On sections of maps from 4-manifolds to the 2-sphere

TL;DR

This paper constructs examples of singular fibrations from 4-manifolds to the 2-sphere with high-genus fibers that lack sections, including certain Lefschetz fibrations and generic maps, revealing limitations in lifting properties.

Contribution

It demonstrates the existence of singular fibrations with high-genus fibers that do not admit sections, expanding understanding of the topology of maps from 4-manifolds to spheres.

Findings

Existence of fibrations with no sections despite high-genus fibers

Examples include achiral Lefschetz fibrations and generic maps

Some disks cannot lift to sections, neither hemisphere lifts

Abstract

This note exhibits singular fibrations over the 2-sphere whose regular fibers are connected surfaces of arbitrarily high genus, but which admit no sections. These include achiral Lefschetz fibrations, as well as generic maps for which some disks cannot lift to sections -- in fact, neither hemisphere lifts.