## Arkiv för Matematik

### A torus theorem for homotopy nilpotent loop spaces

#### 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.

#### Note

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).

#### Article information

Source
Ark. Mat., Volume 56, Number 1 (2018), 53-71.

Dates
Revised: 29 May 2017
First available in Project Euclid: 19 June 2019

https://projecteuclid.org/euclid.afm/1560967220

Digital Object Identifier
doi:10.4310/ARKIV.2018.v56.n1.a5

Mathematical Reviews number (MathSciNet)
MR3800459

Zentralblatt MATH identifier
1396.55006

#### Citation

Costoya, Cristina; Scherer, Jérôme; Viruel, Antonio. A torus theorem for homotopy nilpotent loop spaces. Ark. Mat. 56 (2018), no. 1, 53--71. doi:10.4310/ARKIV.2018.v56.n1.a5. https://projecteuclid.org/euclid.afm/1560967220