Associativity properties of the symplectic sum

TL;DR

This paper explores the associativity of the symplectic sum operation in four-dimensional symplectic manifolds, demonstrating how it affects deformation equivalence and the handling of blow-up points.

Contribution

It establishes an associativity rule for the symplectic sum and shows how blow-up points can be moved without altering deformation classes.

Findings

Certain diffeomorphic symplectic 4-manifolds are deformation equivalent

Blow-up points can be transferred across symplectic sums without changing deformation class

Associativity of the symplectic sum operation is proven

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.