Abstract
In this chapter we generalize two well-known results about towers of fibrations:
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(i)
We will show that, for a (pointed) tower of fibrations {Xn}, the short exact sequence
$$* \to {\mathop {\lim }\limits_ \leftarrow ^1}{\pi _{i + 1}}{X_n} \to {\pi _i}\mathop {\lim }\limits_ \leftarrow {X_n} \to \mathop {\lim }\limits_ \leftarrow {\pi _i}{X_n} \to *$$which 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.
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(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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© 1972 Springer-Verlag Berlin Heidelberg
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Bousfield, A.K., Kan, D.M. (1972). Towers of fibrations. In: Homotopy Limits, Completions and Localizations. Lecture Notes in Mathematics, vol 304. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-38117-4_9
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DOI: https://doi.org/10.1007/978-3-540-38117-4_9
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-06105-2
Online ISBN: 978-3-540-38117-4
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