This is the first in a series of posts where some steps through understanding the $h$-cobordism theorem will be discussed, as I go along. As writing down everything is time-consuming, I will only be mentioning some technicalities that interest me, but will provide the main ideas of other parts as well, so that these posts still have a form of progress through the theorem.

Why $h$-cobordism? I was first interested in the topic when I found out that it was used by Smale to prove the $n$-dimensional topological Poincare conjecture for all $n \geq 5$. It is interesting that cases $n=3,4$ were proved much later using sophisticated techniques (I will surely remark on where Smale’s argument breaks for $n=3,4$ in later posts). The situation is contrary to our idea that working in higher dimensions seem to be harder. This got me more interested in low-dimensional topology as well.

Also I got interested in the $h$-cobordism as I intended to understand Floer homology. Floer homology is said to be (and I still don’t know why) the infinite-dimensional analogue of Morse theory. The $h$-cobordism theorem makes a fair use of Morse theory. So for me, $h$-cobordism has been appealing on its own, and also in line with further interests.

I will mainly be going through Milnor’s Lectures on the $h$-cobordism theorem, but not sure exactly for now; hopefully I will later provide a suggested reading plan for interested readers, having done some trial-and-error.

I will end this post with the statement of the theorem. Let $W$ be a compact smooth manifold having two boundary components $V$ and $V’$ such that $V$ and $V’$ are both deformation retracts of $W$. Then $W$ is said to be an $h$-cobordism between $V$ and $V’$. The $h$-cobordism theorem states:

If $V$ and (hence) $V’$ are simply connected and of dimension greater than 4, then $W$ is diffeomorphic to $V \times [0,1]$, and (consequently) $V$ is diffeomorphic to $V’$.