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In section 16 of the topology book of Bott and Tu, there is a path fibration $\Omega S^2 \to PS^2 \to S^2$. The $E_2$ page of the spectral sequence of this fibration is $$E_2^{p,q}=H^p(S^2,H^q(\Omega S^2)).$$

This is a Cech Cohomology of $S^2$ with values in $H^q(\Omega S^2)$. My question is why all columns in $E_2$ except p=0 and p=2 are zero, why could we show this by using the universal coefficient theorem of singular cohomology? Thanks!

Edit: I know how to calcute Cech Coholomogy of a manifold with a good cover, but this seems to be the direct limit of groups. Since we don't know whether is $H^q(\Omega S^2)$ free we can't get $E_2=H^p(S^2) \otimes H^q(\Omega S^2)$, so how do we get the result?

Haskell Curry
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Proton
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    $S^2$ can be given a CW complex structure, so Cech cohomology with constant coefficient is naturally isomorphic to Singular cohomology with that coefficient. –  Jan 18 '13 at 07:47
  • @Sanchez Thank you! :) – Proton Jan 18 '13 at 08:01
  • I also meet this problem, I wonder if you solve this? thanks – Grey Oct 20 '21 at 00:35
  • @Sanchez By your answer, I can solve this problem, thanks. However, I want to know more about the relationship between CW complex and Cech cohomology, what should I read? – Grey Oct 20 '21 at 07:44

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