# Are there any cool applications of the generalized Atiyah-Hirzebruch(-Serre) spectral sequence?

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?

• 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. – user62675 Mar 28 '14 at 0:32
• 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...? – Eric Peterson Mar 28 '14 at 2:22
• – Drew Heard Mar 28 '14 at 23:36
• Thanks Eric and Drew, I was looking for exactly such things! – Achim Krause Mar 29 '14 at 5:13
• 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. – user43326 Apr 3 '14 at 8:18