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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?

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In the smooth or the topological category? In the smooth case I think you can take an exotic sphere and remove to disjoint discs – mland Dec 31 '12 at 16:24
@mland: The h-cobordism theorem holds smoothly; removing two disks from an exotic sphere yields a manifold diffeomorphic to $S^n \times [0,1]$. You just cannot arrange that the diffeomorphism is the identity on both ends, so you can't conclude your original manifold is diffeomorphic to a standard sphere. – Fabian Lenhardt Dec 31 '12 at 16:32
Of course. I think I mixed things up. But I had in mind that in the smooth case you can give explicit examples of nontrivial h-cobordisms without talking about handle decompositions. But obviously that was not how to do it :) thanks for clarification. – mland Dec 31 '12 at 16:47
I'd prefer answers in the topological category for ease of explanation, but I'm open to answers in the smooth category if they are simpler. – tharris Dec 31 '12 at 17:56
I would really like to know an example of a non-trivial h-cobordism, myself. I have not been able to come up with one on my own. To have an example of any such h-cobordism would already be nice - even if it's not a simple example... – Sam Jan 26 '13 at 21:52
up vote 3 down vote accepted

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!

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