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I am currently taking a course on differential geometry, and have been using Spivak as a reference. On page 69 of "A Comprehensive Introduction to Differential Geometry, Vol. I" Spivak says:

"It is a well-known (hard) theorem of topology that this is impossible (you can't comb the hair on a sphere)."

Obviously Spivak is referencing the hairy ball theorem here. The professor for the course also introduced the hairy ball theorem referring to it as a "hard" theorem. So, what is meant by "hard" in this context? It doesn't seem to mean hard to understand or prove, since I can think of other results even in the same text that are difficult to understand, yet Spivak doesn't refer to them as "hard". Thanks for your time.

grumpy
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Of course it depends on the notion of hard. In Milnor's book "topology from a differentiable" pg 30 you can find a proof. That proof is maybe one of the shortest and most elegant of this fact. The point is that you need the machinery of Brouwer's degree. Now, the book is self contained and doable for one that has studied just Analysis in Euclidean spaces but still it takes 30 pages to define the degree. That is because you have to introduce, the notion of manifold, manifold with boundary, orientability, prove Sard's theorem, you have to classify 1-manifolds, and you have to prove some properties about isotopies and the regular fiber of smooth maps. That's how hard it is.

Having said this, Milnor's book it's one of the most enjoyable you will find in math, thus I recommend, if you are interested, to try to read it and make you own idea of how hard the subject is.

Overflowian
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    Milnor is the author of another, quite short and almost self contained, proof of the hairy ball theorem: https://www.jstor.org/stable/2320860?read-now=1&seq=1#page_scan_tab_contents –  Nov 27 '19 at 00:48