The topological fundamental group and free topological groups

Abstract

The topological fundamental group ${\pi }_1^{top}$ 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 $\Sigma (X_{+})$ and find that ${\pi }_1^{top}(\Sigma (X_{+}))$ 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 ${\pi }_1^{top}(\Sigma (X_{+}))$ is a Hausdorff topological group to some well known classification problems in topology.

33 pages

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Cited by in corpus (11)

• The fundamental group as a topological group
• Topological fundamental groups and small generated coverings
• Free Quasitopological Groups
• On Topological Shape Homotopy Groups
• On the discontinuity of the ${\pi }_1$-action
• On Exact Sequences of the Rigid Fibrations
• A natural pseudometric on homotopy groups of metric spaces
• Adjointness of Suspension and Shape Path Functors
• On locally 1-connectedness of quotient spaces and its applications to fundamental groups
• Comparison of Topologies on Fundamental Groups with Subgroup Topology Viewpoint
• Epitopological and pseudotopological fundamental group functors