Let $f:M\_0\rightarrow M\_1$ be a diffeomorphism between two compact $n$-dimensional manifolds. Let $W^{n+1}$ be an $h$-cobordism between $M\_0$ and $M\_1$. Assume that the cobordism has no torsion and its dimension is high enough so that, by the $s$-cobordism theorem, there is a diffeomorphism $F:M\_0\times I\rightarrow W$. Can $F$ be taken so that $F|\_{M\_0\times 0}=id\_{M\_0}$ and $F|\_{M\_0\times1}=f$?


Thanks in advance!



Diffeomorphism coming from the s-cobordism theorem
The question is unclear, mainly because it's not clear what "can be taken ... up to isotopy" means. Under the only interpretation that seems at all reasonable, the answer is pretty trivially "no".


I'll assume that the given $F$ is assumed to take $M\_0\times 1$ to $M\_0$ by the identity, and that the question is whether there is necessarily another $F$ that is the same on $M\_0\times 0$ and is $f$ on $M\_0\times 1$. The answer is no. For example, let $M\_0=M\_1=S^n$ and let $W$ be $S^n\times I$, with $F$ the identity and $f$ orientation-reversing.


Am I misunderstanding?


