Math 6520 • Algebraic Topology

Let \(S_g\) denote an orientable surface of genus \(g\). So for example \(S_0\) is the
2-sphere and \(S_{1}\) is the torus.
\begin{enumerate}

\item
If \(g>0\) show that every map \(S_0\to S_g\) is null-homotopic.


\item
If \(g>h>0\) show that every map \(S_h\to S_g\) has degree 0, but show
by example that there exist maps that are not null-homotopic.

\end{enumerate}