Commutative complex K-theory: Simon Gritschacher

Abstract: The study of spaces of homomorphisms from a discrete group to a compact Lie group has led to the definition of a new cohomology theory, called commutative K-theory. This theory, which was first introduced by Adem and Gomez, is a refinement of classical topological K-theory. It is defined using vector bundles which can be represented by commuting cocycles. I will begin the talk by discussing some general properties of the "classifying space for commutativity in a Lie group". Specialising to the unitary groups, I will show that the classifying space for commutative complex K-theory is precisely the E-Infinity ring space underlying the ku-group ring of BU(1). If time permits, I will mention some results about the real variant of commutative K-theory.