Chapter IV: A man walking a pig around a lamppost (dog-man-lamppost theorem). (Image courtesy of Talia Blum. Used with permission.)
Winding numbers for differentiable loops are standard content in both topology and complex analysis classes, but their application to real (Lecture 13) and complex (Lecture 14) equations seems to be the part of this course that students find most difficult. Partly, this is because of the lingering impression that such equations are there in order to be solved explicitly, which is what the topological method presented here is intended to avoid. The other stumbling block seems to be that the logic of “this term is small and we can neglect it” sits uneasily with some math students, in spite of many efforts to promote it at various levels of teaching. One could try to ease the transition by spreading out the material over more lectures, but I find that this leads to preliminary discussions which are not particularly meaningful or satisfying by themselves. The last lecture is an excursion into three dimensions. At that point, one could insert an entire chapter on the topology of knots and links, but I am simply not the right person to write such a chapter.
If you want to learn more about winding numbers, a good option is J. Roe, Winding around, Amer. Math. Soc., 2015.
Lecture 13: Equations in Two Variables
