A note on signature of Lefschetz fibrations with planar fiber

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.