# The spectral sequence of a tower of principal fibrations

Assume we have a tower of fibrations (of simplicial sets, let's say):

$$\cdots\rightarrow X_{n+1}\rightarrow X_n\rightarrow\cdots\rightarrow X_0.$$

Let $X=\lim_nX_n$ be the (homotopy) inverse limit. Choose a base point in $X$, which induces base points in all $X_n$ in such a way that bonding maps are based. There is an extended spectral sequence associated to a filtration of $\pi_*(X)$. The $E_1$-term is $E_1^{pq}=\pi_{q-p}F_p$, where $F_n$ is the (homotopy) fiber of $X_n\rightarrow X_{n-1}$ (with the convention $X_{-1}=*$, so $F_0=X_0$). The word 'extended' refers to the fact that $E_r^{p,p+1}$ may be non-abelian groups, $E_r^{p,p}$ are just pointed sets, some differentials are group actions on pointed sets, etc. Bousfield-Kan (and later Bousfield) studied in detail the properties of such spectral sequences, clarifying what's going on in low dimensions, and establishing interesting connections with obstruction theory in case we do not want to choose a base point in $X$.

If all these fibrations happen to be principal fibrations $X_{n}\rightarrow X_{n-1}\rightarrow BF_n$ then $E_r^{p,p+1}$ are abelian and $E_r^{p,p}$ are groups, and probably the whole spectral sequence is better behaved in many ways. In particular, the possiblity of defining $E^{p,p-1}_1=\pi_0BF_p$ and extending accordingly the spectral sequence appears.

I wonder whether there is a reference which deals with this, either in general or a specific case. If there's not such reference, any ideas you can throw in will be appreciated.

• I was looking recently for a discussion of analogous ideas for towers of fibrations of spectra, where similar simplifications ought to occur, but I didn't find much of anything. Oct 12, 2013 at 19:04
• Dan, for spectra I'd recommend you Christensen's thesis. That case is easier because, by stability, all fibrations are principal, etc. Oct 12, 2013 at 22:48
• Is the thesis available somewhere? (I suppose I could just e-mail him.) Oct 14, 2013 at 22:25
• He published a paper in Advances on his thesis. Very well written and readable. Oct 14, 2013 at 22:51