#### Volume 19 (2015)

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Charts, signatures, and stabilizations of Lefschetz fibrations

### Hisaaki Endo, Isao Hasegawa, Seiichi Kamada and Kokoro Tanaka

Geometry & Topology Monographs 19 (2015) 237–267
 arXiv: 1403.7946
##### Abstract

We employ a certain labeled finite graph, called a chart, in a closed oriented surface to describe the monodromy of a(n achiral) Lefschetz fibration over the surface. Applying charts and their moves with respect to Wajnryb’s presentation of mapping class groups, we first generalize a signature formula for Lefschetz fibrations over the $2$–sphere obtained by Endo and Nagami to that for Lefschetz fibrations over arbitrary closed oriented surface. We then prove two theorems on stabilization of Lefschetz fibrations under fiber summing with copies of a typical Lefschetz fibration as generalizations of a theorem of Auroux.

##### Keywords
Lefschetz fibration, chart, signature, fiber sum, stabilization
Primary: 57M15
Secondary: 57N13