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Quasiregular cobordism theorem

Pekka Pankka, Jang-Mei Wu

Abstract

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.

Quasiregular cobordism theorem

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 bilipschitz equivalent to the Euclidean -sphere but which admits a BLD-map to . 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 -dimensional deformation theory originated by S.~Rickman in 1985.

Paper Structure

This paper contains 156 sections, 154 theorems, 547 equations, 37 figures.

Table of Contents

  1. Introduction
  2. Applications
  3. Outline of the proof
  4. Remarks on deformation and partitioning
  5. Preliminaries
  6. Preliminaries on complexes
  7. Cubical complexes
  8. Passing from PL structure to a cubical structure
  9. Standard refinements and standard metrics
  10. Passing from Riemannian structure to cubical structure
  11. Barycentric triangulation of a cubical complex
  12. Barycentric subdivisions of triangulations
  13. Adjacency graph of a complex
  14. Realization of a subgraph
  15. Quotient maps associated to realizations
  16. ...and 141 more sections

Key Result

Theorem 1.0.1

Let $n\geq 3$, $m\geq 2$, $M$ be a compact connected oriented Riemannian $n$-manifold with $m$ boundary components, and $B_1,\ldots, B_m$ be pairwise disjoint closed Euclidean $n$-balls in $\mathbb{S}^n$. Then there exists a constant $\mathsf{K}=\mathsf{K}(n,M, B_1,\ldots, B_m)\ge 1$ such that, for of degree at least $d_0$, which is $\mathsf{K}$-quasiregular in the interior of $M$.

Figures (37)

  • Figure 1.0.1: A degree $2$ Alexander map expanded by a simple cover;
  • Figure 1.2.1: $\mathbb{S}^2\setminus \mathrm{int} (B_1\cup\cdots \cup B_4)$ partitioned by branched sphere $\mathbf{S}$ into $4$ spherical shells.
  • Figure 1.3.1: A shellable cubical $2$-complex $K$, its barycentric triangulation $K^\Delta$, and a star-replacement $K^*$ of $K^\Delta$.
  • Figure 2.1.1: A cubical complex $K$ and its triangulation $K^\Delta$.
  • Figure 2.2.1: Left: the adjacency graph $\Gamma(K)$ of complex $K$ in Example \ref{['ex:adjacency']}. Right: A spanning tree $G$ of $\Gamma(K)$.
  • ...and 32 more figures

Theorems & Definitions (394)

  • Theorem 1.0.1
  • Theorem 1.0.2: Quasiregular cobordism theorem
  • Remark 1.0.3: Remark on the name of the theorem
  • Theorem 1.1.1: Preassigned preimages
  • Theorem 1.1.2: Large local index
  • Theorem 1.1.3: Wildly branching quasiregular map
  • Theorem 1.1.4: Wild Julia set
  • Theorem 1.3.1: Deformation of Alexander maps
  • Theorem 1.3.2: Hopf theorem for Alexander maps
  • Theorem 1.3.3: Lakes of Wada
  • ...and 384 more