Is there a simple proof of Borsuk-Ulam, given Brouwer?

Question

(Brouwer) Any continuous function from a convex compact subset K of a Euclidian space to itself has a fixed point.

Given this lemma, is there a simple proof of:

(Borsuk-Ulam) Any continuous function $f \, : \, S^n \to R^n$ (where $S^n$ is the $n$-sphere) has a point $x$ for which $f(x) = f(-x)$.

?

Answer 1

This doesn't completely answer the question, just a very special case.

Let $S_+^n$ be the (closed) upper hemisphere of $S^n$ and $S_-^n$ the lower hemisphere. Suppose our map $f:S^n\to\mathbb R^n$ besides being continuous, also has the properties

• $f(S^n_+) = f(S^n_-) = B^n$,
• the restriction $f_-:S^n_-\to B^n$ is a homeomorphism.

(Here $f_-:S_-^n\to B^n$ is defined by $f_-(x)=f(x)$. Define $f_+:S_+^n\to B^n$ analogously.)

Next, define the antipodal map $\phi:S^n_-\to S^n_+$ by $\phi(x)=-x$. This gives us a well-defined continuous map $f_+\circ\phi\circ f_-^{-1}:B^n\to B^n$, which has a fixed point by Brouwer's theorem. More specifically, this means that there exists an $x\in B^n$ such that $$f(\phi(f_-^{-1}(x)))=x=f(f_-^{-1}(x)).$$ Writing $y = f_-^{-1}(x)$, this means precisely that $f(-y)=f(y)$, proving Borsuk-Ulam in this special case.

If anyone sees a way to generalize this, I'd be interested to know.