On nondegenerate $\mathbb{Z}_{2}$-harmonic $1$-forms with shrinking branching sets

Abstract

We develop a gluing theorem for non-degenerate $\mathbb{Z}_{2}$-harmonic $1$-forms on compact manifolds, in which non-degenerate $\mathbb{Z}_{2}$-harmonic $1$-forms on $\mathbb{R}^{n}$ are glued to the regular zeros of a non-degenerate $\mathbb{Z}_{2}$-harmonic $1$-form. As an immediate consequence, viewing an ordinary harmonic $1$-form as a $\mathbb{Z}_{2}$-harmonic $1$-form without branching set, we prove that for every compact oriented manifold $M^{n}, n\geq 3$, if the first Betti number $b^{1}(M)>0$, then $M$ admits a family of non-degenerate $\mathbb{Z}_{2}$-harmonic $1$-forms, which resolves a folklore conjecture. We will also discuss several possible applications to special holonomy, in particular, to the field of $G_{2}$-geometry.