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On Einstein Metrics on 4-Manifolds with Finite Cyclic Fundamental Group

The existence or non-existence of Einstein metrics on 4-manifolds with non-trivial fundamental group and the relation with the underlying differential structure are analyzed. For most points $(n,m)$ in a large region of the integer lattice, the manifold $n\mathbb C \mathbb P^2\#m \overline{\mathbb C \mathbb P^2}$ is shown to admit infinitely many inequivalent free actions of finite cyclic groups and there are no Einstein metrics which are invariant under any of these actions. The main tools are Seiberg-Witten theory, cyclic branched coverings of complex surfaces and symplectic surgeries.