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?
