Spinal open books and symplectic fillings with exotic fibers

TL;DR

This paper extends the classification of symplectic fillings of contact 3-manifolds using spinal open books, introducing new models and tools for understanding exotic fibers and monodromy factorizations.

Contribution

It generalizes Wendl's results to minimal symplectic fillings supported by planar spinal open books and introduces explicit local models for singularities at infinity.

Findings

Classified strong fillings of all parabolic torus bundles.

Developed explicit local models for non-compact singularities.

Progressed towards classifying fillings for non-planar open books.

Abstract

Spinal open book decompositions provide a natural generalization of open book decompositions. We show that any minimal symplectic filling of a contact 3-manifold supported by a planar spinal open book is deformation equivalent to the complement of a positive multisection in a bordered Lefschetz fibration, which generalizes a result of Wendl. Along the way, we give an explicit local model for a non-compactly supported "singularity at infinity" in a generalized version of bordered Lefschetz fibrations, given by pseudoholomorphic foliations associated to the spinal open books. This provides new tools to classify symplectic fillings of a contact 3-manifold that is not supported by an amenable spinal open book, by studying monodromy factorizations in the newly defined spinal mapping class group. As an application, we complete the classification of strong fillings of all parabolic torus bundles, and make progress towards classifying symplectic fillings of contact 3-manifolds supported by non-planar open books.