On nondegenerate ${\mathbb{Z}}_2$-harmonic $1$-forms with shrinking branching sets

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 $M^n,n\ge 3$, 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.