New examples of Z2-harmonic 1-forms and their deformations

We collect a number of elementary constructions of Z2 harmonic 1-forms, and of families of these objects. These examples show that the branching set Sigma of a Z2 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 Z2 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.