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 .
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I'm trying to use the Kunnet formula. Am I doing right ? – user52342 Oct 25 '13 at 09:47
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What is your definition of a Hopf surface ? The quotient of $\mathbb C^2-{0}$ by what kind of discrete group ? – Bebop Oct 25 '13 at 10:02
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Yes, sorry my fault. In particular $C^2−{0}/(x,y) /sim (2x,2y)$. I know that it's diffeomorphic to $S^1XS^3$ – user52342 Oct 25 '13 at 10:12
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Also how i prove this last fact, that it's diffeomorphic to $S^1XS^2$ – user52342 Oct 25 '13 at 10:24
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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.
Bebop
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