A new bound on the size of symplectic 4-manifolds with prescribed fundamental group

Abstract

Given any finitely presented group with $g$ generators and $r$ relations, we produce a symplectic 4-manifold of Euler characteristic $10+4(g+r)$ and signature $−2$. This is an improvement on the result in S. Baldridge and P. Kirk, On symplectic 4-manifolds with prescribed fundamental group, and our construction utilizes a construction in R. Fintushel, B. Doug Park and R. J. Stern, Reverse engineering small 4-manifolds.