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Let $M$ be a compact connected three manifold. By an exotic pair of bounding 4-manifolds, I mean two smooth 4-manifolds $X_1,X_2$ such that $X_1$ and $X_2$ are homeomorphic but not diffeomorphic, and $\partial X_1$ and $\partial X_2$ are homeomorphic to $M$.

I imagine that all 3-manifolds have exotic pairs of bounding 4-manifolds. In fact, I would imagine that we could just take some exotic pair of closed 4-manifolds $W_1, W_2$ and any bounding 4-manifold $X$ for $M$ and then just take $X_1 = X \sharp W_1$ and $X_2 = X \sharp W_2$.

Does this process always produce an exotic pair for $M$?

Do all 3-manifolds have an exotic pair that they bound?

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    $\begingroup$ It is not true that your process always produces an exotic pair; some manifolds have a habit of destroying exoticity upon connected sum. In particular, if $X$ has sufficiently many summands of $S^2 \times S^2$, the result will be diffeomorphic. But one hopes that all 3-manifolds do indeed bound exotic pairs. $\endgroup$ – Mike Miller Oct 9 '18 at 19:27
  • $\begingroup$ @Arun I think I fixed it. $\endgroup$ – user101010 Oct 9 '18 at 19:32
  • $\begingroup$ @user101010 The phrasing you probably want is "the $X_i$ are compact connected 4-manifolds, homeomorphic but not diffeomorphic, and $\partial X_i \cong M$." You can break your current formulation by connect summing with some other manifold. $\endgroup$ – Mike Miller Oct 9 '18 at 19:34
  • $\begingroup$ Should work for $S^3$: just take a closed manifold with two different smooth structures and remove the interior of a standard ball from each one. $\endgroup$ – Ian Agol Oct 10 '18 at 23:00
  • $\begingroup$ I was wondering, what if we consider a exotic copy of Manifold with S^3 boundary and then do the boundary sum with a manifold with a given 3 manifold boundary. Equivalently attaching a 1 handle in between those two copies. I am not very good in handeling monopole theory and stuffs. But we can try to compute some Seiberg-Witten invariant for this case. $\endgroup$ – Anubhav Mukherjee Oct 11 '18 at 15:52
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Motivating from your question, we proved that (Theorem 1.1) every closed oriented $3-$manifold bounds a compact simply connected 4 manifold which admits infinitly many smooth structures, rel boundary. https://arxiv.org/abs/1901.07964

Recently an error was pointed out by Gompf on the last page of our final proof. However our result is still true for any 3-manifold which admits a weak symplectic filling. We are trying to fix the error.

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    $\begingroup$ I think it's good practice to actually cite this MathOverflow question in your paper, with a URL link. $\endgroup$ – Sam Hopkins Jan 24 at 16:36
  • $\begingroup$ I did mention it in the acknowledgement. $\endgroup$ – Anubhav Mukherjee Jan 24 at 17:46

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