Four-manifolds admitting hyperelliptic broken Lefschetz fibrations

TL;DR

This paper introduces hyperelliptic broken Lefschetz fibrations, constructs involutions on their total spaces, and shows the fiber's non-trivial homology class, expanding understanding of four-manifold structures.

Contribution

It generalizes hyperelliptic Lefschetz fibrations to broken fibrations, constructs involutions on these spaces, and analyzes their homological properties.

Findings

Constructed involutions on total spaces of hyperelliptic broken Lefschetz fibrations.

Extended involutions induce double branched coverings over sphere bundles.

The fiber of such fibrations represents a non-trivial rational homology class.

Abstract

We introduce hyperelliptic simplified (more generally, directed) broken Lefschetz fibrations, which is a generalization of hyperelliptic Lefschetz fibrations. We construct involutions on the total spaces of such fibrations of genus $g\geq 3$ and extend these involutions to the four-manifolds obtained by blowing up the total spaces. The extended involutions induce double branched coverings over blown up sphere bundles over the sphere. We also show that the regular fiber of such a fibration of genus $g\geq 3$ represents a non-trivial rational homology class of the total space.