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?

• 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. – Mike Miller Oct 9 '18 at 19:27
• @Arun I think I fixed it. – user101010 Oct 9 '18 at 19:32
• @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. – Mike Miller Oct 9 '18 at 19:34
• 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. – Ian Agol Oct 10 '18 at 23:00
• 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. – Anubhav Mukherjee Oct 11 '18 at 15:52

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