De Rham Cohomology of Hopf Surface How I calculate the De Rham coohomology of the Hopf surface? In particular I would like to know why the second Betti number is zero . 

Let $X$ be the complex surface given by $\mathbb C^2-\{0\}/\mathbb Z$ where $\gamma:(z_1,z_2)\mapsto (2z_1,2z_2)$ generate $\mathbb Z$.
The first step is to check that the map $$z=(z_1,z_2)\mapsto \left(\dfrac{z}{\|z\|},\log(\|z\|)\right)$$ induces a diffeomorphism between $\mathbb C^2-\{0\}$ and $S^3\times \mathbb R$ (this is just the polar coordinates system).
Then, the action of $\mathbb Z$ on the $S^3\times \mathbb R$ is exactly given by $(x,t)\in S^3\times \mathbb R\mapsto (x,t+\log(2))$ so the quotient $X=\mathbb C^2-\{0\}/\mathbb Z$ is diffeomorphic to $S^3\times S^1$.
Finally, if you know the cohomology of the $n$-sphere, then $$H_k(X)\simeq H_k(S^3\times S^1) \simeq \bigoplus_{i+j=k} H_i(S^3)\otimes H_j(S^1)$$
and you get $H_2(X)$ easily.
