Lecture notes are available for some topics—look for links in the left margin. Most of these notes are sketchy; most (if not all) of them are also buggy. Feedback is welcome, especially if you find mistakes!
Introduction, history, overview, and administrivia.
The Jordan-Schönflies Theorem, testing whether a point lies inside a polygon
Path homotopy, contractible, simply connected, covering space, universal cover
Shortest homotopic paths: triangulation, crossing sequences, reduction, sleeve, funnel
Testing homotopy between paths in the punctured plane: sentinel points, monotone paths, vertical ranking, rectified paths, sliding brackets
Regular homotopy, winding and turning numbers, the Whitney-Graustein theorem, hexahedral meshing, cube complexes for balls