Elements of higher homotopy groups undetectable by polyhedral approximation

John K. Aceti and Jeremy Brazas

Vol. 322 (2023), No. 2, 221–242

DOI: 10.2140/pjm.2023.322.221

Abstract

When nontrivial 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 Čech homotopy group ${\check{\pi }}_n(X,x_0)$ under the canonical homomorphism ${\mathrm{\Psi }}_n:{\pi }_n(X,x_0)\to {\check{\pi }}_n(X,x_0)$. The subgroup $\ker <!--FUNCTION\; APPLICATION-->({\mathrm{\Psi }}_n)$ 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 $\ker <!--FUNCTION\; APPLICATION-->({\mathrm{\Psi }}_n)$. In particular, we prove that if $X$ is paracompact, Hausdorff, and $L{C}^{n-1}$, then $\ker <!--FUNCTION\; APPLICATION-->({\mathrm{\Psi }}_n)$ 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 ${\mathrm{\Psi }}_n$ is an isomorphism.

Keywords

shape homotopy group, higher Spanier group, $\pi_n$-shape injective, $n$-dimensional earring space

Mathematical Subject Classification

Primary: 55P55, 55Q07, 55Q52

Secondary: 54C56

Milestones

Received: 14 August 2022

Revised: 17 February 2023

Accepted: 18 February 2023

Published: 23 May 2023