An essential relation between Einstein metrics, volume entropy, and   exotic smooth structures
  We show that the minimal volume entropy of closed manifolds remains unaffected when nonessential manifolds are added in a connected sum. We combine this result with the stable cohomotopy invariant of Bauer-Furuta in order to present an infinite family of four-manifolds with the following properties:   1) They have positive minimal volume entropy.   2) They satisfy a strict version of the Gromov-Hitchin-Thorpe inequality, with a minimal volume entropy term.   3) They nevertheless admit infinitely many distinct smooth structures for which no compatible Einstein metric exists. 
Michael Brunnbauer, Masashi Ishida, Pablo Su\'arez-Serrato