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**Chart description and a new proof of the classification theorem of genus one Lefschetz fibrations.**
*(English)*
Zbl 1080.57022

A Lefschetz fibration is a smooth map \(f:M\to B\) from a compact, connected, oriented 4-dimensional manifold to a compact, connected, oriented 2-dimensional manifold where all but a finite number of the fibers \(f^{-1}(b)\) for \(b\in B\) are smooth, oriented surfaces of fixed genus \(g\), and where all of the singular fibers are immersed with a single transverse self-intersection. These fibrations have become a central tool for studying the topology of the total space \(M\), see for instance [R. Gompf and A. Stipsicz, 4-manifolds and Kirby calculus, Graduate Studies in Mathematics. 20. Providence, RI: American Mathematical Society (AMS). (1999; Zbl 0933.57020)]. (It should be noted that a Lefschetz fibration as defined in this paper is very often called an achiral Lefschetz fibration by other authors.) Lefschetz fibrations are typically described in terms of their monodromy representation \(\pi_1(B-S_f)\to {\mathcal M}_g\), where \(S_f\) denotes the set of critical values of \(f\), and \({\mathcal M}_g\) denotes the mapping class group of an oriented genus \(g\) surface. In the 1980s, work of the second named author [J. Math. Soc. Japan, 37, 605-636 (1985; Zbl 0624.57017); Topology, 25, 549-563 (1986; Zbl 0615.14023)] gave a classification of Lefschetz fibrations of fiber genus one, up to fiber-preserving diffeomorphism, in terms of the genus of \(B\) and the number of singular fibers with self-intersection of each sign.

In this paper, the authors introduce an ingenious graphical method for describing the monodromy representations of genus one Lefschetz fibrations. Their basic idea is to describe the monodromy in terms of a chart description, meaning as a particular sort of directed, labeled graph \(\Gamma\) in \(B\). They impose conditions on \(\Gamma\) corresponding to relations in \({\mathcal M}_1\cong SL(2,{\mathbb Z})\), which ensure that any chart description \(\Gamma\) determines a well-defined monodromy representation; and, conversely, they show that the monodromy representation for any Lefschetz fibration over a closed \(B\) can be described by a chart. A chart description for the monodromy is not unique, and they give a set of moves on charts which preserve the induced representation. Using these moves, the classification of genus one Lefschetz fibrations is reproven. While this result is not new, the novelty of the proof, which gives a purely combinatorial description for the monodromies of genus one Lefschetz fibrations, is of interest. (In particular, chart descriptions of monodromies can be generalized to Lefschetz fibrations of any fiber genus, where classifications are presently unknown.)

In this paper, the authors introduce an ingenious graphical method for describing the monodromy representations of genus one Lefschetz fibrations. Their basic idea is to describe the monodromy in terms of a chart description, meaning as a particular sort of directed, labeled graph \(\Gamma\) in \(B\). They impose conditions on \(\Gamma\) corresponding to relations in \({\mathcal M}_1\cong SL(2,{\mathbb Z})\), which ensure that any chart description \(\Gamma\) determines a well-defined monodromy representation; and, conversely, they show that the monodromy representation for any Lefschetz fibration over a closed \(B\) can be described by a chart. A chart description for the monodromy is not unique, and they give a set of moves on charts which preserve the induced representation. Using these moves, the classification of genus one Lefschetz fibrations is reproven. While this result is not new, the novelty of the proof, which gives a purely combinatorial description for the monodromies of genus one Lefschetz fibrations, is of interest. (In particular, chart descriptions of monodromies can be generalized to Lefschetz fibrations of any fiber genus, where classifications are presently unknown.)

Reviewer: Terry Fuller (Northridge)