New examples of Z/2 harmonic 1-forms and their deformations

We collect a number of elementary constructions of ${\mathbb{Z}}_2$ harmonic $1$-forms, and of families of these objects. These examples show that the branching set $\mathrm{\Sigma }$ of a ${\mathbb{Z}}_2$ harmonic 1-form may exhibit the following features: i) $\mathrm{\Sigma }$ may be a non-trivial link; ii) $\mathrm{\Sigma }$ may be a multiple cover; iii) $\mathrm{\Sigma }$ may be immersed, and appear as a limit of smoothly embedded branching loci; iv) there are families of ${\mathbb{Z}}_2$ harmonic $1$-forms whose branching sets $\mathrm{\Sigma }$ have tangent cones filling out a positive dimensional space, even modulo isometries. We show that Features i) and ii) occur already in dimension three, while the remaining ones appear at least in dimension four and higher.