For each of the following assertions, either provide a proof or give and justify a counterexample.

[You may use, without proof, your knowledge of the de Rham cohomology of surfaces.]

(a) A smooth map $f: S^{2} \rightarrow T^{2}$ must have degree zero.

(b) An embedding $\varphi: S^{1} \rightarrow \Sigma_{g}$ extends to an embedding $\bar{\varphi}: D^{2} \rightarrow \Sigma_{g}$ if and only if the map

$$\int_{\varphi\left(S^{1}\right)}: H^{1}\left(\Sigma_{g}\right) \rightarrow \mathbb{R}$$

is the zero map.

(c) $\mathbb{R} \mathbb{P}^{1} \times \mathbb{R P}^{2}$ is orientable.

(d) The surface $\Sigma_{g}$ admits the structure of a Lie group if and only if $g=1$.