Higher homotopy wild sets

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.