Elements of Higher Homotopy Groups Undetectable by Polyhedral Approximation
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 Cech homotopy group ˇ 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.