Algebraic & Geometric Topology

On genus–$1$ simplified broken Lefschetz fibrations

Kenta Hayano

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Auroux, Donaldson and Katzarkov introduced broken Lefschetz fibrations as a generalization of Lefschetz fibrations in order to describe near-symplectic 4–manifolds. We first study monodromy representations of higher sides of genus–1 simplified broken Lefschetz fibrations. We then completely classify diffeomorphism types of such fibrations with connected fibers and with less than six Lefschetz singularities. In these studies, we obtain several families of genus–1 simplified broken Lefschetz fibrations, which we conjecture contain all such fibrations, and determine the diffeomorphism types of the total spaces of these fibrations. Our results are generalizations of Kas’ classification theorem of genus–1 Lefschetz fibrations, which states that the total space of a nontrivial genus–1 Lefschetz fibration over S2 is diffeomorphic to an elliptic surface E(n) for some n1.

Article information

Algebr. Geom. Topol., Volume 11, Number 3 (2011), 1267-1322.

Received: 25 November 2010
Revised: 3 February 2011
Accepted: 14 February 2011
First available in Project Euclid: 19 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 57M50: Geometric structures on low-dimensional manifolds
Secondary: 32S50: Topological aspects: Lefschetz theorems, topological classification, invariants 57R65: Surgery and handlebodies

broken Lefschetz fibration $4$–manifold monodromy representation Kirby diagram chart description


Hayano, Kenta. On genus–$1$ simplified broken Lefschetz fibrations. Algebr. Geom. Topol. 11 (2011), no. 3, 1267--1322. doi:10.2140/agt.2011.11.1267.

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