Document Type

Article

Publication Date

2-2023

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 πn(X, x0) is to consider the image of πn(X, x0) in the nth Cˇ ech homotopy group πˇ n(X, x0) under the canonical homomorphism 9n : πn(X, x0) → πˇ n(X, x0). The subgroup ker(9n) is the obstruction to this tactic as it consists of precisely those elements of πn(X, x0), which cannot be detected by polyhedral approximations to X. In this paper, we use higher dimensional analogues of Spanier groups to characterize ker(9n). In particular, we prove that if X is paracompact, Hausdorff, and LCn−1, then ker(9n) 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 9n is an isomorphism.

Publication Title

Pacific Journal of Mathematics

ISSN

0030-8730

Publisher

Mathematical Sciences Publishers

Volume

322

Issue

2

First Page

221

Last Page

242

DOI

10.2140/pjm.2023.322.221

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