On sections of Lefschetz fibrations and bundles over 2-complexes

TL;DR

This paper investigates conditions for the existence of sections in fibrations over 2-complexes and Lefschetz fibrations, providing algebraic criteria and applications to achiral Lefschetz fibrations.

Contribution

It establishes algebraic criteria for sections in fibrations over 2-complexes and Lefschetz fibrations, including conditions for multiple homologically distinct sections.

Findings

A bundle over a finite 2-complex admits a section iff the fiber inclusion is π₁-injective and the fundamental group sequence splits.

Complete algebraic criteria are given for extending loops to sections in Lefschetz fibrations over the disk.

Criteria are provided for the existence of at least two homologically distinct sections in certain doubled Lefschetz fibrations.

Abstract

We address the question of existence of sections of fibrations in two settings. First, we show that a bundle with base a finite 2-complex admits a section if and only if the inclusion of the fiber is $\pi_1$-injective and the associated short exact sequence of fundamental groups splits. Second, for Lefschetz fibrations over the disk we provide a complete algebraic criterion characterizing which loops in the boundary mapping torus extend to continuous or smooth sections over the disk. Finally, we apply our results to achiral Lefschetz fibrations over the sphere obtained by doubling along the vertical boundary, and give a criterion ensuring the existence of at least two homologically distinct sections.