Construction of $\mathbb{Z}_2$-harmonic 1-forms on closed 3-manifolds with long cylindrical necks

In this paper, we give an explicit construction of families of $\mathbb{Z}_2$-harmonic 1-forms that degenerate to manifolds with cylindrical ends. We do this by considering certain linear combinations of $L^2$-bounded $\mathbb{Z}_2$-harmonic 1-forms and by modifying the metric near the link. This construction can always be done if the homology group that counts $L^2$-bounded $\mathbb{Z}_2$-harmonic 1-forms is sufficiently large. This has the consequence that every smooth link can be obtained as the singular set of a $\mathbb{Z}_2$-harmonic 1-form on some 3-manifold.