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

Let $\pi$ be a discrete group, and let $G$ be a compact connected Lie group. $\mathrm{H}\mathrm{o}\mathrm{m}(\pi ,G{)}_0$ denotes the null-component of the space of homomorphisms from $\pi$ to $G$, and ${\mathrm{m}\mathrm{a}\mathrm{p}}_{\ast }(B\pi ,BG{)}_0$ denotes the null-component of the space of maps from $B\pi$ to $BG$. Since the classifying space functor is continuous, there is a continuous map $\mathrm{\Theta }\text{\hspace{0.17em}⁣}:\mathrm{H}\mathrm{o}\mathrm{m}(\pi ,G{)}_0\to {\mathrm{m}\mathrm{a}\mathrm{p}}_{\ast }(B\pi ,BG{)}_0$. Atiyah and Bott studied this map when $\pi$ is a surface group, and proved surjectivity in rational cohomology. In this paper, we obtain the condition that the map $\mathrm{\Theta }$ is surjective or not in rational cohomology when $\pi$ is ${\mathbf{Z}}^m$ for $m\ge 3$ and $G$ is a compact connected Lie group.