Mapping homotopic to the identity map has a fixed point Suppose $\phi:\mathbb{S}^2\to\mathbb{S}^2$ is a mapping, homotopic to the identity map.  Show that there is a fixed point $\phi(p)=p$.

Alternatively, this follows from the Lefschetz fixed point theorem. As $\phi$ is homotopic to the identity map, the Lefschetz number of $\phi$ is just the Euler characteristic of $S^2$, which is 2. As $2 \neq 0$, it follows that $\phi$ has at least one fixed point. 
