Nontrivial h-cobordism

I'm learning the h-cobordism theorem as I want to use it in a talk. I'd like to be able to give an example of an h-cobordism that isn't a cylinder, if possible by drawing a picture. What is the simplest such example?

Solution 1:

I learned about such an example recently through a conversation with Kirby:

Given an h-cobordism $W:M_1\to M_0$ between simply-connected 4-manifolds, it is actually a product outside of an "Akbulut cork".

Visual elaboration: There is a contractible 4-manifold $C_1$ with boundary (called the cork) sitting inside of $M_1$, and there is its involution $C_0$ $(\approx C_1)$ sitting inside of $M_0$. Now it turns out that there exists an involution of $\partial C_1$ which doesn't extend to a diffeomorphism of $C_1$. Furthermore, we have a cobordism $A:C_1\to C_0$ which is diffeomorphic to a 5-ball (but not relative-boundary of course).

So the picture is that any such $W$ can be viewed as $(M_1-C_1)\times[0,1]$ outside of $C_1$, and looks like $A$ inside of $C_1$. Amazing!