Math 6520 • Algebraic Topology

Let \(S^n\) be the \(n\)-sphere, for some integer \(n\ge 1\), and \(f:S^n\to S^n\) a continuous map.
\begin{enumerate}

\item
Show that if \(f\) is not surjective, then it is homotopic to a constant map.


\item
Construct an example of a surjective map \(f\) that is homotopic to a constant map.


\item
Is every map \(f\) as above homotopic to a constant? Either give a proof or a counterexample.

\end{enumerate}