Let $M$ be an oriented smooth $n$-manifold with non-empty boundary $\partial M$ where $n\geq 5$, and let $\Sigma\in \Theta_n$ be a homotopy sphere of dimension $n$. Does there exist a diffeomophism $M \cong M\# \Sigma$ by "sliding the exotic disk out to the boundary". We can assume that $M$ is simply connected and compact if needed.
I have tried to use the $h$-cobordism theorem but with no success. Any help would be appreciated.
Best Answer
Using the reduction to the problem of if $(\partial(M) \times [0,1]) \sharp \Sigma \cong \partial(M) \times [0,1]$, we see that this is true if, $(\partial(M) \times [0,1]) \sharp \Sigma$ is an s-cobordism. It is certainly an h-cobordism since connect summing with a sphere does not change the homotopy type. But a similar analysis shows this is an s-cobordism as well:
For $W$ an h-cobordism, the question of if $M' \rightarrow W$ is simple can be phrased purely topologically. Hence, if we have even a homeomorphism $W \cong M' \times [0,1]$ (commuting with the inclusions $M' \rightarrow W$ and $M' \rightarrow M' \times [0,1])$, then we conclude that $M' \rightarrow W$ is simple because the latter clearly is. So we have reduced the problem to if connect summing with $S^n$ results in a homeomorphic manifold, where the homeomorphism is the identity on the boundary. This is clearly true since it is easily seen to be true for a disk.
Hence, the map $\partial(M) \rightarrow (\partial(M) \times [0,1]) \sharp \Sigma$ is an s-cobordism, so we apply the s-cobordism theorem to deduce the diffeomorphism $(\partial(M) \times [0,1]) \sharp \Sigma \cong \partial(M) \times [0,1]$. Meaning that yes, if $M$ has nonempty boundary, connect summing with an exotic sphere does not affect diffeomorphism type (in dimensions where s-cobordism holds).
To explicitly spell out the rest of the argument: removing an open collar of the boundary results in a diffeomorphic manifold to the original, so by equipping our manifold with this new choice of open collar obtained from the diffeomorphism we just proved, we obtain a manifold diffeomorphic to our original $M$, since we have removed where the connect sum has happened, but also diffeomorphic to $M \sharp \Sigma$ by our comments just now.