John K. Aceti | Searcharxiv

arXiv2024

On The Whisker Topology

The purpose of this paper is to explore properties of the whisker topology, which is a topology endowed on the fundamental group and whose utility is to detect locally complicated phenomena in pathological topological spaces. We show that the whisker topology preserves products, resolve an open question regarding the existence of a space which makes $π_1^{wh}(X,x_0)$ a non-discrete, non-abelian, and Hausdorff topological group, and show the whisker topology is not separable on the earring group $π_1(\Er^1,x_0)$.

math.GN↗

arXiv2023

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 $π_n(X,x_0)$ is to consider the image of $π_n(X,x_0)$ in the $n$-th Čech homotopy group $\checkπ_n(X,x_0)$ under the canonical homomorphism $Ψ_{n}:π_n(X,x_0)\to \checkπ_n(X,x_0)$. The subgroup $\ker(Ψ_n)$ is the obstruction to this tactic as it consists of precisely those elements of $π_n(X,x_0)$, which cannot be detected by polyhedral approximations to $X$. In this paper, we use higher dimensional analogues of Spanier groups to characterize $\ker(Ψ_n)$. In particular, we prove that if $X$ is paracompact, Hausdorff, and $LC^{n-1}$, then $\ker(Ψ_n)$ 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 $Ψ_{n}$ is an isomorphism.

math.AT↗