Abstract
In this chapter we establish the following simple and useful sufficient conditions on a tower of fibrations {Ys}, in order that it can be used to obtain the homotopy type of the R-completion of a given space X:
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(i)
If f: X → {Ys is a map which induces, for every R-module M, an isomorphism
$$\mathop {\lim }\limits_ \to H*\left( {{Y_S};M} \right) \approx H*\left( {X;M} \right)$$then f induces a homotopy equivalence \({R_\infty }X \simeq \mathop {\lim }\limits_ \to {R_\infty }{Y_S}\).
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(ii)
If, in addition, each Ys is R-complete (Ch.I, 5.1) , then the space \(\mathop {\lim }\limits_ \to {Y_S}\) already has the same homotopy type as R∞X.
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(iii)
If, in addition, each Ys satisfies the even stronger condition of being R-nilpotent (4.2), then, in a certain precise sense, the tower {Ys} has the same homotopy type as the tower {Rs}
Keywords
- Spectral Sequence
- Nilpotent Group
- Full Subcategory
- Homotopy Type
- Inverse Limit
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.
An erratum to this chapter is available at http://dx.doi.org/10.1007/978-3-540-38117-4_14
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© 1972 Springer-Verlag Berlin Heidelberg
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Bousfield, A.K., Kan, D.M. (1972). Tower lemmas. 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_3
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DOI: https://doi.org/10.1007/978-3-540-38117-4_3
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-06105-2
Online ISBN: 978-3-540-38117-4
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