Higher homotopy wild sets

TL;DR

This paper investigates the properties of higher homotopy wild sets in topological spaces, establishing their homotopy invariance and exploring their possible topological types, extending known results from the one-dimensional case.

Contribution

It proves that the homotopy type of higher homotopy wild sets is a homotopy invariant and shows their potential topological diversity in Peano continua.

Findings

Homotopy type of $oldsymbol{w}_n(X)$ is a homotopy invariant.

Homeomorphism type of $oldsymbol{w}_n(X)$ is a homotopy invariant for certain spaces.

The $oldsymbol{ ext{pi}}_n$-wild set of a Peano continuum can be any compact metric space.

Abstract

The $\pi_n$-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 $\pi_n$-shape injective metric spaces, the homeomorphism type of $\mathbf{w}_{n}(X)$ is a homotopy invariant of $X$. We also prove that the $\pi_n$-wild set of a Peano continuum can be homeomorphic to any compact metric space.