TITLE: Fixed point theorem on spheres
QUESTION [10 upvotes]: In Milnor's book Topology from the Differentiable Viewpoint there's the following problem:
Problem $6$ (Brouwer). Show that any map $S^n\to S^n$ with degree different from $(-1)^{n+1}$ must have a fixed point.
My solution: Assume that the map $f:S^n\to S^n$ has no fixed points. Let $a:S^n\to S^n$ denote the antipodal map $a(x)=-x$. Then the map $a\circ f$ is homotopic to the identity as follows:
Since $x$ is never mapped to $-x$ (by assumption), there exist a unique shortest great circle arc from $a(f(x))$ to $x$, simply take the straight line homotopy flowing along such arches to get a homotopy $a\circ f\simeq\operatorname{id}$.
Now we have:
$$1=\deg(\operatorname{id})=\deg(a\circ f)=\deg(a)\deg(f)=(-1)^{n+1}\deg(f)$$
and thus $\deg(f)=(-1)^{n+1}$.
My questions:

Is this solution correct?
How to show that the homotopy is continuous? It seems intuitively true to me, but I'm not sure about how to proceed to show it. Maybe saying that it is the flow of some good tangent field?
Are there other (elegant/short/interesting) proofs for this fact?

REPLY [3 votes]: Best solution: Notice that $\|f(x)-(a \circ f) (x)\| < 2$ for every $x.$ But then a problem above this one in the book, asks to prove that such maps must be homotopic (smoothly so if maps themselves are smooth.) Then your computation goes thru with nothing on conscience!