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We show that each of the topological 4-manifolds $CP^2#k\bar{CP^2}, for $k =6, 7$ admits a smooth structure which has an Einstein metric of scalarcurvature $s > 0$, a smooth structure which has an Einstein metric with $s < 0$and infinitely many non-diffeomorphic smooth structures which do not admitEinstein metrics. We show that there are infinitely many manifolds homeomorphicnon-diffeomorphic to $CP^2#5\bar{CP^2}$ which do not admit an Einstein metric.We also exhibit new higher dimensional examples of manifolds carrying Einsteinmetrics of both positive and negative scalar curvature. The main ingredientsare recent constructions of exotic symplectic or complex manifolds with smalltopological numbers.

Smooth Structures and Einstein Metrics on $CP^2#5,6,7\bar{CP^2}$