Characteristic classes for TC structures

In this article we study the construction of characteristic classes for principal $G$-bundles equipped with an additional structure called transitionally commutative structure (TC structure). These structures classify, up to homotopy, possible trivializations of a principal $G$-bundle, such that the induced cocycle have functions that commute in the intersections of their domains. We focus mainly on the cases where the structural group $G$ equals $SU(n)$, $U(n)$ or $\mathrm{Sp}(n)$. Our approach is an algebraic-geometric construction that relies on the so called power maps defined on the space $B_{\mathrm{com}}G$, the classifying space for commutativity in the group $G$.