Title: 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 Niederkrüger 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.