Using a construction derived from the descending central series of the free groups, we produce filtrations by infinite loop spaces of the classical infinite loop spaces , , , , , and . We show that these infinite loop spaces are the zero spaces of nonunital –ring spectra. We introduce the notion of –nilpotent K–theory of a CW–complex for any , which extends the notion of commutative K–theory defined by Adem and Gómez, and show that it is represented by , where is the term of the aforementioned filtration of .
For the proof we introduce an alternative way of associating an infinite loop space to a commutative –monoid and give criteria for when it can be identified with the plus construction on the associated limit space. Furthermore, we introduce the notion of a commutative –rig and show that they give rise to nonunital –ring spectra.
"Infinite loop spaces and nilpotent K–theory." Algebr. Geom. Topol. 17 (2) 869 - 893, 2017. https://doi.org/10.2140/agt.2017.17.869