I'm trying to understand Proposition 4.3 (page 562) in S. Morita's article *Characteristic Classes of Surface Bundles*, which can be found on Andy Putman's website [here](http://math.rice.edu/~andyp/teaching/Math541/MoritaPaper.pdf). I don't think that my question is terribly particular to the situation at hand, but I'll try and give some context in case particular details turn out to be important.

We have an oriented fiber bundle $\pi:M \to X$ with fiber a closed oriented surface of genus $g$, which I'll call $\Sigma\_g$. The base $X$ need not be simply-connected. We are interested in the cohomology with coefficients in $\mathbb Z / m$ (but I don't think this particular point is essential). Part of our construction of $M \to X$ ensures that the coefficient system is trivial, and so we have the Serre spectral sequence with $E\_2$-page 
$$
E\_2^{p,q} = H^p(X; H^q(\Sigma\_g; \mathbb Z/m)).
$$

> 
> Partway through his argument, Morita makes the following claim:
>  $$
> E\_\infty^{2,0} = \operatorname{Im}(H^2(X;\mathbb Z / m) \to H^2(M; \mathbb Z)).
> $$
>  Why is this?
> 
> 
> 

There is a filtration (I'll suppress coefficients here for simplicity)
$$
0 \subset F^2\_2 \subset F^2\_1 \subset F^2\_0 = H^2(M)
$$
with $E\_\infty^{2-i,i} = F^2\_i / F^2\_{i+1}$. The filtration comes by taking (at least following Allen Hatcher's construction in his spectral sequences book)
$$
F^2\_i = \operatorname{Ker}(H^2(M) \to H^2(M\_{i})),
$$
where $M\_i$ denotes the fiber of the $i$-skeleton of $X$. This should mean that
$$
E\_\infty^{2,0} = F^2\_2 = \operatorname{Ker}(H^2(M) \to H^2(M\_{1})).
$$
Why is this also realizable as the image of the pullback of the projection map? Is there some exact sequence lurking somewhere?