#### 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
##### Publication
Received: 17 April 2015
Accepted: 20 April 2015
Published: 29 December 2015
##### Authors
 Hisaaki Endo Department of Mathematics Tokyo Institute of Technology 2-12-1 Oh-okayama Meguro-ku Tokyo 152-8551 Japan Isao Hasegawa Ministry of Health, Labour and Welfare 1-2-2 Kasumigaseki Chiyoda-ku Tokyo 100-8916 Japan Seiichi Kamada Department of Mathematics Osaka City University 3-3-138 Sugimoto Sumiyoshi-ku Osaka 558-8585 Japan Kokoro Tanaka Department of Mathematics Tokyo Gakugei University 4-1-1 Nukuikita-machi Koganei-shi Tokyo 184-8501 Japan