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?



Which 3-manifolds are known to admit exotic pairs of bounding 4-manifolds?
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.


