2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/58719In this paper, we prove a number of inequalities between the signature and the Betti numbers of a 4-manifold with even intersection form. Furthermore, we introduce a new geometric group invariant and discuss some of its properties.Revised version, to appear in Math. Proc. Cambridge Phil. SocGeometric TopologyAlgebraic Topology57M99 (Primary) 57M05 (Secondary)text