Examples of $\mathbb{Z}/2$-Harmonic 1-Forms

Abstract

In this paper, we develop methods for constructing $\mathbb{Z}/2$-harmonic 1-forms in dimension three by varying the background metric. On $\mathbb{R}^3$, we construct an example for which the complement of the smooth locus in the singular set is a Cantor set. We realize every finite graph with positive even valence at each vertex as the monodromy locus of a $\mathbb{Z}/2$-harmonic 1-form on $B^3$. We also construct desingularization models for every critical $\mathbb{Z}/2$-eigensection, agreeing exactly with the associated homogeneous model outside a compact set. Finally, we construct a nondegenerate $\mathbb{Z}/2$-harmonic 1-form on every closed connected oriented three-manifold for a suitable smooth metric.