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Am I right that de Rham cohomology $H^k(S^2\setminus \{k~\text{points}\})$ of $2-$dimensional sphere without $k$ points are $$H^0 = \mathbb{R}$$ $$H^2 = \mathbb{R}^{N}$$ $$H^1 = \mathbb{R}^{N+k-1}?$$

I received this using Mayer–Vietoris sequence. And I want only to verify my result.

If you know some elementery methods to compute cohomology of this manifold, I am grateful to you.

Calculation:

Let's $M = S^2$, $U_1$ - set consists of $k$ $2-$dimensional disks without boundary and $U_2 = S^2\setminus \{k~\text{points}\}$. $$M = U_1 \cup U_2$$ each punctured point of $U_2$ covered by disk (which contain in $U_1$). And $$U_1\cap U_2$$ is a set consists of $k$ punctured disks (which homotopic to $S^1$). Than collection of dimensions in Mayer–Vietoris sequence $$0\to H^0(M)\to\ldots\to H^2(U_1 \cap U_2)\to 0$$ is $$0~~~~~1~~~~~k+\alpha~~~~~k~~~~~0~~~~~\beta~~~~~k~~~~~1~~~~~\gamma~~~~~0~~~~~0$$ whrer $\alpha, \beta, \gamma$ are dimensions of $0-$th, $1-$th and $2-$th cohomolody respectively. $$1 - (k+\alpha) + k = 0,$$ so $$\alpha = 1.$$ $$\beta - k + 1 - \gamma = 0,$$ so $$\beta = \gamma + (k-1).$$ So $$H^0 = \mathbb{R}$$ $$H^2 = \mathbb{R}^{N}$$ $$H^1 = \mathbb{R}^{N+k-1}$$

Thanks a lot!

  • 0
    Mariano is right, your result is not correct. Could you outline your calculation, so we can help find where you went wrong?2012-03-11
  • 0
    @you: it may take some time2012-03-11
  • 0
    @you: I have done2012-03-11
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    What is $N{}{}$? (Your result is correct for exactly *one* value of $N$ :) )2012-03-11
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    @MarianoSuárez-Alvarez: I need $H^{*}(S^{n}\setminus\{k~\text{points}\})$ to calculate the cohomology of sphere with $k$ handles. It is enough to satisfy the equality $\dim H^{1}(S^{n}\setminus\{k~\text{points}\}) - \dim H^{2}(S^{n}\setminus\{k~\text{points}\}) = k-1$2012-03-11
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    @MarianoSuárez-Alvarez: but from Mayer–Vietoris sequence $\ldots\to\mathbb{R}^{1}\to\mathbb{R}^{\gamma}\to\mathbb{R}^{0}\to\ldots$ clear that $\gamma = 0$ or $1$2012-03-11
  • 0
    Actually computing the cohomology of a sphere minus a finite set is an exercise you should really complete at some point!2012-03-11
  • 0
    I'm considering computing $\mathop C\limits^\vee ech-Cohomology$ of $S^2\setminus \{k~\text{points}\}$.Then by the isomorphism between de Rham Cohomology and Cech Cohomology,we could get the de Rham Cohomology of $S^2\setminus \{k~\text{points}\}$. But how to find a good cover of it?For example,find a good cover of $S^2$-2 points.2012-03-12
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    @Aspirin: also, you probably need to know that $N$ is *finite* for that to work, no?2012-03-12

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