Symplectically replacing plumbings with Euler characteristic $2 \: 4$‑manifolds

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 $\mathbb{C}P^2 \# 6 \overline{\mathbb{C}P^2}$.

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Received 12 October 2016

Accepted 11 October 2019

Published 29 October 2020