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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?

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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. –  user27126 Jan 18 '13 at 7:47
@Sanchez Thank you! :) –  Jiangnan Yu Jan 18 '13 at 8:01

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