A stabilization result for Z/2-harmonic 1-forms by constructing solutions on closed 3-manifolds with long cylindrical necks

Abstract

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 works if number of $L^2$-bounded $\mathbb{Z}_2$-harmonic 1-forms is strictly more than twice the number of connected components of the link. This can always be done if we consider a connected sum with a 3-manifold with sufficiently large $b_1$.