Fundamental groups of reduced suspensions are locally free

In this paper, we analyze the fundamental group ${\pi }_1(\mathrm{\Sigma }X,\overline{x_0})$ of the reduced suspension $\mathrm{\Sigma }X$ where $(X,x_0)$ is an arbitrary based Hausdorff space. We show that ${\pi }_1(\mathrm{\Sigma }X,\overline{x_0})$ is canonically isomorphic to a direct limit ${\mathrm{}}_{A\in \mathcal{P}}{\pi }_1(\mathrm{\Sigma }A,\overline{x_0})$ where each group ${\pi }_1(\mathrm{\Sigma }A,\overline{x_0})$ is isomorphic to a finitely generated free group or the infinite earring group. A direct consequence of this characterization is that ${\pi }_1(\mathrm{\Sigma }X,\overline{x_0})$ is locally free for any Hausdorff space $X$. Additionally, we show that $\mathrm{\Sigma }X$ is simply connected if and only if $X$ is sequentially $0$-connected at $x_0$.