Hairy ball theorem for beginners

Question

I am intended to teach about the hairy ball theorem to some students with little background in proper courses in mathematics. Only linear algebra and some vector calculus would be assumed.(and formalization/proofs in mathematics)

I only want to talk about the particular situation for vector fields in the sphere, so I am thinking that using the winding number for vector fields and Euler characteristic would be the most appropriate way to get up to the proof of the theorem.

I am trying to find a textbook reference that helps me guiding these students along the way, and provide examples and exercises for them to keep up. I do not want to talk about manifolds or generalization of the theorem to higher-dimensional spaces, seven want to deal with other surfaces rather than the sphere.

Is there any suggestion of textbooks that deal with the winding number, euler characteristic and the index theorem just at the level of the sphere or surfaces, and which is accessible to underground student with no big background.

If you want to avoid diff geometry and/ or alg topology, how about Milnor's analytical proof? — Matematleta, Sep 18 '19 at 03:35
@Matematleta do you happen to know a good reference for that? I might take a look at that approach — C. Zhihao, Sep 18 '19 at 11:03
Milnor's proof is linked in https://math.stackexchange.com/questions/90737/shortest-proof-for-hairy-ball-theorem?rq=1. — Moishe Kohan, Sep 18 '19 at 23:42

score 3 · Answer 1 · 2019-09-18T18:36:55.477

3

Guilleman and Pollack is pretty good. We used it when I had Rosenlicht for differential topology as an undergraduate at uc Berkeley. It was a nice course.

It (the hairy-ball theorem) is a corollary of the Poincaré-Hopf index theorem, which they prove using basic ideas. Mainly linear algebra and calculus. Everything is done in $\Bbb R^n$.

edited Sep 18 '19 at 18:36

answered Sep 18 '19 at 03:36

Hairy ball theorem for beginners

1 Answers1