Arkiv för Matematik

Volume 56 (2018)

Number 1

A torus theorem for homotopy nilpotent loop spaces

Pages: 53 – 71

DOI: http://dx.doi.org/10.4310/ARKIV.2018.v56.n1.a5

Authors

Cristina Costoya (Computación, Facultad de Informática, Universidade da Coruña, A Coruña, Galicia, Spain)

Jérôme Scherer (Laboratoire pour la topologie et les neurosciences UPHESS, École polytechnique fédérale de Lausanne (EPFL), Switzerland)

Antonio Viruel (Álgebra, Geometría y Topología, Universidad de Málaga, Spain)

Abstract

Nilpotency for discrete groups can be defined in terms of central extensions. In this paper, the analogous definition for spaces is stated in terms of principal fibrations having infinite loop spaces as fibers, yielding a new invariant between the classical LS cocategory and the more recent notion of homotopy nilpotency introduced by Biedermann and Dwyer. This allows us to characterize finite homotopy nilpotent loop spaces in the spirit of Hubbuck’s Torus Theorem, and obtain corresponding results for $p$-compact groups and $p$-Noetherian groups.

Keywords

nilpotent, homotopy nilpotent, cocategory, algebraic theory, Goodwillie calculus, excisive functor, $p$-compact group

2010 Mathematics Subject Classification

Primary 55P35. Secondary 18C10, 55M30, 55P65.

Full Text (PDF format)

The authors are supported by Xunta de Galicia grant EM2013/016. The first author is supported by Ministerio de Economía y Competitividad (Spain), grant MTM2016-79661-P (AEI/FEDER, UE, support included). The second author is supported by Ministerio de Economía y Competitividad (Spain), grant MTM2016-80439-P. The third author is supported by Ministerio de Economía y Competitividad (Spain), grants MTM2013-41768-P and MTM2016-78647-P (AEI/FEDER, UE, support included).

Received 25 May 2016

Received revised 29 May 2017