Planar open books, monodromy factorizations, and symplectic fillings

TL;DR

This paper investigates the symplectic fillings of contact structures supported by planar open books, establishing uniqueness results and characterizing fillability on specific 3-manifolds using monodromy factorizations.

Contribution

It introduces a method based on monodromy factorizations and Wendl's theorem to classify fillings of planar open book-supported contact structures, including new results on L(p,1) and Seifert fibered spaces.

Findings

Virtually overtwisted contact structures on L(p,1) have unique fillings.

Characterization of fillable and non-fillable tight contact structures on certain Seifert fibered spaces.

Application of monodromy factorizations to classify symplectic fillings.

Abstract

We study fillings of contact structures supported by planar open books by analyzing positive factorizations of their monodromy. Our method is based on Wendl's theorem on symplectic fillings of planar open books. We prove that every virtually overtwisted contact structure on L(p,1) has a unique filling, and describe fillable and non-fillable tight contact structures on certain Seifert fibered spaces.