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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
Mathematical Subject Classification 2010
Primary: 57M15
Secondary: 57N13
References
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