[proofplan] We reduce the statement to Perelman's finite-extinction theorem for closed three-manifolds with finite fundamental group. The only topological input needed is that a simply connected manifold has one-element, hence finite, fundamental group. Perelman's construction of surgery parameters then supplies a Ricci flow with surgery to which the finite-extinction theorem applies, giving a finite time after which the evolving manifold is empty. [/proofplan] [step:Declare the surgical Ricci flow and the meaning of extinction] Let $g_0$ denote the given smooth Riemannian metric on the closed simply connected three-manifold $M$. Let $(I_j)_{j\in J}$ denote the collection of maximal smooth time intervals of the surgical flow, where $J$ is the finite or countable index set of smooth pieces. For each $j\in J$, let $\mathcal{M}_j$ be the smooth time-slice map \begin{align*} \mathcal{M}_j: I_j \to \{\text{closed smooth three-manifolds with Riemannian metrics}\}. \end{align*} For $t\in I_j$, this map is given by \begin{align*} \mathcal{M}_j(t)=(M_t,g(t)). \end{align*} Here $M_t$ is the time-slice manifold of the surgical flow; if the time-slice is denoted $M(t)$ elsewhere, then $M_t=M(t)$ by definition. Define $\mathcal{M}$ to be the full surgical flow datum consisting of the family $(\mathcal{M}_j)_{j\in J}$ together with Perelman's transition rule at surgery times. The flow $\mathcal{M}$ is the Ricci flow with surgery starting from $(M,g_0)$ and constructed with surgery parameters, cutoff scales, canonical-neighborhood assumptions, pinching estimates, and noncollapsing controls chosen according to Perelman's construction. Thus, for every $j\in J$ and every $t\in I_j$, $\mathcal{M}_j(t)=(M_t,g(t))$ is a smooth solution of the Ricci flow equation \begin{align*} \partial_t g(t)=-2\operatorname{Ric}_{g(t)}, \end{align*} and at surgery times the flow is continued by Perelman's prescribed cutoff and cap replacement procedure. We say that $\mathcal{M}$ becomes extinct in finite time if there exists $T<\infty$ such that $M_t=\varnothing$ for every $t\geq T$. [/step] [step:Use simple connectedness to verify the topological hypothesis] The fundamental group of $M$ based at any point $p\in M$ has exactly one element because $M$ is simply connected. Therefore \begin{align*} \pi_1(M,p) = \{e\}, \end{align*} where $e$ denotes the identity element of the fundamental group. In particular $\pi_1(M,p)$ is finite. [guided] The external Ricci-flow input we will use is a finite-extinction theorem whose topological hypothesis is finiteness of the fundamental group. We therefore isolate exactly where simple connectedness enters. Choose a base point $p\in M$. Since $M$ is simply connected, every loop in $M$ based at $p$ is homotopic to the constant loop at $p$. Hence the fundamental group contains only the identity homotopy class: \begin{align*} \pi_1(M,p)=\{e\}. \end{align*} A one-element group is finite. Thus the simply connected hypothesis gives precisely the finite-fundamental-group condition required by Perelman's finite-extinction theorem. [/guided] [/step] [step:Apply Perelman's finite-extinction theorem with the verified hypotheses] We apply Perelman's finite-extinction theorem in its finite-fundamental-group form, cited here as an external result not yet represented by a separate Androma theorem page: if $N$ is a closed three-manifold, $h_0$ is a smooth Riemannian metric on $N$, $\pi_1(N,q)$ is finite for some base point $q\in N$, and the Ricci flow with surgery from $(N,h_0)$ is constructed with Perelman's surgery parameters, cutoff scales, canonical-neighbourhood assumptions, pinching estimates, and noncollapsing controls, then the surgical flow becomes extinct in finite time. These hypotheses hold here with $N=M$, $h_0=g_0$, and $q=p$: $M$ is closed and three-dimensional by assumption, $g_0$ is smooth by assumption, $\pi_1(M,p)$ is finite by the preceding step, and the surgery parameters and accompanying controls are those prescribed by Perelman's construction in the theorem statement. The theorem therefore gives a time $T<\infty$ such that the time-$t$ manifold of the surgical flow is empty for every $t\geq T$: \begin{align*} M_t=\varnothing \qquad \text{for all } t\geq T. \end{align*} This is exactly finite-time extinction of the Ricci flow with surgery starting from $(M,g_0)$. [guided] The only deep input in the proof is Perelman's finite-extinction theorem in the finite-fundamental-group form. We cite it as an external result not yet represented by a separate Androma theorem page, rather than linking to the present simply connected corollary. Its hypotheses are: a closed three-manifold $N$, a smooth Riemannian metric $h_0$ on $N$, a base point $q\in N$ for which $\pi_1(N,q)$ is finite, and a Ricci flow with surgery from $(N,h_0)$ constructed using Perelman's surgery parameters, cutoff scales, canonical-neighbourhood assumptions, pinching estimates, and noncollapsing controls. Its conclusion is finite-time extinction of that surgical flow. We now match these hypotheses to the present situation. Take \begin{align*} N=M, \qquad h_0=g_0, \qquad q=p. \end{align*} The manifold $M$ is closed and three-dimensional by the theorem statement. The metric $g_0$ is smooth by construction in the first step. The fundamental group $\pi_1(M,p)$ is finite by the previous step, where simple connectedness gave $\pi_1(M,p)=\{e\}$. Finally, the surgical flow $\mathcal{M}$ was defined using precisely Perelman's surgery parameters and accompanying geometric controls. Therefore every hypothesis of the finite-fundamental-group extinction theorem is satisfied. Perelman's theorem gives a finite time $T<\infty$ such that the time-slice manifold is empty after that time: \begin{align*} M_t=\varnothing \qquad \text{for all } t\geq T. \end{align*} This statement is exactly the definition of finite-time extinction for the Ricci flow with surgery $\mathcal{M}$ starting from $(M,g_0)$. [/guided] [/step]