A Torus Theorem for homotopy nilpotent groups

TL;DR

This paper introduces a new invariant for homotopy nilpotent groups based on principal fibrations with infinite loop space fibers, extending classical nilpotency concepts to topological spaces and characterizing finite homotopy nilpotent loop spaces.

Contribution

It defines a novel homotopy nilpotency invariant using principal fibrations and compares it with existing notions like cocategory and homotopy nilpotency, leading to a homotopy-theoretic analogue of the Torus Theorem.

Findings

Characterization of finite homotopy nilpotent loop spaces

Comparison of the new invariant with classical cocategory

Results for p-compact and p-Noetherian groups

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 we compare with classical cocategory, but also with 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 corresponding results for $p$-compact groups and $p$-Noetherian groups.