Both the Atiyah-Hirzebruch and the Serre spectral sequence can be constructed from a skeletal filtration of a CW-complex:

We can construct the Atiyah-Hirzebruch spectral sequence by filtering $X$ by its skeleta and applying an exotic cohomology theory $h$. This gives an exact couple, and the spectral sequence associated to this runs $H^p(X; h^q(pt)) \Rightarrow h^{p+q}(X)$.

Given a fibration $E\rightarrow B$ with typical fiber $F$, we can filter the base $B$ by skeleta. This induces (by taking preimages) a filtration on $E$, and under mild assumptions, we arrive at the Serre spectral sequence running $H^p(B; H^q(F))\Rightarrow H^{p+q}(E)$ (by applying ordinary cohomology to the filtration).

Now we can combine those two constructions into one, obtaining a spectral sequence running $H^p(B; h^q(F))\Rightarrow h^{p+q}(E)$. This specializes to the Serre spectral sequence by setting $h$ to be $H$, and to the Atiyah-Hirzebruch spectral sequence by considering the fibration $X \rightarrow X$, where $F$ is just a point.

(This is pointed out, for example, in Hatcher's "Spectral sequences in Algebraic Topology", where he mentions this as possible construction of the AHSS)

I have never seen any application where this tells you substantially more than, for example, first computing $H^{p+q}(E)$ via the ordinary Serre SS, and then $h^{p+q}$ via the ordinary AHSS. I do know some examples where the answer is slightly different from what you'd expect at first, for example the Hopf fibration $S^3\rightarrow S^2$ and $h^*=KO^*$. However, in these examples it seems to me that obtaining the actual structure of the spectral sequence requires considerable work with the ordinary ones.

Does anyone know of a striking example where having the generalized spectral sequence tells you more than just the two special cases?

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    $\begingroup$ Maybe math.uiuc.edu/K-theory/0627/ah.pdf would be useful - it applies the Atiyah-Hirzebruch spectral sequence to motivic cobordisms. I believe that the paper mathcs.emory.edu/~dzb/math/papers/sss.pdf may contain the same information as Hatcher's "Spectral sequences in Algebraic Topology" - however, it still may be helpful. $\endgroup$
    – user62675
    Mar 28 '14 at 0:32
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    $\begingroup$ I don't know how striking this is, but I like the example of computing $E^* BZ/p$ for $E$ a $p$-complete complex-orientable theory via the fiber sequence $S^1 \to BZ/p \to CP^\infty \xrightarrow{p} CP^\infty$. The spectral sequence has a single differential, which upon choice of coordinate is given by the $p$-series of the orientation. Probably it's possible to state this as "First compute $(HZ_p)_*(BZ/p)$, then compute the AHSS," but it seems that justifying this description of the differential in that setting might be more cumbersome...? $\endgroup$ Mar 28 '14 at 2:22
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    $\begingroup$ Maybe ruhr-uni-bochum.de/imperia/md/content/mathematik/lehrstuhlxiii/…? $\endgroup$
    – Drew Heard
    Mar 28 '14 at 23:36
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    $\begingroup$ Thanks Eric and Drew, I was looking for exactly such things! $\endgroup$ Mar 29 '14 at 5:13
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    $\begingroup$ John Hunton's computation of Morava K-theories of extended power, for example. One can compute the ordinary Serre SS, but it is not clear if one can compute the AHSS afterwards. $\endgroup$
    – user43326
    Apr 3 '14 at 8:18

A very nice generalized AHSS calculation that deserves to be better known is in

Vershinin, V. V. and Gorbunov, V. G. Multiplicative spectra that do not have torsion in homology. (Russian) Mat. Zametki 41 (1987), no. 1, 87–92, 121. [MR0886171 (88f:55012)]


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