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I am following Hatcher's notes on spectral sequences. On page 522 an exact couple $(A, E, i, j, k)$ is defined. You can construct a derived exact couple easily from this to get your desired $E'$ however constructing the new differential from this information is said to be not so easy in practice. I would like to know why this is, to me it seems that $d'$ is simply $j'k'$. Is this a difficult thing to calculate or is it simply tedious? It seems to be the same opinion everywhere that given a term in a spectral sequence $(E^r, d_r)$, calculating $E^{r+1}$ is easy and $d_{r+1}$ just isn't practically possible with the given information.

I'd like to think there is some algorithm where you give a pair $(E^r, d_r)$ and it spits out a pair $(E^{r+1},d_{r+1})$. So what is the reason this isn't the case?

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    $\begingroup$ A term of a spectral sequence is the homology of the previous one, but the differential cannot be computed from the previous terms and deferentials. It's just not defined like that. $\endgroup$
    – Fernando Muro
    Commented Sep 11, 2017 at 18:20
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    $\begingroup$ @FernandoMuro That's what I thought until Hatcher said: "This information is contained in the original exact couple, but often in a way which is difficult to extract, so in practice one usually seeks other ways to compute the subsequent differentials." $\endgroup$
    – John Smith
    Commented Sep 11, 2017 at 18:23
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    $\begingroup$ Your hope that «there is some algorithm that from $(E_r,d_r)$ spits out $(E_{r+1},d_{r+1})$ is simply not realized. It is trivial to produce examples of spectral sequences which coincide only up to some page. $\endgroup$
    – Mariano Suárez-Álvarez
    Commented Sep 11, 2017 at 18:25
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    $\begingroup$ If a SS comes from an exact couple then all information is contained therein, but a page of a spectral sequence can only compute the terms of the next page. $\endgroup$
    – Fernando Muro
    Commented Sep 11, 2017 at 18:25
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    $\begingroup$ I think perhaps the point is that many spectral sequences come in a form where we know the exact couple exists, and we know E_r and perhaps d_r, but we do not know the rest of the exact couple (indeed, we are often trying to compute A or something like it). $\endgroup$
    – Tyler Lawson
    Commented Sep 11, 2017 at 22:41

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Expanding on Tyler Lawson's comment, the point of a spectral sequence is often that we know what $E$ is concretely, and we want to use this to compute $A$. The issue is that if we want to explicitly figure out what $d'$ is, we need to understand $A$ itself, which becomes quite circular. Indeed, to compute $d'$, the procedure is to apply $k$, compute $i^{-1}$ of that, then apply $j$. But $i$ is a map $A \to A$, and in general if we want to understand this map, we need to know something about $A$.

For example, if $h_*$ is a (generalized) homology theory and $X$ is a CW complex, then the Atiyah-Hirzebruch spectral sequence is a spectral sequence converging to $h_*(X)$ whose $E_1$ page is the cellular chain complex of $X$ with coefficients in $h_*(*)$. If we are given a cellular decomposition of $X$, then we have some fairly solid grasp on what $E_1$ is, and the differentials $d_1$ are just the differentials we use to compute cellular homology, which can be calculated using local degrees. These are things we can actually work with.

However, if we wanted to compute $d_2$, we need to understand $i$. In this case, $i$ is given by the inclusion map $h_*(X^{(n)}) \to h_*(X^{(n+1)})$, where $X^{(n)}$ is the $n$th skeleton of $X$. But our original goal was to compute what $h_*(X)$ is! If we have some concrete information about $i$, chances are we already know what $h_*(X)$ is, and the whole computation is moot.

What the machinery of spectral sequences gives us is that we can often use other means to deduce the nature of the differentials $d_r$, without resorting to taking apart the construction (since that is circular). The most common way is that if $d_r$ maps to or from the zero group, then $d_r$ must be zero. At other times, we use certain multiplicative structures on the spectral sequence, or we use the naturality of the constructions to compare the spectral sequence with others.

One might object that even understanding $d_1$ requires knowledge about $A$, since $j$ and $k$ also pass through $A$. This is true, and the hard part of setting up a useful spectral sequence is often to identify what $d_1$ is doing and give a concrete description of the $E_2$ page. In the successful cases, we can get away with sufficiently high-level arguments that don't depend on what the details of the space we are working with.

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