Infinite loop spaces and nilpotent K–theory

Abstract

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 $BSU$, $\mathit{BU}$, $BSO$, $\mathit{BO}$, $BSp$, $B{GL}_{\infty }{\left(R\right)}^{+}$ and $Q_0\left({\mathbb{S}}^0\right)$. We show that these infinite loop spaces are the zero spaces of nonunital $E_{\infty }$–ring spectra. We introduce the notion of $q$–nilpotent K–theory of a CW–complex $X$ for any $q\ge 2$, which extends the notion of commutative K–theory defined by Adem and Gómez, and show that it is represented by $\mathbb{Z}\times B\left(q,U\right)$, where $B\left(q,U\right)$ is the $q^{th}$ term of the aforementioned filtration of $\mathit{BU}$.

For the proof we introduce an alternative way of associating an infinite loop space to a commutative $\mathbb{I}$–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 $\mathbb{I}$–rig and show that they give rise to nonunital $E_{\infty }$–ring spectra.