H-cobordisms, Manifolds and Poincare

 The h-cobordism theorem is essentially the following result: Let $M$ be a simply connected $N-$cobordism (with $N\geq 6$) between $V_{0}^{N-1}$ and $V_{1}^{N-1}$. Then, $M\xrightarrow{\;\cong\;}V_{0}^{N-1}\times [0, 1]$. Then, if $M$ is a contractible manifold, one has $M\xrightarrow{\;\cong\;}\mathbb{D}^{N}$. We can prove this as follows: Let $\mathfrak{G}$ be an embedding of $\mathbb{D}^{N}$ into $M$ and identify the interior $\mathrm{Int}(\mathbb{D})$, for which $M-\mathfrak{G}\left(\mathrm{Int}(\mathbb{D}) \right)$ is a cobordism $\partial M\Longleftrightarrow \mathbb{S}^{N-1}$. If we piece these sections back, we would have $M$ from $\mathbb{D}^{N-1}$ and the cobordism $M-\mathfrak{G}\left(\mathrm{Int}\left(\mathbb{D} \right) \right)\equiv \mathcal{B}$. The following pushout diagram shows this decomposition:

Here, $\mathcal{B}$ is simply connected (due to homotopy cofiber sequence $\mathbb{S}^{N-1}\to \mathcal{B}\to M$), and therefore, one can has $\mathbb{S}^{N-1}\xrightarrow{\;H\cong\;}\mathcal{B}$, where $A\xrightarrow{\;H\cong\;}B$ denotes homotopic equivalence. Now, one can use the h-cobordism result from theorem to identify that $\mathcal{B}\xrightarrow{\;\cong\;} \mathbb{S}^{N-1}\times [0, 1]$. Gluing $\partial \mathbb{D}$ back, one sees that $M\xrightarrow{\;\cong\;}\mathbb{D}^{N-1}$, completing the proof. Then, we note the follows: Given a map $\mathfrak{G}$ between $\mathbb{S}^{N-1}$, one can identify that there exists a homeomorphism $\mathfrak{F}$ between $\mathbb{D}^{N}$:

Then, the Poincare conjecture can be rewritten into the follows: An $N-$manifold that is homotopy equivalent to the $\mathbb{S}^{N}$ is also homeomorphic to $\mathbb{S}^{N}$. As you can guess at this point, the proof of this is somewhat straightforward in terms of the decomposition of the primes $\mathbb{D}^{N}_{\pm }$ and $\mathbb{S}^{N-1}_{\pm }$ under the assumption of the h-cobordism theorem. The proof is as follows. The first diagram shows the decomposition of $M=\mathbb{D}_{0}^{N}\sqcup \mathcal{B}\sqcup \mathbb{D}_{1}^{N}$ by identifying the boundaries of the disks and $M-\mathfrak{G}\left(\mathrm{Int}\left(\mathbb{D}_{0}^{N}\sqcup \mathbb{D}_{1}^{N} \right) \right)\equiv \mathcal{B}$. Next, using the Alexandrov trick to induce $\mathfrak{F}$ on $\mathbb{D}^{N}$ from $\mathfrak{G}$ homeomorphism induced on $\mathbb{S}$, we would get the second diagram below:

From the second diagram where we used the Alexandrov trick, we see that there is a homeomorphism between $\mathbb{S}^{N}$ and $M$ by identifying the maps of the $N-$disks. However, in using the h-cobordism theorem, one has to be sure that the inclusion maps \textit{indeed} have the homotopy equivalence nature. This can be found as a lemma: 

Lemma: If $M$ is homotpic to $\mathbb{S}^{N}$ and $\mathcal{B}$ is a cobordism between $\mathbb{S}_{0}^{N-1}$ and $\mathbb{S}_{1}^{N-1}$ as obtained from the subtraction of the $N-$disk images $M-\mathfrak{G}\left(\mathrm{Int}\left(\mathbb{D}_{0}^{N}\sqcup \mathbb{D}_{1}^{N} \right) \right)$, then $\mathbb{S}_{0}^{N-1}\hookrightarrow \mathcal{B}$ is a homotopy equivalence.

Due to the above lemma, one can use the h-cobordism theorem to show that there is a homeomorphism between $\mathbb{S}^{N}$ and $M$, concluding our proof. This proof works (with several subtleties) in $N\geq 6$. This discussion is taken from my notes. However, the Ricci flow approach is much more interesting, since it has a lot to do with the finite-time existence of singularities. Based on the nature of prime decomposition, the works of Hamilton and others, and most importantly Perelman, showed that one can do surgery and continue Ricci flow. To illustrate this, consider the neckpinch characterized by the nature of a pullback being a shrinking cylinder soliton. To prove Poincare from Ricci flow, one has to show the finite time existence of the connected sum decomposition, which forms the basis of Perelman's work.