On the Second Homotopy Group of Spaces of Commuting Elements in Lie Groups

Let G be a compact connected Lie group and n >= 1 an integer. Consider the space of ordered commuting n-tuples in G, Hom(Z(n), G), and its quotient under the adjoint action, Rep (Z(n), G) := Hom(Z(n), G)/ G. In this article, we study and in many cases compute the homotopy groups pi(2)(Hom(Z(2), G)). For G simply connected and simple, we show that pi(2)(Hom(Z(2), G)) congruent to Z and pi(2)(Rep(Z(2), G)) congruent to Z and that on these groups the quotient map Hom(Z(2), G) -> Rep(Z(2), G) induces multiplication by the Dynkin index of G. More generally, we show that if G is simple and Hom(Z(2), G)(1) subset of Hom(Z(2), G) is the path component of the trivial homomorphism, then H-2(Hom(Z(2), G)(1); Z) is an extension of the Schur multiplier of pi(1)(G)(2) by Z. We apply our computations to prove that if B(com)G(1) is the classifying space for commutativity at the identity component, then pi(4)(B(com)G(1)) congruent to Z circle plus Z, and we construct examples of non-trivial transitionally commutative structures on the trivial principal G-bundle over the sphere S-4.