The MO-question asks why the classifying space of a group is not necessarily rationally a product of Eilenberg-Maclane spaces. I am looking for classes of examples of connected topological groups/connected associative H-spaces such that their classifying space happens to be rationally a product of Eilenberg-Maclane spaces. I'm the most interested in examples of groups that have an interesting action on a closed manifold.

So far I know that this holds for

• compact groups, because the cohomology ring of their classifying space is a polynomial ring
• Lie groups since they are homotopy equivalent to their maximal compact subgroup
• the identity component of the homotopy automorphisms of an aspherical space, because its classifying space is already an Eilenberg-Maclane space

The MO-question asks why the classifying space of a group is not necessarily rationally a product of Eilenberg–MacLane spaces. I am looking for classes of examples of connected topological groups/connected associative H-spaces such that their classifying space happens to be rationally a product of Eilenberg–MacLane spaces. I'm the most interested in examples of groups that have an interesting action on a closed manifold.

So far I know that this holds for

• compact groups, because the cohomology ring of their classifying space is a polynomial ring
• Lie groups since they are homotopy equivalent to their maximal compact subgroup
• the identity component of the homotopy automorphisms of an aspherical space, because its classifying space is already an Eilenberg–MacLane space.

The MO-question asks why the classifying space of a group is not necessarily rationally a product of Eilenberg-Maclane spaces. I am looking for classes of examples of connected topological groups/connected associative H-spaces such that their classifying space happens to be rationally a product of Eilenberg-Maclane spaces. I'm the most interested in examples of groups that have an interesting action on a closed manifold.

So far I know that this holds for

• compact groups, because the cohomology ring of their classifying space is a polynomial ring
• Lie groups since they are homotopy equivalent to their maximal compact subgroup
• the identity component of the homotopy automorphisms of an aspherical space, because its classifying space is already an Eilenberg-Maclane space

The MO-question asks why the classifying space of a group is not necessarily rationally a product of Eilenberg-Maclane spaces. I am looking for classes of examples of connected topological groups/connected associative H-spaces such that their classifying space happens to be rationally a product of Eilenberg-Maclane spaces. I'm the most interested in examples of groups that have an interesting action on a closed manifold.

So far I know that this holds for

• compact groups, because the cohomology ring of their classifying space is a polynomial ring
• Lie groups since they are homotopy equivalent to their maximal compact subgroup
• the identity component of the homotopy automorphisms of an aspherical space, because its classifying space is already an Eilenberg-Maclane space