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On the $ p$-adic completions of nonnilpotent spaces


Author: A. K. Bousfield
Journal: Trans. Amer. Math. Soc. 331 (1992), 335-359
MSC: Primary 55P60; Secondary 20E18, 20J05
DOI: https://doi.org/10.1090/S0002-9947-1992-1062866-4
MathSciNet review: 1062866
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Abstract: This paper deals with the $ p$-adic completion $ {F_{p\infty }}X$ developed by Bousfield-Kan for a space $ X$ and prime $ p$. A space $ X$ is called $ {F_p}$-good when the map $ X \to {F_{p\infty }}X$ is a $ \bmod$-$ p$ homology equivalence, and called $ {F_p}$-bad otherwise. General examples of $ {F_p}$-good spaces are established beyond the usual nilpotent or virtually nilpotent ones. These include the polycyclic-by-finite spaces. However, the wedge of a circle with a sphere of positive dimension is shown to be $ {F_p}$-bad. This provides the first example of an $ {F_p}$-bad space of finite type and implies that the $ p$-profinite completion of a free group on two generators must have nontrivial higher $ \bmod$-$ p$ homology as a discrete group. A major part of the paper is devoted to showing that the desirable properties of nilpotent spaces under the $ p$-adic completion can be extended to the wider class of $ p$-seminilpotent spaces.


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DOI: https://doi.org/10.1090/S0002-9947-1992-1062866-4
Article copyright: © Copyright 1992 American Mathematical Society

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