New examples of Z/2 harmonic 1-forms and their deformations

TL;DR

This paper presents new elementary constructions of $ ext{Z}_2$ harmonic 1-forms, illustrating diverse features of their branching sets, including non-trivial links, multiple covers, immersions, and continuous families of tangent cones across dimensions.

Contribution

It introduces explicit examples demonstrating complex branching set behaviors of $ ext{Z}_2$ harmonic 1-forms, expanding understanding of their geometric and topological properties.

Findings

Branching sets can be non-trivial links in dimension three.

Branching sets can be multiple covers in dimension three.

Families of harmonic 1-forms can have tangent cones forming positive-dimensional spaces.

Abstract

We collect a number of elementary constructions of $\Z_2$ harmonic $1$-forms, and of families of these objects. These examples show that the branching set $\Sigma$ of a $\Z_2$ harmonic 1-form may exhibit the following features: i) $\Sigma$ may be a non-trivial link; ii) $\Sigma$ may be a multiple cover; iii) $\Sigma$ may be immersed, and appear as a limit of smoothly embedded branching loci; iv) there are families of $\Z_2$ harmonic $1$-forms whose branching sets $\Sigma$ have tangent cones filling out a positive dimensional space, even modulo isometries. We show that Features i) and ii) occur already in dimension three, while the remaining ones appear at least in dimension four and higher.