TITLE: A kind of uniqueness for the double of a manifold
QUESTION [8 upvotes]: Given two smooth, connected manifolds, M, N, with the same boundary. If their doubles, D(M) and D(N) are diffeomorphic, does it follow that M and N are diffeomorphic ? The condition on the boundary is to exclude Mazur manifolds..

REPLY [5 votes]: Here's an example in dimension 4: let $M$ and $N$ be the 4-dimensional 2-handlebodies associated to $-1$-surgery along the trefoil knot and $+1$-surgery along the figure-eight knot respectively.
They have the same boundary, as the two surgeries are diffeomorphic. On the other hand, Corollary 5.1.6 in Gompf and Stipsicz's 4-manifolds and Kirby calculus tells us that $D(M)$ and $D(N)$ are both diffeomorphic to $\mathbb{CP}^2\#\overline{\mathbb{CP}}^2$.
However, $M$ and $N$ can't even be homeomorphic, since the intersection form on $M$ is negative-definite and the intersection form on $N$ is positive-definite.