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.
Categories: math.DG math.GT