A classifying space for commutativity in Lie groups

In this article we consider a space B(com)G assembled from commuting elements in a Lie group G first defined by Adem, Cohen and Torres-Giese. We describe homotopy-theoretic properties of these spaces using homotopy colimits, and their role as a classifying space for transitionally commutative bundles. We prove that Z x BcomU is a loop space and define a notion of commutative K-theory for bundles over a finite complex X, which is isomorphic to [X, Z x BcomU]. We compute the rational cohomology of B(com)G for G equal to any of the classical groups SU(r), U(q) and Sp(k), and exhibit the rational cohomologies of BcomU, BcomSU and B(com)Sp as explicit polynomial rings.