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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. $\endgroup$
    – mme
    Oct 9, 2018 at 19:27
  • $\begingroup$ @Arun I think I fixed it. $\endgroup$
    – user101010
    Oct 9, 2018 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. $\endgroup$
    – mme
    Oct 9, 2018 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, 2018 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, 2018 at 15:52

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Here is a list of 3-manifolds $Y$ that are boundaries of exotic 4-manifolds https://arxiv.org/pdf/1901.07964.pdf

  1. If either $Y$ or $-Y$ (i.e with reverse orientation) has a contact structure with non-trivial contact invariant.

  2. If $Y$ or $-Y$ has weak symplectic filling.

  3. If $Y$ bounds both positive and negative definite 4 manifolds.

In case of 1) and 2) those manifolds bounds simply-connected manifolds with infinitely many exotic structures and in case of 3rd case we cannot gurantee the simply-connected condition.

The above classes cover all Seifert fibered 3-manifolds, all 3 manifolds that admits Taut folitaion, all irreducible 3 manifolds with 1st Betti number strictly bigger than zero or $M\# -M$.

Conjecturally we covered all the irreducible 3-manifolds in the above 3 cases. One obstruction when dealing with reducible manifold is that most of the 4-manifolds invariant vanishes under connected sum. So it is still an open problem if all 3-manifolds bounds exotic 4-manifolds. Hope in some near future we (or someone else) will find some clever way to deal with all 3-manifolds.

In Theorem 1.13 above we gave a general construction which holds for every 3-manifolds (because all 3-manifolds admit contact structures). But we do not know how to prove that they all are not diffeomorphic in general.

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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, 2019 at 16:36
  • $\begingroup$ I did mention it in the acknowledgement. $\endgroup$
    – Anubhav Mukherjee
    Jan 24, 2019 at 17:46

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