Perturbation and Pruning of Nondegenerate $\mathbb{Z}/2$ Harmonic 1-forms

TL;DR

This paper demonstrates that nondegenerate  harmonic 1-forms can be perturbed to have discrete zero sets, and explores the topological and geometric implications of such forms on 3-manifolds, including their leaf space structures.

Contribution

It introduces a method to perturb nondegenerate  harmonic 1-forms to achieve discrete zero sets and analyzes the resulting geometric and topological properties.

Findings

Existence of metric perturbations producing discrete zero sets.

Generic forms have leaf spaces that are -trees.

On rational homology spheres, forms can be adjusted to have exactly two connected singular components.

Abstract

We prove that for any nondegenerate $\mathbb{Z}/2$ harmonic $1$-form, there exists a metric perturbation producing a new nondegenerate $\mathbb{Z}/2$ harmonic $1$-form whose ordinary zero set is discrete. As an application, we show that for generic smooth nondegenerate $\mathbb{Z}/2$ harmonic $1$-forms, the leaf spaces are $\mathbb{Z}$-trees. Moreover, we show that if a $3$-dimensional rational homology sphere admits a smooth nondegenerate $\mathbb{Z}/2$-harmonic $1$-form, then there exists another nondegenerate $\mathbb{Z}/2$-harmonic $1$-form whose singular locus has exactly two connected components.