Stably exotic fillings of 3-manifolds

We investigate which 3-manifolds bound 4-manifolds that are homeomorphic but not stably diffeomorphic, where stabilising means taking connected sum with copies of $S^2\times S^2$. We show that every closed, orientable 3-manifold admits such fillings, as do certain families of nonorientable 3-manifolds. In contrast we show that for a 3-manifold containing a 2-sided ${\mathbb{R}\mathbb{P}}^2$, any two smooth, homeomorphic fillings are stably diffeomorphic.