Determine which 3-manifolds admit exotic fillings

Characterize which closed, smooth 3-manifolds admit exotic fillings, namely fillings that are homeomorphic relative to the boundary but not diffeomorphic relative to the boundary.

Background

The paper studies the stronger stable version of exotic fillings and notes that the corresponding unstabilized classification problem is also unresolved. An exotic filling is a smooth compact 4-manifold filling whose homeomorphism class relative to the boundary contains more than one diffeomorphism class. The paper does not resolve the general existence question.