A note on signature of Lefschetz fibrations with planar fiber

TL;DR

This paper derives a signature formula for allowable Lefschetz fibrations with planar fibers, providing an alternative proof of a key theorem on symplectic fillings of contact 3-manifolds, and linking it to Stein structures.

Contribution

It introduces a new signature formula for Lefschetz fibrations with planar fibers, connecting it to existing theorems and providing alternative proofs.

Findings

Derived a signature formula for Lefschetz fibrations with planar fiber.

Provided an alternative proof of Etnyre's theorem on symplectic fillings.

Linked the signature formula to Stein structures on Lefschetz fibrations.

Abstract

Using theorems of Eliashberg and McDuff, Etnyre [Et] proved that the intersection form of a symplectic filling of a contact 3-manifold supported by planar open book is negative definite. In this paper, we prove a signature formula for allowable Lefschetz fibrations over $D^2$ with planar fiber by computing Maslov index appearing in Wall's non-additivity formula. The signature formula leads to an alternative proof of Etnyre's theorem via works of Niederkr\"uger and Wendl [NWe] and Wendl [We]. Conversely, Etnyre's theorem, together with the existence theorem of Stein structures on Lefschetz fibrations over $D^2$ with bordered fiber by Loi and Piergallini [LP], implies the formula.