Elements of higher homotopy groups 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 πnpX, x0q is to consider the image of πnpX, x0q in the n-th Čech homotopy group πnpX, x0q under the canonical homomorphism Ψn : πnpX, x0q Ñ πnpX, x0q. The subgroup kerpΨnq is the obstruction to this tactic as it consists of precisely those elements of πnpX, x0q, which cannot be detected by polyhedral approximations to X. In this paper, we use higher dimensional analogues of Spanier groups to characterize kerpΨnq. In particular, we prove that if X is paracompact, Hausdorff, and U V n´1 , then kerpΨnq 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 Ψn is an isomorphism.