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| Autor principal: | |
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| Formato: | Preprint |
| Publicado: |
2025
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| Materias: | |
| Acceso en línea: | https://arxiv.org/abs/2510.07678 |
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- 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.