## Algebraic & Geometric Topology

### 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 $BSU$, $BU$, $BSO$, $BO$, $BSp$, $BGL∞(R)+$ and $Q0(S0)$. We show that these infinite loop spaces are the zero spaces of nonunital $E∞$–ring spectra. We introduce the notion of $q$–nilpotent K–theory of a CW–complex $X$ for any $q ≥ 2$, which extends the notion of commutative K–theory defined by Adem and Gómez, and show that it is represented by $ℤ × B(q,U)$, where $B(q,U)$ is the $qth$ term of the aforementioned filtration of $BU$.

For the proof we introduce an alternative way of associating an infinite loop space to a commutative $I$–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 $I$–rig and show that they give rise to nonunital $E∞$–ring spectra.

#### Article information

Source
Algebr. Geom. Topol., Volume 17, Number 2 (2017), 869-893.

Dates
Revised: 16 September 2016
Accepted: 29 September 2016
First available in Project Euclid: 19 October 2017

https://projecteuclid.org/euclid.agt/1508431447

Digital Object Identifier
doi:10.2140/agt.2017.17.869

Mathematical Reviews number (MathSciNet)
MR3623675

Zentralblatt MATH identifier
1360.55003

Keywords
K-theory Nilpotent K-theory

#### Citation

Adem, Alejandro; Gómez, José; Lind, John; Tillmann, Ulrike. Infinite loop spaces and nilpotent K–theory. Algebr. Geom. Topol. 17 (2017), no. 2, 869--893. doi:10.2140/agt.2017.17.869. https://projecteuclid.org/euclid.agt/1508431447

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