\bar{\Omega }_j$ denotes $\Omega _j$ with each $\epsilon$-horn one point compactified.