## Algebraic & Geometric Topology

### On genus–$1$ simplified broken Lefschetz fibrations

Kenta Hayano

#### Abstract

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 $n≥1$.

#### Article information

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

Dates
Revised: 3 February 2011
Accepted: 14 February 2011
First available in Project Euclid: 19 December 2017

https://projecteuclid.org/euclid.agt/1513715228

Digital Object Identifier
doi:10.2140/agt.2011.11.1267

Mathematical Reviews number (MathSciNet)
MR2801419

Zentralblatt MATH identifier
1229.57017

#### Citation

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. https://projecteuclid.org/euclid.agt/1513715228

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