TITLE: Are there exotic balls?
QUESTION [12 upvotes]: 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).

REPLY [5 votes]: This is an addendum regarding dimensions 4 and 5: While the (smooth) h-cobordism argument for 5-dimensional cobordisms no longer works, Milnor proves in his "h-cobordism" book (pages 110-111, Proposition C) that if $M$ is a 5-dimensional smooth compact manifold homeomorphic to $D^5$, whose boundary is diffeomorphic to $S^4$, then $M$ is diffeomorphic to $D^5$. According to theorems of Milnor-Kervaire and Wall, if $M^4$ is an exotic $S^4$, then it bounds a smooth contractible 5-manifold $W$.  The manifold $W$ then will have to be homeomorphic to $D^5$ (by the topological h-cobordism theorem for 5-dimensional h-cobordisms, due to Freedman). Thus, the existence of an exotic $D^5$ is equivalent to the existence of an exotic $S^4$.
In dimension 4 one would need a bit more: If $W$ is a smooth 4-manifold homeomorphic to $D^4$, then its double $DW$ is homeomorphic to $S^4$. If $DW$ is diffeomorphic to $S^4$, assuming, in addition, smooth Schoenflies conjecture in dimension 4, one would obtain that $W$ is diffeomorphic to $D^4$. If smooth Schoenflies conjecture fails, then you would obtain a smooth submanifold inside $S^4$ which is homeomorphic but not diffeomorphic to $D^4$. (The same deal if the smooth Poincare conjecture fails in dimension 4.) So, "in essence," in dimension 4, the problem is equivalent to Poincare+Schoenflies.