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A note on the topology of Lefschetz fibrations

Sierra Knavel

TL;DR

The paper derives a tight global bound on the first Betti number for nontrivial genus-$g$ Lefschetz fibrations and proves that a transitive monodromy forces simple connectivity of the total space. It extends the bound to bases of higher genus and analyzes how vanishing cycles (separating vs nonseparating) control $b_1(X)$ and the abelianization of the fundamental group. In the genus-2 hyperelliptic case, it obtains strong constraints on the vanishing-cycle data, including a refinement of $2n-s\le 5$ and the identification of an indecomposable family with $b_1=2$ and $b_2=n+s-2$, as well as connections via lantern substitutions to other $(n,s)$-types. The results contribute to conjectures that genus-2 fibrations have abelian fundamental groups with at most two generators and provide tools for exploring large fundamental groups in symplectic 4-manifolds through explicit monodromy computations.

Abstract

We prove an upper bound for the first Betti number of a nontrivial genus-$g$ Lefschetz fibration. We also show that if the monodromy of a Lefschetz fibration is transitive with respect to the mapping class group, the Lefschetz fibration is simply connected. Lastly, we discuss a potential family of indecomposable genus-2 Lefschetz fibrations with maximally non-trivial first homology which would be candidates for large fundamental group computations, if they exist.

A note on the topology of Lefschetz fibrations

TL;DR

The paper derives a tight global bound on the first Betti number for nontrivial genus- Lefschetz fibrations and proves that a transitive monodromy forces simple connectivity of the total space. It extends the bound to bases of higher genus and analyzes how vanishing cycles (separating vs nonseparating) control and the abelianization of the fundamental group. In the genus-2 hyperelliptic case, it obtains strong constraints on the vanishing-cycle data, including a refinement of and the identification of an indecomposable family with and , as well as connections via lantern substitutions to other -types. The results contribute to conjectures that genus-2 fibrations have abelian fundamental groups with at most two generators and provide tools for exploring large fundamental groups in symplectic 4-manifolds through explicit monodromy computations.

Abstract

We prove an upper bound for the first Betti number of a nontrivial genus- Lefschetz fibration. We also show that if the monodromy of a Lefschetz fibration is transitive with respect to the mapping class group, the Lefschetz fibration is simply connected. Lastly, we discuss a potential family of indecomposable genus-2 Lefschetz fibrations with maximally non-trivial first homology which would be candidates for large fundamental group computations, if they exist.

Paper Structure

This paper contains 7 sections, 6 theorems, 22 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Lefschetz fibrations
  4. Hyperelliptic Lefschetz fibrations
  5. Indecomposability of Lefschetz fibrations
  6. Results on nontrivial Lefschetz fibrations
  7. Applications to genus-2 Lefschetz fibrations

Key Result

Theorem 1.1

Let $X$ be a smooth, closed, compact 4-manifold and $f: X \rightarrow S^2$ be a nontrivial genus-$g$ Lefschetz fibration, then

Figures (3)

  • Figure 1: Lantern relation, $\tau_{\partial_1} \tau_{\partial_2}\tau_{\partial_3} \tau_{\partial_4} = \tau_A \tau_B \tau_C$. Note that the separating curve depends on the embedding of the four-punctured sphere into the fiber surface.
  • Figure 2: The standard curves on the genus-2 surface.
  • Figure 3: Genus-2 Lefschetz fibrations of type $(n,s)$. Red line is $2n-s=3$, Blue line is $2n-s=5$, the mark $\newmoon$ indicates a type $(n,s)$ Lefschetz fibration whose monodromy has been explicitly found, the mark $\fullmoon$ indicates a possible $(n,s)$ whose monodromy has not been found.

Theorems & Definitions (16)

  • Theorem 1.1
  • Theorem 1.3
  • Remark 1.4
  • Lemma 1.5
  • Remark 1.6
  • Theorem 1.7
  • proof : Proof of Theorem \ref{['ThmTransitivity']}
  • proof : Proof of Theorem \ref{['ThmBettinum']}
  • Remark 3.2
  • Theorem 3.3
  • ...and 6 more