Symplectically replacing plumbings with Euler characteristic 2 4-manifolds

TL;DR

This paper introduces new symplectic cut-and-paste techniques generalizing rational blowdowns, classifies certain replaceable plumbings, and constructs exotic symplectic 4-manifolds with specific topological properties.

Contribution

It defines and classifies $k$-replaceable plumbings, especially 2-replaceable ones, and constructs new exotic symplectic 4-manifolds using these methods.

Findings

Classified 2-replaceable linear plumbings

Constructed 2-replaceable plumbing trees

Built a symplectic exotic #6

Abstract

We introduce new symplectic cut-and-paste operations that generalize the rational blowdown. In particular, we will define $k$-replaceable plumbings to be those that, heuristically, can be symplectically replaced by Euler characteristic $k$ 4-manifolds. We will then classify 2-replaceable linear plumbings, construct 2-replaceable plumbing trees, and use one such tree to construct a symplectic exotic $\mathbb{C}P^2\#6\overline{\mathbb{C}P^2}$.