The Topological Fundamental Group and Hoop Earring Spaces

TL;DR

This paper investigates the topological fundamental group of hoop earring spaces, revealing that it is not always a topological group and providing counterexamples to existing assumptions about its functorial properties.

Contribution

It computes the topological fundamental group for hoop earring spaces derived from totally path-disconnected spaces and demonstrates that it is not a topological group in general.

Findings

The topological fundamental group of hoop earring spaces can be non-Hausdorff.

The paper provides a factorization of the quotient map through a free topological monoid.

Counterexamples show $ ext{pi}_1^{top}$ is not always a topological group.

Abstract

The topological fundamental group $\pi_{1}^{top}$ is a topological invariant that assigns to each space a quasi-topological group and is discrete on spaces which are well behaved locally. For a totally path-disconnected, Hausdorff, unbased space $X$, we compute the topological fundamental group of the "hoop earring" space of $X$, which is the reduced suspension of $X$ with disjoint basepoint. We do so by factorizing the quotient map $\Omega(\Sigma X_{+},x)\to \pi_{1}^{top}(\Sigma X_{+},x)$ through a free topological monoid with involution $M(X)$ such that the map $M(X)\shortrightarrow \pi_{1}^{top}(\Sigma X_{+},x)$ is also a quotient map. $\pi_{1}^{top}(\Sigma X_{+},x)$ is T1 and an embedding $X\shortrightarrow \pi_{1}^{top}(\Sigma X_{+},x)$ illustrates that $\pi_{1}^{top}(\Sigma X_{+},x)$ is not a topological group when $X$ is not regular. These hoop earring spaces provide a simple class of counterexamples to the claim that $\pi_{1}^{top}$ is a functor to the category of topological groups.