# Geometric Analysis and Topology Seminar

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

**Speaker:**
Ioana Suvaina, NYU

**Location:**
Warren Weaver Hall 813

**Date:**
Friday, November 10, 2006, 11 a.m.

**Synopsis:**

The existence or non-existence of Einstein metrics on a topological 4-manifold is strongly related to the differential structure considered. We show that there exist infinitely many topological 4-manifolds such that each manifold admits a smooth structure which supports an Einstein metric and has infinitely many other structures on which no Einstein metric can exist. This result is known for simply connected manifolds, our contribution is to exhibit it for manifolds with finite cyclic fundamental group. We complete this result with theorems about non-existence of Einstein metrics on manifolds with finitely presented fundamental group and discuss some nice corollaries. The main tools are Seiberg-Witten theory, cyclic coverings of complex surfaces and symplectic surgeries.