Pull-back of differential forms by multi-valued Sobolev maps, and the quasiregularity of the multi-valued inverse of a quasiregular map

We use Almgren’s framework of multi-valued maps to construct a multi-valued inverse $F:f(\mathrm{\Omega })\to {\mathcal{A}}_d({\mathbb{R}}^n)$ of a quasiregular map $f:\mathrm{\Omega }\to {\mathbb{R}}^n$ of finite degree $d$. We then develop a pull-back theory of differential forms on ${\mathcal{A}}_d({\mathbb{R}}^n)$ by Sobolev maps, and use it to show that the multi-valued inverse is a quasiregular $\omega$-curve (in the sense of Pankka) with respect to a natural $n$-form $\omega$ (suitably interpreted). The pull-back theory is of independent interest, and allows us to conclude e.g. higher Sobolev integrability and quasiminimality of the multi-valued inverse.