Suppose $M$ is a compact smooth manifold with boundary, which is homeomorphic to the compact ball $\mathbb{B}^d\subset \mathbb{R}^d$. Must $M$ be diffeomorphic to $\mathbb{B}^d$ or are there exotic smooth structures?
I suspect that the answer is well known or might follow from a simple argument that experts (which I am certainly not) have up their sleeves.
A natural candidate in $d=4$ would be to take $M$ to be the compact unit-ball in an exotic $\mathbb{R}^4$, but I don't know how to check whether $M$ is exotic (or even whether it has a smooth boundary to be honest).
from Hot Weekly Questions - Mathematics Stack Exchange
Jan Bohr
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