Fixed point theorem on spheres

Question

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:

1. Is this solution correct?
2. 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?
3. Are there other (elegant/short/interesting) proofs for this fact?

Answer 1

I think, what you gave is (a sketch of) the simplest proof, unless you know about the Lefshetz fixed point theorem. Using the latter, then the argument goes like this: $H_k(S^n)$ is nonzero only in dimensions $0$ and $n$ and $H_0(S^n)\cong H_n(S^n)\cong Z$. The action of $f$ on $H_0$ is trivial and the action on $H_n$ is by multiplication by $d=\deg(f)$. The Lefschetz number of $f$ then equals $$ \Lambda_f= (-1)^0 + (-1)^n (d)= 1+ d(-1)^n. $$ This number is nonzero unless $$ d= (-1)^{n+1} $$ as required. If $\Lambda_f\ne 0$ then $f$ has a fixed point (this is the Lefschetz fixed point theorem).

As for your solution, do not bother with geodesic flow, that's unnecessarily complicated; you can use linear algebra instead. For $t\in [0,1]$ and $x, y\in S^n$ non-antipodal, define $$ z_t= \frac{tx+(1-t)y}{|tx+(1-t)y|}\in S^n. $$ This is a point on the shortest arc of the great circle connecting $x$ and $y$. The rest you can figure out yourself.

Answer 2

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!