In this lecture we generalise the notion of "degree" from Lecture 5.

Our first main result is that the antipodal map $S^n \to S^n$ has degree $(-1)^n$. We then prove a rather deeper theorem that says any map $f \colon S^n \to S^n$ which commutes with the antipodal map has odd degree.

As an application we deduce that there does not exist a nowhere vanishing vector field on an even-dimensional sphere. This result is called the "Hairy Ball Theorem" because it can be interpreted as saying that if you try and "comb" a hairy ball flat, there will always be at least one tuft sticking up.

This has the following "real-life" consequence. Think of the surface of the earth as the sphere, and think of the wind as a vector field. Then there is always somewhere on the planet where there is no wind. You can try searching for this point here (updated every three hours). Pro Tip: When googling the Hairy Ball Theorem, make sure "safe search" is turned on.