Elements of higher homotopy groups undetectable by polyhedral approximation

• Published in: arXiv.org
• Main Authors: Aceti, John K, Brazas, Jeremy
• Format: Paper Journal Article
• Language: English
• Published: Ithaca Cornell University Library, arXiv.org 24.01.2023
• Subjects: Homomorphisms; Isomorphism; Mathematics - Algebraic Topology; Subgroups
• ISSN: 2331-8422
• DOI: 10.48550/arxiv.2208.06645

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 Čech 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.