Commutative K-theory

The bar construction BG of a topological group G has a subcomplex BcomG ⊂ BG assembled from spaces of commuting elements in G. If G = U;O (the infinite unitary / orthogonal groups) then BcomU and BcomO are E∞-ring spaces. The corresponding cohomology theory is called commutative K-theory. In this work we study properties of the spaces BcomG 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 BcomG and apply these to study the question when the inclusion map BcomG ⊂ BG admits a section up to homotopy. In Chapter 2 we show that BcomU 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 BcomO. Our results include a computation of the torsionfree part of the homotopy groups of BcomO 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 {hocolim𝕀B(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 hocolim𝕀B(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, Γg,1) ⊂ BΓg;1, where Γg;1 is the mapping class group of a genus g surface with one boundary component.