Fundamental groups of reduced suspensions are locally free

In this paper, we analyze the fundamental group $π_1(ΣX,\overline{x_0})$ of the reduced suspension $ΣX$ where $(X,x_0)$ is an arbitrary based Hausdorff space. We show that $π_1(ΣX,\overline{x_0})$ is canonically isomorphic to a direct limit $\varinjlim_{A\in\mathscr{P}}π_1(ΣA,\overline{x_0})$ where each group $π_1(ΣA,\overline{x_0})$ is isomorphic to a finitely generated free group or the infinite earring group. A direct consequence of this characterization is that $π_1(ΣX,\overline{x_0})$ is locally free for any Hausdorff space $X$. Additionally, we show that $ΣX$ is simply connected if and only if $X$ is sequentially $0$-connected at $x_0$.

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