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

We collect a number of elementary constructions of $\Z_2$ harmonic $1$-forms, and of families of these objects. These examples show that the branching set $\Sigma$ of a $\Z_2$ harmonic 1-form may exhibit the following features: i) $\Sigma$ may be a non-trivial link; ii) $\Sigma$ may be a multiple cover; iii) $\Sigma$ may be immersed, and appear as a limit of smoothly embedded branching loci; iv) there are families of $\Z_2$ harmonic $1$-forms whose branching sets $\Sigma$ 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.