On the comparison of stable and unstable $p$-completion
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- by Tobias Barthel and A. K. Bousfield PDF
- Proc. Amer. Math. Soc. 147 (2019), 897-908 Request permission
Abstract:
In this note we show that a $p$-complete nilpotent space $X$ has a $p$-complete suspension spectrum if and only if its homotopy groups $\pi _*X$ are bounded $p$-torsion. In contrast, if $\pi _*X$ is not all bounded $p$-torsion, we locate uncountable rational vector spaces in the integral homology and in the stable homotopy groups of $X$. To prove this, we establish a homological criterion for $p$-completeness of connective spectra. Moreover, we illustrate our results by studying the stable homotopy groups of $K(\mathbb {Z}_p,n)$ via Goodwillie calculus.References
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Additional Information
- Tobias Barthel
- Affiliation: Department of Mathematical Sciences, University of Copenhagen, Universitets- parken 5, 2100 København Ø, Denmark
- MR Author ID: 1015635
- Email: tbarthel@math.ku.dk
- A. K. Bousfield
- Affiliation: Department of Mathematics, Statistics and Computer Sciences, University of Illinois at Chicago, 851 S. Morgan Street (M/C 249), Chicago, Illinois 60607-7045
- MR Author ID: 198766
- Email: bous@uic.edu
- Received by editor(s): December 22, 2017
- Received by editor(s) in revised form: May 28, 2018
- Published electronically: November 5, 2018
- Additional Notes: The first author has been partially supported by the DNRF92.
- Communicated by: Michael A. Mandell
- © Copyright 2018 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 147 (2019), 897-908
- MSC (2010): Primary 55P60, 55P42
- DOI: https://doi.org/10.1090/proc/14250
- MathSciNet review: 3894926