Algebraic & Geometric Topology

Infinite loop spaces and nilpotent K–theory

Alejandro Adem, José Gómez, John Lind, and Ulrike Tillmann

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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 × 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

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

Received: 2 September 2015
Revised: 16 September 2016
Accepted: 29 September 2016
First available in Project Euclid: 19 October 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 55N15: $K$-theory [See also 19Lxx] {For algebraic $K$-theory, see 18F25, 19- XX} 55R35: Classifying spaces of groups and $H$-spaces

K-theory Nilpotent K-theory


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.

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