Show that every map \(S^n \to S^n\) can be homotoped to have a fixed point if \(n > 0\).
Solution
Let \[ f:S^n\to S^n, \qquad n>0. \] Choose a point \(x_0\in S^n\).
1. There exists a rotation \(R\in SO(n+1)\) such that\[ R(f(x_0))=x_0. \]
The group \(SO(n+1)\) acts transitively on \(S^n\) for \(n>0\), so some orientation-preserving orthogonal transformation carries \(f(x_0)\) to \(x_0\).
2. The rotation \(R\) is joined to the identity by a path in \(SO(n+1)\).
The group \(SO(m)\) is path connected for every \(m\ge2\). Here \(m=n+1\ge2\), so there is a continuous path \[ R_t\in SO(n+1), \qquad R_0=I,\quad R_1=R. \]
3. The maps\[ f_t=R_t\circ f \]form a homotopy from \(f\) to a map having a fixed point.
Clearly \(f_0=f\). At the other endpoint, \[ f_1(x_0)=R(f(x_0))=x_0, \] so \(x_0\) is a fixed point of \(f_1\).
Therefore every map \(S^n\to S^n\) with \(n>0\) is homotopic to a map with a fixed point.