Cohomology of Product of Spheres I am trying to compute the De Rahm cohomology of $M = S^2 \times S^2$. I was able to compute $H^0(M), H^1(M),$ and $H^4(M)$, but am having trouble computing $H^2(M)$ and $H^3(M)$. I am using the Mayer Vietoris sequence, and I took $U = S^2 \times (S^2 \setminus \{p\})$ and $V  = S^2 \times (S^2 \setminus \{q\})$, so $U \cap V = S^2 \times S^1$. But I haven't been able to find any of the maps needed to compute $H^2$ or $H^3$. How can compute these groups?

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 $H^3(S^2\times S^2)=0$.
