The space of commuting elements in a Lie group and maps between classifying spaces

Let π be a discrete group, and let G be a compact-connected Lie group. Then, there is a map Θ: Hom(π, G)0 → map_∗(Bπ, BG)0 between the null components of the spaces of homomorphisms and based maps, which sends a homomorphism to the induced map between classifying spaces. Atiyah and Bott studied this map for π a surface group, and showed that it is surjective in rational cohomology. In this paper, we prove that the map Θ is surjective in rational cohomology for π = Z^m and the classical group G except for SO(2n), and that it is not surjective for π = Z^m with m ≥ 3 and G = SO(2n) with n ≥ 4. As an application, we consider the surjectivity of the map Θ in rational cohomology for π a finitely generated nilpotent group. We also consider the dimension of the cokernel of the map Θ in rational homotopy groups for π = Z^m and the classical groups G except for SO(2n).