I'm reading chapter about the Borsuk-Ulam theorem in Allen Hatcher's "Algebraic Topology" and can't understand proof of the following corollary. To be more precise, it is unclear to me why restriction of $f$ to the equator is nullhomotopic. Can anyone explain end of the proof given more details?

Allen Hatcher "Algebraic Topology"

P.S. I take a look through lots of similar questions but none of them satisfy me since they mostly about giving proof for particular case $n=2$ or using some advanced homology technique. It seems that here Hatcher means more elementary proof, because if we are able to show that some odd degree map is nullhomotopic then there is a contradiction and function $g(x)-g(-x)$ has a zero, so we are done.


I'll write out the nullhomotopy explicitly in the case of $n = 2$. (The argument works for higher $n$ in the same way. I just want to avoid overwhelming you with notation!)

So we have constructed an odd map $f : S^2 \to S^1$. Let's use spherical polar coordinates $(\theta, \phi)$ on the $S^2$. Then the restriction of $f$ to the equator is a map $f_{\rm equator} : S^1 \to S^1$, defined by $$ f_{\rm equator} (\phi) = f(\theta =\tfrac \pi 2, \phi).$$

Now, I'm going to define a new map $f_{\rm North \ Pole} : S^1 \to S^1$ to be the constant map obtained by simply evaluating $f$ at the North Pole of the $S^2$, i.e. $$ f_{\rm North \ Pole} (\phi) = f(\theta = 0).$$ [Technically, the spherical polar coordinate system breaks down at the North Pole, but I hope you can understand what I'm writing nonetheless...]

A homotopy between $f_{\rm North \ Pole }$ and $f_{\rm equator}$ is then given by: $$ [0,1 ] \times S^1 \to S^1 \\ (s, \phi) \mapsto f(\theta = \tfrac \pi 2 s, \phi). $$

The key point is that $f_{\rm North \ Pole }$ is a constant map, so it has degree zero. Hence $f_{\rm equator}$, being homotopic to $f_{\rm North \ Pole }$, must also have degree zero. However, $f_{\rm equator}$ is constructed in such a way that it maps antipodal points to antipodal points, and by the preceding proposition in Hatcher, this implies that $f_{\rm equator}$ has odd degree. As zero is not an odd number, we have a contradiction.

For $n > 2$, we can use an almost identical argument, using higher-dimensional analogues of spherical coordinates.


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