Which 3-manifolds are known to admit exotic pairs of bounding 4-manifolds? 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?
 A: Here is a list of 3-manifolds $Y$ that are boundaries of exotic 4-manifolds https://arxiv.org/pdf/1901.07964.pdf

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


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


*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.
