Category Archives: Topological groups

Topologized Fundamental Groups: The Quotient Topology Part 3 (Why isn’t it always a topological group?)

In Part 1, I mentioned that one of the surprising things about the natural quotient topology on the fundamental group is that the resulting group with topology often fails to be a topological group. In fact, I’d say it’s usually … Continue reading

Posted in compact-open topology, earring group, Free groups, Fundamental group, quotient topology, topological fundamental group, Topological groups, Uncategorized | Tagged , , , , , , , , , , | 1 Comment

Topologized Fundamental Groups: The Quotient Topology Part 1

Next up for topologies on the fundamental group is what I’d consider the most “natural” one. It’s almost certainly the topology you’d most often get if you asked random topologists on the street to construct one for you. This is … Continue reading

Posted in Algebraic Topology, compact-open topology, Fundamental group, Quasitopological groups, quotient topology, Topological groups, Uncategorized | Tagged , , , , , | 3 Comments

Topologized Fundamental Groups: The Whisker Topology, Part 3

In Part 1 and Part 2, I gave detailed introductory exposition about the whisker topology on the fundamental group. In general, this topologized fundamental group is a left topological group and therefore a homogeneous space. Moreover, whenever this group is … Continue reading

Posted in Covering Space Theory, Fundamental group, Higher Homotopy groups, homotopically Hausdorff, Infinite Group Theory, Infinite products, topological fundamental group, Topological groups | 1 Comment