Category Archives: Homotopy theory

Topologized Fundamental Groups: The Whisker Topology, Part 3

Posted on June 30, 2022 by Jeremy Brazas

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

Visualizing the Hopf Fibration

Posted on August 21, 2021 by Patrick Gillespie

This is a guest post by Patrick Gillespie, who is currently a 2nd year Ph.D. student at the University of Tennessee Knoxville. The Hopf map is a classical example of a non-trivial fiber bundle. There are many great visualizations of … Continue reading →

Posted in Algebraic Topology, Higher Homotopy groups, Homotopy theory | Tagged , , , , , | Leave a comment

Homotopically Reduced Paths (Part III)

Posted on December 26, 2020 by Jeremy Brazas

In this last post about reduced paths, I’m going to work through the details of one of the most useful results in wild topology. Writing this post helped me work out my own way of proving this result and hopefully … Continue reading →

Posted in Algebraic Topology, Dendrite, Fundamental group, Fundamental groupoid, Homotopy theory, Inverse Limit, one-dimensional spaces, reduced paths | Leave a comment