Category Archives: Homotopy theory

Visualizing the Hopf Fibration

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)

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

Homotopically Reduced Paths (Part II)

Understanding this post requires reading Part I where I explain what reduced paths are and how we might go about proving they exist. In this second part, I’ll go into detail about how to use Zorn’s lemma to identifying a … Continue reading

Posted in Fundamental groupoid, reduced paths | Tagged , , , , , , , | 1 Comment