(i) Let $f: X \rightarrow Y$ be a smooth map between manifolds without boundary. Define critical point, critical value and regular value. State Sard's theorem.

(ii) Explain how to define the degree modulo 2 of a smooth map $f$, indicating clearly the hypotheses on $X$ and $Y$. Show that a smooth map with non-zero degree modulo 2 must be surjective.

(iii) Let $S$ be the torus of revolution obtained by rotating the circle $(y-2)^{2}+z^{2}=1$ in the $y z$-plane around the $z$-axis. Describe the critical points and the critical values of the Gauss map $N$ of $S$. Find the degree modulo 2 of $N$. Justify your answer by means of a sketch or otherwise.