Quasiregular cobordism theorem

In this article we prove that, for an oriented PL $n$-manifold $M$ with $m$ boundary components and $d_0\in \mathbb N$, there exist mutually disjoint closed Euclidean balls and a $\mathsf K$-quasiregular mapping $M \to \mathbb S^n \setminus \mathrm{int}(B_1\cup \cdots \cup B_m)$ of degree at least $d_0$. The result is quantitative in the sense that the distortion $\mathsf K$ of the mapping does not depend on the degree. As applications of this construction, we obtain Rickman's large local index theorem for quasiregular maps to all dimensions $n\ge 4$. We also construct, in dimension $n=4$, a version of a wildly branching quasiregular map of Heinonen and Rickman, and a uniformly quasiregular map of arbitrarily large degree whose Julia set is a wild Cantor set. The existence of a wildly branching quasiregular map yields an example of a metric $4$-sphere $(\mathbb S^4,d)$, which is not bilipschitz equivalent to the Euclidean $4$-sphere $\mathbb S^4$ but which admits a BLD-map to $\mathbb S^4$. For the proof of the main theorem, we develop a dimension-free deformation method for cubical Alexander maps. For cubical and shellable Alexander maps this completes the $2$-dimensional deformation theory originated by S.~Rickman in 1985.