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 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 $\pi$ a surface group, and showed that it is surjective in rational cohomology. In this paper, we prove that the map $\Theta$ is surjective in rational cohomology for $\pi =\mathbb {Z}^m$ and the classical group $G$ 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 an application, we consider the surjectivity of the map $\Theta$ in rational cohomology for $\pi$ a finitely generated nilpotent group. We also consider the dimension of the cokernel of the map $\Theta$ in rational homotopy groups for $\pi =\mathbb {Z}^m$ and the classical groups $G$ except for $SO(2n)$.

We consider the Birman–Hilden inclusion $\phi\colon\Br_{2g+1}\to\Gamma_{g,1}$ of the braid group into the mapping class group of an orientable surface with boundary, and prove that $\phi$ is stably trivial in homology with twisted coefficients in the symplectic representation $H_1(\Sigma_{g,1})$ of the mapping class group; this generalises a result of Song and Tillmann regarding homology with constant coefficients. Furthermore we show that the stable homology of the braid group with coefficients in $\phi^*(H_1(\Sigma_{g,1}))$ has only 4-torsion.

We show that various classes of products of manifolds do not support transitive Anosov diffeomorphisms. Exploiting the Ruelle–Sullivan cohomology class, we prove that the product of a negatively curved manifold with a rational homology sphere does not support transitive Anosov diffeomorphisms. We extend this result to products of finitely many negatively curved manifolds of dimension at least three with a rational homology sphere that has vanishing simplicial volume. As an application of this study, we obtain new examples of manifolds that do not support transitive Anosov diffeomorphisms, including certain manifolds with non-trivial higher homotopy groups and certain products of aspherical manifolds.