Commutative K-theory

Thesis

The bar construction BG of a topological group G has a subcomplex B_comG ⊂ BG assembled from spaces of commuting elements in G. If G = U;O (the infinite unitary / orthogonal groups) then B_comU and B_comO are E_∞-ring spaces. The corresponding cohomology theory is called commutative K-theory.

In this work we study properties of the spaces B_comG and of infinite loop spaces built from them, with an emphasis on the cases G = U,O. The content of this thesis is organised as follows:

In Chapter 1 we consider a family of self-maps of B_comG and apply these to study the question when the inclusion map B_comG ⊂ BG admits a section up to homotopy.

In Chapter 2 we show that B_comU is a model for the E_∞-ring space underlying the ku-group ring of ℂP^∞. Thus we provide a complete description of complex commutative K-theory. We also study the space B_comO. Our results include a computation of the torsionfree part of the homotopy groups of B_comO and a long exact sequence relating real commutative K-theory to singular mod-2 homology.

Chapter 3 is self-contained. We prove a result about the acyclicity of the "comparison map" M_∞ → ΩBM in the group-completion theorem and apply this to compare the infinite loop space associated to a commutative 𝕀-monoid with the Quillen plus-construction.

Chapter 4 is concerned with a previously known filtration of Ω₀^∞S^∞ by certain infinite loop spaces {hocolim_𝕀B(q, Σ_)}_q≥2. For each term in this filtration we construct another filtration on the spectrum level, whose subquotients we describe. Our set-up is more general, but the space hocolim_𝕀B(q, Σ_) will serve as our main example.

Appendix A is an excerpt from the author's Oxford transfer thesis. There we gave a construction of an infinite loop space associated to certain subspaces B(q, Γ_g,1) ⊂ BΓ_g;1, where Γ_g;1 is the mapping class group of a genus g surface with one boundary component.

Authors: Gritschacher, S

• Type of award: DPhil
• Level of award: Doctoral
• Awarding institution: University of Oxford