Hopefully an easy question on spectral sequences

Question

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. 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?

Answer 1

One way to see this is that the Serre spectral sequence is natural (we only need this naturality in fiber bundles over $X$). There is a map from the bundle $\Sigma_g \to M \to X$ to the trivial bundle $* \to X \to X$, and so you get an associated map of spectral sequences: $$ H^p(X; H^q(*; \mathbb{Z}/m)) \to H^p(X; H^q(\Sigma_g; \mathbb{Z}/m)) $$ For the right-hand spectral sequence, $E_\infty^{p,0}$ is a subgroup of $H^p(M)$ coming from the filtration you list. On the left, however, the spectral sequence is computing $H^*(X)$ and degenerates at the $E^2$-term to $H^p(X; \mathbb{Z}/m)$, concentrated on the $q=0$ line.

This induces a map of $E_\infty$ terms, which is a filtration of the map $H^*(X) \to H^*(M)$. For $q=0$ it is a surjection from $H^p(X;\mathbb{Z}/m)$ to some subobject of $H^*(M)$. This exhibits the $q=0$ line in the $E_\infty$ term as the image of $H^*(X)$.

Answer 2

At each stage (or page, if you prefer) of the spectral sequence there are differentials $d_r: E_r^{p,q} \to E_r^{p+r, q - r + 1}.$ The cohomology of these on $E_r^{p,q}$ gives $E_{r+1}^{p,q}$. "Passing to the limit" over all $r$ gives $E_{\infty}^{p,q}$. Your spectral sequence (which is a case of what I would call the Leray spectral sequence) is a first quadrant s.s., and so there are no convergence issues: if $r$ is large enough (with respect to any fixed $p,q$) the source and target of $d_r$ are zero (draw the picture!) and so the $E_r^{p,q}$ stablize.

Now $E_r^{2,0}$ sits along the $p$-axis, and so $d_r$ maps out of the first quadrant (since $r \geq 2$) and is necessarily zero. Thus in this case every element of $E_r^{2,0}$ is a cocycle for $d_r$, and so $E_{r+1}^{2,0}$ is the quotient of $E_r^{2,0}$ by the image of $d_r$. Thus $E_{\infty}^{2,0}$ is a quotient of $E_2^{2,0} = H^2(X)$ which is also a subobject of $H^2(M)$. Now a consideration of the construction of the s.s. (which may or may not be easy, depending on how you think of it as being constructed) shows that in fact it is the image of the natural map $H^2(X) \to H^2(M)$. (And what else could it be?!)

---

Thinking in terms of skeleta, as you do, is normally not helpful. There are lots of ways to build this s.s.; using skeleta and a simplicial/singular approach is one way; using sheaves is another. Delving into any particular construction normally will lead to madness. (This is not a definitive rule, of course, but I have found it to be a useful general principle.) It is normally better to use the "internal logic" of the spectral sequence itself, i.e. the knowledge that there are various pages $E_r$ linked by the differentials $d_r$ and their cohomology. (As a vague justification for this philosophy, the whole point of a s.s. is to wrap up the complicated details of some situation into a nice package and delving too much into the construction just unwraps that package, which defeats the purpose to some extent.)

In general, the $d_r$ are hard to compute in cases where you can't identify them for some easy reason, but in the case of $E_r^{p,0}$ and $E_r^{0,q}$, where the $d_r$ mapping out or in necessarily vanishes, one is usually in better shape, and the resulting map of $E_2^{p,0}$ to the limit of the s.s., or of the limit to $E_2^{q,0}$ can often be described. (These maps, which describe the two extreme "ends" of the filtration on the limit of the s.s., are called edge maps, because they come terms on the two edges of the quadrant.)

So in this case, the edge map $E_2^{p,0} \to H^p(M)$ is just the pull-back $H^p(X) \to H^p(M),$ while the edge map $H^q(M) \to E_2^{0,q}$ is the natural map $H^q(M) \to H^q(\Sigma_g)$ given by restricting a cohomology class to a fibre.

When you are trying to understand a spectral sequence, working out the edge maps is a good first step. It is reasonable to expect them to be given by certain natural maps in the given situation. The other differentials are usually harder, and one shouldn't typically expect to compute them (unless you can show that they are zero, say, by showing that their source or target vanishes).