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Infinite loop spaces and nilpotent K–theory

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

Algebraic & Geometric Topology 17 (2017) 869–893

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.

K-theory, Nilpotent K-theory
Mathematical Subject Classification 2010
Primary: 55N15, 55R35
Received: 2 September 2015
Revised: 16 September 2016
Accepted: 29 September 2016
Published: 14 March 2017
Alejandro Adem
Department of Mathematics
University of British Columbia
1984 Mathematics Road, Room 121
Vancouver BC V6T 1Z2
José Manuel Gómez
Departmento de Matemáticas
Universidad Nacional de Colombia
AA 3840
John A Lind
Department of Mathematics
Reed College
3203 SE Woodstock Blvd.
Portland, OR 97202
United States
Ulrike Tillmann
Mathematical Institute
Oxford University
United Kingdom