Associativity properties of the symplectic sum

Abstract

In this note we apply a 4-fold sum operation to develop an associativity rule for the pairwise symplectic sum. This allows us to show that certain diffeomorphic symplectic $4$-manifolds made out of elliptic surfaces are in fact symplectically deformation equivalent. We also show that blow-up points can be traded from one side of a symplectic sum to another without changing the symplectic deformation class of the resulting manifold.