[Math] Are there any cool applications of the generalized Atiyah-Hirzebruch(-Serre) spectral sequence

Question

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?

Answer 1

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)]