We prove that there are infinitely many pairs of homeomorphic non-diffeomorphic smooth 4–manifolds, such that in each pair one manifold admits an Einstein metric and the other does not. We also show that there are closed 4–manifolds with two smooth structures which admit Einstein metrics with opposite signs of the scalar curvature.
"Einstein metrics and smooth structures." Geom. Topol. 2 (1) 1 - 10, 1998. https://doi.org/10.2140/gt.1998.2.1