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 Q 0 (S 0 ). We show that these infinite loop spaces are the zero spaces of non-unital 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-Gómez, and show that it is represented by Z × 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 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 I-rig and show that they give rise to non-unital E ∞ -ring spectra.

Commutative cocycles and stable bundles over surfaces

“…Naturality of the isomorphisms {±I} × SO(2n + 1) ∼ = O(2n + 1) with respect to the aforementioned inclusions and the induced isomorphisms at the level of π 2 also give the claimed isomorphisms in part (3).…”

“…Similar considerations with unitary matrices yield complex commutative K-theory. There is a natural forgetful map KO com (X) → KO 0 (X), and in Adem-Gómez-Lind-Tillman [3] it was shown that this map admits a splitting (additively), and the same holds in the complex case.…”

“…The kernel of this map is the "non-standard" part of the commutative K-theory of X. In [3], it is shown that there is a splitting of infinite loop spaces B com O ≃ BO × E com O. In fact, it follows from [3] that the natural map…”

“…Since the map π 2 (B com O) → π 2 (BO) induced by ι is surjective and π 1 (B com O) ∼ = π 1 (BO) (see [3,Lemma 4.3]), E com O is simply connected, and then the last term in the above sequence is trivial.…”

Colimits of abelian groups

“…A key property of BG is that it classifies principal G-bundles; there is a bijection between the set of isomorphism classes of principal G-bundles over a CW-complex X and the set of homotopy classes of maps X → BG. It is shown in [2] that B(q, G) is a classifying space for principal G-bundles of certain type, so called principal G-bundles of transitional nilpotency class less than q. It is a natural problem to study the homotopy type of B(q, G).…”