Suppose $X$, $Y$ are two $2n$-dimensional manifolds with handles of index $0$ and $n$. In particular, $X$ and $Y$ have boundary. Suppose that $X$, $Y$ have isomorphic homology and intersection form and that their boundaries are diffeomorphic. Does this imply that $X$ and $Y$ are diffeomorphic? Less precisely, I am wondering if the intersection form and boundary determine the framing of the n handles.
Let me point out that there are examples of manifolds with the same homology and diffeomorphic boundary but are not diffeomorphic. Their intersection forms have the same rank but not the same signature (although their signatures are always congruent modulo some large integer).