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 $BSU$, $BU$, $BSO$, $BO$, $BSp$, $BGL_{\infty}(R)^{+}$ and $Q_0(\mathbb{S}^{0})$. We show that these infinite loop spaces are the zero spaces of non-unital $E_\infty$-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-G\'omez, and show that it is represented by $\mathbb Z\times B(q,U)$, were $B(q,U)$ is the $q$-th term of the aforementioned filtration of $BU$. For the proof we introduce an alternative way of associating an infinite loop space to a commutative $\mathbb{I}$-monoid and give criteria 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 non-unital $E_\infty$-ring spectra.