  In this article we apply the technique of Luttinger surgery to study the
complexity of the fundamental group of symplectic $4$-manifolds with
holomorphic Euler number $\chi_h=1$. We discuss the topology of symplectic
$4$-manifolds with $b^+=1$ and provide various constructions of symplectic
$4$-manifolds with $b^+=1$ and prescribed $c_1^2$.
