Prescribed Singular Sets for Z/2-Harmonic 1-Forms on R^n

Abstract

Z/2 harmonic 1-forms arise naturally as singular limits in gauge theory and calibrated geometry, but the topology that can occur in their singular sets is poorly understood. In this paper we study the flexibility of these singular sets when the ambient Riemannian metric is allowed to vary. We prove that every compact smoothly embedded codimension-two submanifold with trivial normal bundle in a Euclidean space of dimension at least three can be realized as the singular set of a nondegenerate Z/2 harmonic 1-form for a complete metric that is Euclidean outside a compact set. As an application, we use Calabi surgery to construct Z/2-harmonic 1-forms with prescribed local singular sets on closed manifolds of positive first Betti number.