Quasiregular cobordism theorem
Pekka Pankka, Jang-Mei Wu

TL;DR
This paper proves the existence of certain quasiregular maps on manifolds with boundary, leading to new results in quasiregular map theory, including large local index theorems and examples of wild Julia sets.
Contribution
It introduces a dimension-free deformation method for cubical Alexander maps and constructs novel quasiregular maps with wild topological properties.
Findings
Existence of degree-at-least-d0 quasiregular maps with controlled distortion
Extension of Rickman's large local index theorem to all dimensions n≥4
Construction of wild Julia sets and metric spheres with unique properties
Abstract
In this article we prove that, for an oriented PL -manifold with boundary components and , there exist mutually disjoint closed Euclidean balls and a -quasiregular mapping of degree at least . The result is quantitative in the sense that the distortion 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 . We also construct, in dimension , 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 -sphere , which is not…
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Taxonomy
TopicsGeometric Analysis and Curvature Flows · Analytic and geometric function theory · Geometric and Algebraic Topology
