On the second homotopy group of the classifying space for commutativity in Lie groups

In this note we show that the second homotopy group of B(2, G), classifying space for commutativity a compact Lie G, contains direct summand isomorphic to $$\pi _1(G)\oplus \pi _1([G,G])$$ , where [G, G] is commutator subgroup G. It follows from similar statement E(2, fiber canonical inclusion $$B(2,G)\hookrightarrow BG$$ . As consequence our main result obtain if G) 2-connected, then simply-connected. This last completes how higher connectivity resembles