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

TL;DR

This paper investigates the existence of Einstein metrics on certain 4-manifolds with finite cyclic fundamental groups, showing that many such manifolds admit no invariant Einstein metrics despite having multiple group actions.

Contribution

It demonstrates the non-existence of invariant Einstein metrics on a broad class of 4-manifolds with cyclic fundamental groups using advanced geometric and topological tools.

Findings

Most manifolds admit infinitely many inequivalent cyclic group actions.

No Einstein metrics are invariant under these cyclic actions.

The results connect differential topology with geometric structures on 4-manifolds.

Abstract

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.