Elements of higher homotopy groups undetectable by polyhedral approximation

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 ${\mathrm{\Psi }}_n:{\pi }_n(X,x_0)\to {\check{\pi }}_n(X,x_0)$. The subgroup $\ker ({\mathrm{\Psi }}_n)$ 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 $\ker ({\mathrm{\Psi }}_n)$. In particular, we prove that if $X$ is paracompact, Hausdorff, and $L{C}^{n-1}$, then $\ker ({\mathrm{\Psi }}_n)$ 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.