I know this is a simple exercise but I am stuck unfortunately.
Question: Use de Rham cohomology to prove that the sphere $S^2$ is not diffeomorphic to the torus $T$. You may assume that $H^1(\mathbb{R}^2) = \{0\}$.
Answer: For the sphere $S^2$, one can show that $H^1(S^2) \cong \{0\}$, for the torus one can show that $H^1(T) \cong \mathbb{R}^2$. Now I don't know how to proceed, what I vaguely understand is that different cohomology implies the manifolds cannot be diffeomorphic. How can I make this precise ? In particular, I must be missing something because I don't know how to make use of the given fact $H^1(\mathbb{R^2}) \cong \{0\}$.
Many thanks for your help!