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 $x_0$.
Categories: math.AT