= Solution
For a <sphere> self-map, the <Lefschetz number> is $L(f)=1+(-1)^n\deg f$. At degree zero this is $1$, so the <Lefschetz fixed-point theorem> supplies $x$ with $f(x)=x$.
Let $A:S^n\to S^n$ be the <antipodal map>, $A(z)=-z$, of degree $(-1)^{n+1}$. The composite $Af$ also has degree zero, hence Lefschetz number one, so it has a fixed point $y$. The equation $A(f(y))=y$ is $f(y)=-y$. Consequently
$$
\boxed{\exists x,y\in S^n:\ f(x)=x,\quad f(y)=-y.}
$$
The two points are distinct, since a unit vector is never its own negative.
Back to article page
