How large is the braid monodromy of low-genus Lefschetz fibrations?

TL;DR

The paper analyzes how large the liftable braid subgroup Br$(\pi)$ can be inside the spherical braid group Mod$(S^2,Δ)$ for low-genus Lefschetz fibrations. It develops a Hurwitz-action framework on monodromy data, relates Br$(\pi)$ to stabilizers of monodromy classes and to SL$_2$-character varieties in genus 1, and proves infinite-index results in key cases (genus 1, self-fiber sums, and holomorphic genus-2 fibrations with nonseparating vanishing cycles). It also introduces general criteria via Moishezon spiders and uses Auroux’ lemma to extend infinite-index conclusions to broad classes, while providing explicit finite-index disk-examples and computational tools; the results illuminate the often-dominant size of the braid monodromy and its implications for Lefschetz-fibration classification. The work connects geometric topology with representation theory and character varieties, offering concrete algorithms and open questions about generation, presentation, and torsion in Br$(\pi)$.

Abstract

Given a genus $g$ smooth Lefschetz fibration $π: M \to S^2$ with singular locus $Δ\subseteq S^2$, we describe the subgroup $\operatorname{Br}(π)$ of the spherical braid group $\operatorname{Mod}(S^2,Δ)$ consisting of braids admitting a lift to a fiber-preserving diffeomorphism of $M$. We develop general methods for showing that the index $[\operatorname{Mod}(S^2,Δ) : \operatorname{Br}(π)]$ is infinite. As an application of our methods, we prove that $[\operatorname{Mod}(S^2,Δ) : \operatorname{Br}(π)] = \infty$ when $g = 1$, when $π$ is expressible as a self-fiber sum when $g \geq 2$, or when $π$ is a holomorphic genus $g = 2$ Lefschetz fibration whose vanishing cycles are nonseparating. In the genus $g = 1$ case, we relate the subgroup $\operatorname{Br}(π)$ to the action of $\operatorname{Mod}(S^2,Δ)$ on the $\operatorname{SL}_2$-character variety for $S^2 \setminus Δ$ and provide an alternate proof of the first application via recent work of Lam--Landesman--Litt.

Figures (4)

• Figure 1: The braids $r = \sigma_1\cdots \sigma_{n-1}$ given by rotation about the equator and the Garside half-twist $T_n = (\sigma_1\cdots \sigma_{n-1})(\sigma_1\cdots \sigma_{n-2})\cdots (\sigma_1\sigma_2)\sigma_1$ given by rotation about a longitude. $r$ lies in $\mathop{\mathrm{Br}}\nolimits(\pi)$ while $T_n$ does not. In each case, the left figure depicts a spherical braid with its associated mapping class and the right figure depicts the corresponding element in the braid group $B_n$.
• Figure 2: Hurwitz orbits of $\phi_{q_n}$ for $n = 3$ and $4$. Vertex $0$ is $\phi_{q_n}$, and missing edges indicate representations fixed by the corresponding $\sigma_i$. When $n=3$ the orbit has size $8$, and when $n = 4$ it has size $27$. Figure (B) is the center portion of Figure (A).
• Figure 3: The Local Calculation of \ref{['lemma:int-1-to-int-2']}.
• Figure 4: Curves $\alpha_1,\ldots,\alpha_{2g+1}$ on $\Sigma_g$

Theorems & Definitions (29)

• Theorem 1.1: Index of $\mathop{\mathrm{Br}}\nolimits(\pi)$
• Theorem 1.3
• Theorem 1.4
• Proposition 1.5
• Theorem 2.1: Moishezon moishezon, Matsumuto matsumuto
• Remark 2.2
• Theorem 2.3: Moishezon moishezon
• Theorem 3.1
• proof
• Corollary 3.2