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 $E^1_{jk}=\pi_jF_k\Longrightarrow \pi_{j+k}X_\infty$. 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 $\pi_*(X_\infty,a_\infty)$, 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 $\pi_*(X_n)$ are not defined until we have chosen $a_n$. Has any theory of this type been worked out in detail?
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As requested, I am reposting this comment as an answer. Bousfield covers this material in "Homotopy spectral sequences and obstructions," Israel J. Math 66. The discussion is specific to cosimplicial objects (e.g. the discussion of obstruction cocycles in Section 5) and the general method of obtaining "partially" defined spectral sequences without basepoints would have to be extracted. However, I believe that all the necessary content is already there. |
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