Fundamental groups of reduced suspensions are locally free

TL;DR

This paper characterizes the fundamental group of reduced suspensions of Hausdorff spaces as a direct limit of free or earring groups, proving it is always locally free and linking simple connectivity to sequential 0-connectedness.

Contribution

It provides a canonical isomorphism for the fundamental group of reduced suspensions and establishes its local freeness for any Hausdorff space.

Findings

Fundamental group is a direct limit of free or earring groups.

Fundamental group is locally free for any Hausdorff space.

Reduced suspension is simply connected iff the space is sequentially 0-connected.

Abstract

In this paper, we analyze the fundamental group $\pi_1(\Sigma X,\overline{x_0})$ of the reduced suspension $\Sigma X$ where $(X,x_0)$ is an arbitrary based Hausdorff space. We show that $\pi_1(\Sigma X,\overline{x_0})$ is canonically isomorphic to a direct limit $\varinjlim_{A\in\mathscr{P}}\pi_1(\Sigma A,\overline{x_0})$ where each group $\pi_1(\Sigma A,\overline{x_0})$ is isomorphic to a finitely generated free group or the infinite earring group. A direct consequence of this characterization is that $\pi_1(\Sigma X,\overline{x_0})$ is locally free for any Hausdorff space $X$. Additionally, we show that $\Sigma X$ is simply connected if and only if $X$ is sequentially $0$-connected at $x_0$.