Pairings and monomorphisms of classifying spaces

We consider the maps between classifying spaces of compact Lie groups of the form B K × B L → B G . If the restriction map B L → B G is a weak epimorphism, then the restriction on BK is known to factor through the classifying spaces of the center of the compact Lie group G. Suppose H is a semi-simple subgroup of a connected compact Lie group G with rank ( H ) = rank ( G ) . Replacing the weak epimorphism B L → B G by the map B H → B G , analogous results are obtained. We also consider some monomorphisms of classifying spaces of compact Lie groups, such as BSO ( n ) → BSU ( n ) . Our proof will make use of admissible maps.