Cohomology of Product of Spheres

Solution 1:

You can use the Kunneth formula $H^2(S^2\times S^2)=H^0(S^2)\otimes H^2(S^2)\oplus H^1(S^2)\otimes H^1(S^2)\oplus H^0(S^2)\otimes H^2(S^2)=\mathbb{R}\oplus \mathbb{R}$,

the same formula gives $H^3(S^2\times S^2)=0$.

Solution 2:

The fact that this is deRham cohomology is not so relevant once you understand how the maps are defined in terms of closed forms. So you know that $H^j(U) \cong H^j(S^2)$. What you need to do is figure out $H^j(U\cap V) \cong H^j(S^2\times S^1)$ by doing a Mayer-Vietoris decomposition on that first. You can do that either of two ways — either you need to know the cohomology of the torus ($S^1\times S^1$) or you break it down into $S^2\times \{p\}$ and $S^2\times\{p,q\}$, both of which you know.

Note that you'll be able to do $S^m\times S^n$ by such an inductive process. :)