  After introducing the simplicial manifolds, such as the different ways of
defining the differential forms on them, we summarized a canonical way of
calculating the characteristic classes of a $G$-principal bundle by computing
them on the classifying bundle $EG\longrightarrow BG$. Finally, we calculated
the first Pontryagin class on the classifying bundle of the Lie matrix groups
and showed that for certain of them, the computed form is equal to the
symplectic form on $BG$ given by some authors up to a constant coefficient.
