# Unbased spectral sequences

Suppose we have a tower of fibrations of spectra $\{X_k\}_{k\in\mathbb{N}}$ with inverse limit $X_\infty$, and let $F_k$ be the fibre of the map $X_k\to X_{k-1}$. There is then a spectral sequence $E^1_{jk}=\pi_j F_k \Longrightarrow \pi_{j+k} X_\infty$. If we instead have a tower of fibrations of based spaces, then we still have something like a spectral sequence except that some of the entries may be nonabelian groups or just pointed sets, and the sense in which $E^{r+1}$ is the homology of $E^r$ must be modified to take account of this. The details are in the book 'Homotopy limits, completions and localizations' by Bousfield and Kan. Now suppose we have a tower of fibrations of unbased spaces. I think I have heard it said that there is still some kind of spectral sequence building up to $\pi_\ast(X_\infty,a_\infty)$, where the basepoint $a_\infty$ is not given in advance but is chosen iteratively by lifting basepoints $a_n\in X_n$ as we work through the spectral sequence. This makes life difficult because $F_{n+1}$ and $\pi_\ast(X_n)$ are not defined until we have chosen $a_n$. Has any theory of this type been worked out in detail?

• I have the impression that there is a paper of Bousfield that does something like this. – Tom Goodwillie Jul 15 '11 at 11:15
• Tom, might you be thinking of Bousfield's "Homotopy spectral sequences and obstructions" (Israel J. Math vol. 66)? It does something like this for unbased cosimplicial spaces, rather than a general tower of fibrations. – Tyler Lawson Jul 15 '11 at 12:57
• @Tyler: thanks for the pointer. If you want to promote that comment to an answer, then I will accept it. – Neil Strickland Jul 15 '11 at 16:35

• Interesting! Am I right if I say that treating the case of cosimplicial spaces is enough since the spectral sequence of a tower of fibrations would be the same as the spectral sequence of the associated cosimplicial space constructed as a kind of André-Quillen resolution using the nerve of the category $\mathbb N$? – Fernando Muro Aug 24 '13 at 22:28