On symplectic fillings of spinal open book decompositions II: Holomorphic curves and classification

TL;DR

This paper classifies symplectic and Stein fillings of certain contact 3-manifolds using Lefschetz fibrations, extending previous results to a broader class with applications to contact invariants and deformation theory.

Contribution

It extends classification results of symplectic and Stein fillings to contact manifolds with planar spinal open books, including non-orientable cases and new criteria for invariants.

Findings

Classified fillings of all oriented circle bundles with specific contact structures.

Established new vanishing criteria for ECH contact invariant and algebraic torsion.

Proved a general symplectic quasiflexibility result in four dimensions.

Abstract

In this second paper of a two-part series, we prove that whenever a contact 3-manifold admits a uniform spinal open book decomposition with planar pages, its (weak, strong and/or exact) symplectic and Stein fillings can be classified up to deformation equivalence in terms of diffeomorphism classes of Lefschetz fibrations. This extends previous results of the third author to a much wider class of contact manifolds, which we illustrate here by classifying the strong and Stein fillings of all oriented circle bundles with non-tangential $S^1$-invariant contact structures. Further results include new vanishing criteria for the ECH contact invariant and algebraic torsion in SFT, classification of fillings for certain non-orientable circle bundles, and a general "symplectic quasiflexibility" result about deformation classes of Stein structures in real dimension four.