# Manifolds with diffeomorphic boundaries

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).

• Can you provide some more details? I agree it is a question of framing. I need to show that the boundary remembers the framing. I think the intersection form only knows part of the framing data? – user39598 Aug 3 '17 at 22:36
• I deleted the comment because I didn't realize you wanted to allow an arbitrary number of n-handles. I don't know the answer, but surely you want to look at Wall's paper and the various citations to and from. – user98602 Aug 3 '17 at 22:36
• I think even for a single handle it might not be true. – user39598 Aug 3 '17 at 22:38
• I remember thinking about this some time ago and thought it had been; my apologies. In that case you are precisely interested in the homomorphism $\pi_n SO(n-1) \to \Gamma_{2n-1}$ and whether or not it has kernel. But I don't remember whether or not this is true. Kervaire and Milnor would know, somewhere... – user98602 Aug 3 '17 at 22:49

No. There are plenty of pairs of knots with distinct $n$-traces with the same $n$-surgery. There are many papers by people like Yasui and Akbulut on this. For instance this one : https://arxiv.org/abs/1505.02551 .