We recall the definition of the fundamental group develop in the previous lecture then prove that it is indeed a group. Finally, we show that the fundamental group of the circle is isomorphic to Z, the integers.

We give a quick review of group theory then discuss homotopy of paths building up to the definition of the fundamental group.

When are two shapes the “same”? Topics covered include deformation retract, homotopy of maps, and the homotopy equivalence of spaces.

A nice walk from homotopy to Lie theory to category theory to combinatorics.

Proving a topology theorem by using category theory to translate it into a group theory problem.

I’m a fifth-year Ph.D. student in the Department of Mathematics at The Ohio State University. I’m interested in algebraic topology, semigroup theory, and computer science.

via: https://www.youtube.com/watch?v=CxGtAuJdjYI

An animated explainer on homotopy groups. Discusses geometric intuition for π₁/π₂/π₃ using loopspaces of metric spaces. Includes an original, nonstandard visualization of the Hopf map.

Beautiful typesetting and animation. Homotopy is really weird; surprised that people turned this stuff into type theory.

Ranked #1 among over 400 entries in the 2025 Summer of Math Exposition (SoME4) competition

!!

I think this is the first highly technical Topos Institute blog post where I walked away and felt like I understood everything I was meant to understand. Heck yeah.

Great introduction to sheaf theory.

Knot theory is far more complicated than I initially gave it credit. Would love to learn how they found the counter example via computer search.

The inscribed square/rectangle problem, solved using Möbius strips and Klein bottles.

Excellent intuition-building for topology. I jumped out of my chair when I recognized the Möbius strip construction :P.

via: ~azurylite

This video singlehandedly helped me understand coverings and Lie algebras way better than any Wikipedia article I’ve ever read. :P