The topological fundamental group and free topological groups

The topological fundamental group $π_{1}^{top}$ is a homotopy invariant finer than the usual fundamental group. It assigns to each space a quasitopological group and is discrete on spaces which admit universal covers. For an arbitrary space $X$, we compute the topological fundamental group of the suspension space $Σ(X_+)$ and find that $π_{1}^{top}(Σ(X_+))$ either fails to be a topological group or is the free topological group on the path component space of $X$. Using this computation, we provide an abundance of counterexamples to the assertion that all topological fundamental groups are topological groups. A relation to free topological groups allows us to reduce the problem of characterizing Hausdorff spaces $X$ for which $π_{1}^{top}(Σ(X_+))$ is a Hausdorff topological group to some well known classification problems in topology.

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