A new bound on the size of symplectic 4-manifolds with prescribed fundamental group (Q360425)

In [Ann. Math. (2) 142, No. 3, 527--595 (1995; Zbl 0849.53027)], \textit{R. E. Gompf} proved that any finitely presented group is the fundamental group of a closed symplectic 4-manifold. In [Comment. Math. Helv. 82, No. 4, 845--875 (2007; Zbl 1155.57024)], \textit{S. Baldridge} and \textit{P. Kirk} improved upon Gompf's construction, giving a smaller closed symplectic 4-manifold subject to the restriction of having a specified fundamental group, where the size of a manifold is measured by the magnitude of its signature and Euler characteristic.NEWLINENEWLINENEWLINEIn the paper under review the author proves the followingNEWLINENEWLINETheorem: Let \(G\) be a group with a presentation with \(g\) generators and \(r\) relations. Then there is a symplectic \(4\)-manifold \(X\) with \(\pi_1(X)=G\) and \(\chi(X)=10+4(g+r)\) and \(\sigma(X)=-2\). NEWLINENEWLINENEWLINEThis is an improvement on the aforementioned result of Baldridge and Kirk.