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Let $\pi$ be a discrete group, and let $G$ be a compact connected Lie group.Then there is a map$\Theta\colon\mathrm{Hom}(\pi,G)_0\to\mathrm{map}_*(B\pi,BG)_0$ between thenull-components of the spaces of homomorphism and based maps, which sends ahomomorphism to the induced map between classifying spaces. Atiyah and Bottstudied this map for $\pi$ a surface group, and showed that it is surjective inrational cohomology. In this paper, we prove that the map $\Theta$ issurjective in rational cohomology for $\pi=\mathbb{Z}^m$ and the classicalgroup $G$ except for $SO(2n)$, and that it is not surjective for$\pi=\mathbb{Z}^m$ with $m\ge 3$ and $G=SO(2n)$ with $n\ge 4$. As anapplication, we consider the surjectivity of the map $\Theta$ in rationalcohomology for $\pi$ a finitely generated nilpotent group. We also consider thedimension of the cokernel of the map $\Theta$ in rational homotopy groups for$\pi=\mathbb{Z}^m$ and the classical groups $G$ except for $SO(2n)$.

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