Existence of nondegenerate $\mathbb{Z}_2$ harmonic 1-forms via $\mathbb{Z}_3$ symmetry

TL;DR

This paper establishes a topological criterion for the existence of nondegenerate $bZ_2$ harmonic 1-forms on Riemannian manifolds using $bZ_3$ symmetry, with applications to links and homology 3-spheres.

Contribution

It introduces a new topological condition leveraging $bZ_3$ symmetry for the existence of $bZ_2$ harmonic 1-forms, including specific cases for links and homology spheres.

Findings

Existence of $bZ_2$ harmonic 1-forms for links with zero determinant.

Infinite rational homology 3-spheres admit such harmonic forms.

Application of $bZ_3$ symmetry to topological existence problems.

Abstract

Using $\mathbb{Z}_3$ symmetry, we present a topological condition for the existence of the $\mathbb{Z}_2$ harmonic 1-forms over Riemannian manifold. As a corollary, if $L$ is an oriented link on $S^3$ with determinant zero, then there exists a nondegenerate $\mathbb{Z}_2$ harmonic 1-form over the 3-cyclic branched covering of $L$. Furthermore, we found infinite number of rational homology 3-spheres that admit a nondegenerate $\mathbb{Z}_2$ harmonic 1-form.