Degree under suspension

If $f:S^m\to S^m$, then its suspension $\Sigma f:S^{m+1}\to S^{m+1}$ has the same degree. This follows from naturality of the suspension isomorphism in reduced homology.

Every integer $m$ is the mapping degree of a continuous map $S^n\to S^n$ in every dimension $n\ge 1$. Start with the circle map $z\mapsto z^m$, whose argument winds $m$ times, and use degree under suspension repeatedly. Negative $m$ reverses the winding, and $m=0$ is constant before suspending. The degrees of maps of the zero-sphere are a genuine exceptional case.