Dihedral branched covers of four-manifolds

TL;DR

This paper develops formulas and conditions for irregular dihedral branched covers of four-manifolds, including signature calculations, intersection form constraints, and constructions of new examples, advancing understanding of four-manifold topology.

Contribution

It introduces a signature formula for irregular dihedral covers, establishes necessary conditions on intersection forms, and constructs infinite families of such covers, especially with two-bridge slice singularities.

Findings

Signature formula for irregular dihedral covers

Necessary conditions on intersection forms

Construction of infinite families of covers

Abstract

Given a closed oriented PL four-manifold $X$ and a closed surface $B$ embedded in $X$ with isolated cone singularities, we give a formula for the signature of an irregular dihedral cover of $X$ branched along $B$. For $X$ simply-connected, we deduce a necessary condition on the intersection form of a simply-connected irregular dihedral branched cover of $(X, B)$. When the singularities on $B$ are two-bridge slice, we prove that the necessary condition on the intersection form of the cover is sharp. For $X$ a simply-connected PL four-manifold with non-zero second Betti number, we construct infinite families of simply-connected PL manifolds which are irregular dihedral branched coverings of $X$. Given two four-manifolds $X$ and $Y$ whose intersection forms are odd, we obtain a necessary and sufficient condition for $Y$ to be homeomorphic to an irregular dihedral $p$-fold cover of $X$, branched over a surface with a two-bridge slice singularity.