I'm studying for an algebraic topology exam, and the following question has me stumped.
Problem. For $n \geq 1$, prove there does not exist a continuous map $f : S^n \rightarrow S^{n-1}$ such that $f(-x) = -f(x)$ for all $x \in S^n$.
The case $n = 1$ is fairly obvious since $S^0$ is disconnected, but I don't expect this to help for $n \geq 2$.
I'm interested in any answer, but one based in algebraic topology would be especially helpful.