Fixed-point-free sphere maps are homotopic to the antipodal map
= Fixed-point-free sphere maps are homotopic to the antipodal map
{title2=$\deg f=(-1)^{d+1}$}
If a continuous map $f:S^d\to S^d$ has no fixed point, the formula $H_t(x)=((1-t)f(x)-tx)/\|(1-t)f(x)-tx\|$ is a <homotopy> from $f$ to the <antipodal map>. The denominator could vanish only at $t=1/2$ and $f(x)=x$, which is excluded. The <homotopy invariance of mapping degree> therefore gives $\deg f=(-1)^{d+1}$.