Finite Cyclic Fundamental Groups and Exotic Universal Covers from the Cartwright--Steger Surface

Abstract

We use a particular nodal curve on the Cartwright-Steger surface to construct minimal symplectic four-manifolds with arbitrary nontrivial finite cyclic fundamental group and fixed topological invariants $(e,σ,χ_h,c_1^2)=(43,1,11,89)$. Using these symplectic four-manifolds as building blocks, we construct a symplectic 4-manifold that is homeomorphic to $(22m-1)\mathbb{CP}^2\#(21m-1)\overline{\mathbb{CP}}^{\,2}$ for any integer $m\geq 2$.