Category Archives: Homotopy theory

Local n-connectivity vs. the LC^n Property, Part 2

Posted on October 17, 2023 by Jeremy Brazas

In Part 1, I defined local n-connectivity, which says that a space has a basis of n-connected open sets, and the porperty, which says that for all , “small” maps from the -sphere contract using “small” homotopies of “relatively the … Continue reading →

Posted in Algebraic Topology, Examples, General topology, Higher Homotopy groups, Homotopy theory, locally n-connected spaces, locally path connected, Peano continuum, Uncategorized | Tagged , , , , , , | Leave a comment

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 | 2 Comments

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