Infinite loop spaces and nilpotent K–theory

Adem, Alejandro¹ ; Gómez, José² ; Lind, John³ ; Tillmann, Ulrike⁴

¹Department of Mathematics, University of British Columbia, 1984 Mathematics Road, Room 121, Vancouver BC V6T 1Z2, Canada
²Departmento de Matemáticas, Universidad Nacional de Colombia, Medellín, AA 3840, Colombia
³Department of Mathematics, Reed College, 3203 SE Woodstock Blvd., Portland, OR 97202, United States
⁴Mathematical Institute, Oxford University, Oxford, OX2 6GG, United Kingdom

Algebraic and Geometric Topology, Tome 17 (2017) no. 2, pp. 869-893

Résumé

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.

Détail

DOI : 10.2140/agt.2017.17.869

Classification : 55N15, 55R35
Keywords: K-theory, Nilpotent K-theory

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