Geography of simply connected nonspin symplectic 4-manifolds with positive signature (Q1951192)

The existence of symplectic structures on smooth \(4\)-manifolds is known as the symplectic geography problem. In the simply-connected case, it is understood in the negative signature case. This paper addresses the case of non-negative signature by constructing new families of symplectic manifolds of non-negative signature. The new construction begins with a surface of genus \(g\) admiting a semifree \(\mathbb{Z}_p\) action. Given such a surface, \(\Sigma_g\), one blows up the product \(\Sigma_g\times\Sigma_g\) at the points where the graphs of left multiplication by elements of \(\mathbb{Z}_p\) intersect. The \(p\)-fold branched cover of \(\Sigma_g\times\Sigma_g\) branched along the proper transform of the union of the graphs is an interesting building block that may be combined with the fiber sum operation and knot surgery to construct large new families of simply-connected \(4\)-manifolds having non-negative signature. This is exactly what is done in this nice paper.