Homotopy Limits, Completions and Localizations pp 249-264 | Cite as

# Towers of fibrations

Chapter

## Abstract

In this chapter we generalize two well-known results about towers of fibrations:

- (i)We will show that, for a (pointed) tower of fibrations {X
_{n}}, the short exact sequencewhich is “well known” for i ≥ 1, also exists for i = 0, if one uses a suitable notion of \({\mathop {\lim }\limits_ \leftarrow ^1}\) for not necessarily abelian groups.$$* \to {\mathop {\lim }\limits_ \leftarrow ^1}{\pi _{i + 1}}{X_n} \to {\pi _i}\mathop {\lim }\limits_ \leftarrow {X_n} \to \mathop {\lim }\limits_ \leftarrow {\pi _i}{X_n} \to *$$ - (ii)
We will generalize the usual homotopy spectral sequence of a (pointed) tower of fibrations, to an “extended” homotopy spectral sequence, which in dimension 1 consists of (possibly non-abelian) groups, and in dimension 0 of pointed sets, acted on by the groups is dimension 1.

## Keywords

Abelian Group Spectral Sequence Short Exact Sequence Homotopy Type Homotopy Group
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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## Copyright information

© Springer-Verlag Berlin Heidelberg 1972