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.

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

!!

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