Comparing star surgery to rational blow-down

TL;DR

This paper compares star surgery and rational blow-down operations in symplectic 4-manifolds, showing star surgery's greater generality and its implications for mapping class monoids.

Contribution

It demonstrates that star surgery is a strictly more general operation than rational blow-down, providing new relations in planar mapping class monoids.

Findings

Star surgery shares properties useful for constructing small exotic symplectic 4-manifolds.

There exists an infinite family of star surgeries inequivalent to rational blow-downs.

Star surgery yields relations in planar mapping class monoids not generated by rational blow-down relations.

Abstract

We compare the star surgery operations introduced in [KS] to the generalized rational blow-down. We show that star surgery shares the properties that make rational blow-down useful for constructions of small exotic symplectic 4-manifolds. Then we show that star surgery operations provide a strictly more general class of operations by proving that there is an infinite family of star surgeries which are inequivalent to any sequence of generalized symplectic rational blow-downs. This answers a question posed to the author by Ozbagci. It also demonstrates that the monodromy substitutions coming from star surgery operations yield relations in planar mapping class monoids which cannot be positively generated by the relations determined in [EMVHM11] which come from the generalized rational blow-downs.