(i) Show that if $E \rightarrow T$ is a covering map for the torus $T=S^{1} \times S^{1}$, then $E$ is homeomorphic to one of the following: the plane $\mathbb{R}^{2}$, the cylinder $\mathbb{R} \times S^{1}$, or the torus $T$.

(ii) Show that any continuous map from a sphere $S^{n}(n \geqslant 2)$ to the torus $T$ is homotopic to a constant map.

[General theorems from the course may be used without proof, provided that they are clearly stated.]