A torus theorem for homotopy nilpotent loop spaces

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.

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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

Accepted 8 June 2017

Published 30 April 2018