Pullback of cohomology classes under smooth maps

Let $T^{2}$ be the two-dimensional torus and let $\varphi : S^{2} \to T^{2}$ be smooth map. Show that for any top de Rham cohomology class $[v] \in H^{2}(T^{2})$, we have $\varphi^{*}[v] = 0$.

Suppose $F : M \to N$ is a smooth map between differentiable manifolds, and is homotopically trivial. Show that in this case, $F^*\omega$ will be exact whenever $\omega$ is a closed 1-form on $N$. [Note: $F$ is homotopically trivial if it extends to a smooth mapping $\bar F : M \times [0,1] \to N$ such that $\bar F(x,0) = F(x)$, for all $x \in M$, while $\bar F(x,1) \equiv q \in N$ ($q$ constant) for all $x$.]