Gompf realization theorem

Every finitely presented group is the fundamental group of a closed symplectic four-manifold. The construction starts from products of surfaces, realizes relator words by embedded loops after stabilization, and takes symplectic sums with copies of the rational elliptic surface along square-zero tori. Since the complement of a regular elliptic fiber in the rational elliptic surface is simply connected, van Kampen's theorem imposes the desired generators and relators.