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Higher homotopy wild sets

Jeremy Brazas, Atish Mitra

TL;DR

This work extends the concept of wild points to higher homotopy, defining the $\pi_n$-wild set $\mathbf{w}_n(X)$ as the locus of points admitting fully essential maps from the $n$-dimensional infinite earring $\mathbb{E}_n$. It proves that the homotopy type of $\mathbf{w}_n(X)$ is a homotopy invariant of $X$, and for certain $n$-dimensional, $\pi_n$-shape injective Peano continua, the homeomorphism type of $\mathbf{w}_n(X)$ is also a homotopy invariant. A central contribution is a construction showing that any compact metric space can occur as $\mathbf{w}_n(X)$ for some Peano continuum, via shrinking point-attachments and wildification of copies of $\mathbb{E}_n$. The paper also develops a rigidity framework: completely $\pi_n$-rigid spaces have wild sets that are preserved under homotopy equivalence, with concrete results in the inverse-limit and 2D ($n=2$) cases, linking shape injectivity, rigidity, and wild sets to distinguish homotopy types beyond traditional homotopy invariants.

Abstract

The $π_n$-wild set $\mathbf{w}_{n}(X)$ of a topological space $X$ is the subspace of $X$ consisting of the points at which there exists a shrinking sequence of essential based maps $S^n\to X$. In this paper, we show that the homotopy type of $\mathbf{w}_{n}(X)$ is a homotopy invariant of $X$ and, in analogy to the known one-dimensional case, we show that for certain $n$-dimensional $π_n$-shape injective metric spaces, the homeomorphism type of $\mathbf{w}_{n}(X)$ is a homotopy invariant of $X$. We also prove that the $π_n$-wild set of a Peano continuum can be homeomorphic to any compact metric space.

Higher homotopy wild sets

TL;DR

This work extends the concept of wild points to higher homotopy, defining the -wild set as the locus of points admitting fully essential maps from the -dimensional infinite earring . It proves that the homotopy type of is a homotopy invariant of , and for certain -dimensional, -shape injective Peano continua, the homeomorphism type of is also a homotopy invariant. A central contribution is a construction showing that any compact metric space can occur as for some Peano continuum, via shrinking point-attachments and wildification of copies of . The paper also develops a rigidity framework: completely -rigid spaces have wild sets that are preserved under homotopy equivalence, with concrete results in the inverse-limit and 2D () cases, linking shape injectivity, rigidity, and wild sets to distinguish homotopy types beyond traditional homotopy invariants.

Abstract

The -wild set of a topological space is the subspace of consisting of the points at which there exists a shrinking sequence of essential based maps . In this paper, we show that the homotopy type of is a homotopy invariant of and, in analogy to the known one-dimensional case, we show that for certain -dimensional -shape injective metric spaces, the homeomorphism type of is a homotopy invariant of . We also prove that the -wild set of a Peano continuum can be homeomorphic to any compact metric space.

Paper Structure

This paper contains 6 sections, 34 theorems, 6 equations, 5 figures.

Table of Contents

  1. Introduction
  2. The $\pi_n$-wild set of a space
  3. Basic Properties of $\pi_n$-wild sets
  4. Homotopy invariance of $\pi_n$-wild sets
  5. Constructing Spaces with Prescribed Wild Sets
  6. Rigidity of wild sets

Key Result

Theorem 1.2

Let $n\geq 1$. If $X$ is a Peano continuum, then $\mathbf{w}_{n}(X)$ is a compact metric space. Moreover, if $C$ is any compact metric space, then there exists a Peano continuum $X$ such that

Figures (5)

  • Figure 1: A space obtained by attaching copies of $\mathbb{E}_1$ to $[0,1]$ along the points $1/k$ (in the weak topology).
  • Figure 2: The Warsaw circle in the $xy$-plane with a sequence of circles of shrinking radius attached along a dense subset of the non-compact path component of the topologist's sine curve (illustrated in gray).
  • Figure 3: The stages $Q_1$ and $Q_2$ in the construction of the $2$-dimensional case of the Peano continuum $Q_{\infty}$.
  • Figure 4: A one-dimensional Peano continuum with a non-simply connected $\pi_1$-wild set (left) and the filled-in version (right), which is not homotopy equivalent to any one-dimensional space.
  • Figure 5: Attaching a shrinking sequence of $2$-spheres along the dyadic rationals in $[0,1]$.

Theorems & Definitions (91)

  • Theorem 1.2
  • Theorem 1.3
  • Corollary 1.4
  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • Definition 2.4
  • Remark 2.5
  • Example 2.6
  • Example 2.7
  • ...and 81 more