Symplectic Rational Blow-Up and Embeddings of Rational Homology Balls

Abstract

We define the symplectic rational blow-up operation, for a family of rational homology balls \(B_n\), which appeared in Fintushel and Stern's rational blow-down construction. We do this by exhibiting a symplectic structure on a rational homology ball \(B_n\) as a standard symplectic neighborhood of a certain 2-dimensional Lagrangian cell complex. We also study the obstructions to symplectically rationally blowing up a symplectic 4-manifold, i.e. the obstructions to symplectically embedding the rational homology balls \(B_n\) into a symplectic 4-manifold. First, we present a couple of results which illustrate the relative ease with which these rational homology balls can be smoothly embedded into a smooth 4-manifold. Second, we prove a theorem and give additional examples which suggest that in order to symplectically embed the rational homology balls \(B_n\), for high \(n\), a symplectic 4-manifold must at least have a high enough \(c^2_1\) as well.