The fundamental group as a topological group

Abstract

This paper is devoted to the study of a natural group topology on the fundamental group which remembers local properties of spaces forgotten by covering space theory and weak homotopy type. It is known that viewing the fundamental group as the quotient of the loop space often fails to result in a topological group; we use free topological groups to construct a topology which promotes the fundamental group of any space to topological group structure. The resulting invariant, denoted ${\pi }_1^{\tau }$, takes values in the category of topological groups, can distinguish spaces with isomorphic fundamental groups, and agrees with the quotient fundamental group precisely when the quotient topology yields a topological group. Most importantly, this choice of topology allows us to naturally realize free topological groups and pushouts of topological groups as fundamental groups via topological analogues of classical results in algebraic topology.

23 pages

References in corpus (6)

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• On small homotopies of loops
• The topological fundamental group and free topological groups
• Discreteness and Homogeneity of the Topological Fundamental Group
• On Topological Homotopy Groups of $n$-Hawaiian like spaces
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Cited by in corpus (18)

• Topological fundamental groups and small generated coverings
• Topological and uniform structures on universal covering spaces
• On the Spanier Groups and Covering and Semicovering Spaces
• Test map characterizations of local properties of fundamental groups
• Free Quasitopological Groups
• On Topological Shape Homotopy Groups
• Homotopical Algebra in Categories with Enough Projectives
• On the discontinuity of the ${\pi }_1$-action
• On quasitopological homotopy groups of Inverse Limit Spaces
• A natural pseudometric on homotopy groups of metric spaces
• On Subgroup Topologies on Fundamental Groups
• Open subgroups of free topological groups
• On zero dimensional sequential spaces
• Adjointness of Suspension and Shape Path Functors
• Small Loop Transfer Spaces with Respect to Subgroups of Fundamental Groups
• Epitopological and pseudotopological fundamental group functors
• On locally 1-connectedness of quotient spaces and its applications to fundamental groups
• Comparison of Topologies on Fundamental Groups with Subgroup Topology Viewpoint