In this talk I will define the space BcomG arising from commuting tuples in G originally presented by A. Adem, F. Cohen and E. Torres. This space sits inside the classifying space BG and classifies principal G-bundles with a transitionally commutative structure. I will focus on describing the space BcomG for G=SU(2), U(2) and O(2), via its integral and mod 2 cohomology ring together with its Steenrod algebra. If time permits, for the Lie groups above, I’ll describe the homotopy type of the homotopy fiber of the inclusion BcomG into BG, denoted EcomG, whose homotopy groups are in bijection with transitionally commutative structures on the trivial G-bundle over spheres.

This is joint work with O. Antolín and S. Gritschacher.