Elements of higher homotopy groups undetectable by polyhedral approximation

TL;DR

This paper characterizes the elements of higher homotopy groups undetectable by polyhedral approximation using higher Spanier groups, providing conditions under which the canonical homomorphism is an isomorphism.

Contribution

It introduces higher Spanier groups to describe the kernel of the homomorphism from homotopy groups to Čech homotopy groups, generalizing previous theorems.

Findings

Kernel of the canonical homomorphism equals the higher Spanier group under certain conditions.

Provides a generalization of Kozlowski-Segal's theorem on when the homomorphism is an isomorphism.

Characterizes higher homotopy elements undetectable by polyhedral approximation.

Abstract

When non-trivial local structures are present in a topological space $X$, a common approach to characterizing the isomorphism type of the $n$-th homotopy group $\pi_n(X,x_0)$ is to consider the image of $\pi_n(X,x_0)$ in the $n$-th \v{C}ech homotopy group $\check{\pi}_n(X,x_0)$ under the canonical homomorphism $\Psi_{n}:\pi_n(X,x_0)\to \check{\pi}_n(X,x_0)$. The subgroup $\ker(\Psi_n)$ is the obstruction to this tactic as it consists of precisely those elements of $\pi_n(X,x_0)$, which cannot be detected by polyhedral approximations to $X$. In this paper, we use higher dimensional analogues of Spanier groups to characterize $\ker(\Psi_n)$. In particular, we prove that if $X$ is paracompact, Hausdorff, and $LC^{n-1}$, then $\ker(\Psi_n)$ is equal to the $n$-th Spanier group of $X$. We also use the perspective of higher Spanier groups to generalize a theorem of Kozlowski-Segal, which gives conditions ensuring that $\Psi_{n}$ is an isomorphism.