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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.
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    $\begingroup$ Someone explained to me, but I can't remember the details of, the orthography of Mac Lane's name; he latterly went by Mac Lane, but originally by MacLane. (Maybe that's all there is to the explanation, but I seem to remember more.) But it was never Maclane, so I edited accordingly. $\endgroup$
    – LSpice
    Commented Mar 8 at 21:20
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    $\begingroup$ I believe it was for the sake of his first wife, Dorothy Jones, who found it easier to type it with a space. $\endgroup$
    – Dave Benson
    Commented Mar 8 at 21:31
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    $\begingroup$ @LSpice: you missed the title. $\endgroup$
    – Ryan Budney
    Commented Mar 9 at 6:46
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    $\begingroup$ It will be true of any topological group (or even space) whose rational cohomology is a free associative, graded-commutative algebra over $\mathbb{Q}$. Simply choose a set of primitive generators to obtain a rational equivalence with a product of EM spaces. It's easy to verify that the obstructions for this to be an $A_\infty$-equivalence vanish. On the other hand, these cohomological conditions clearly need to hold for the outcome to be true. They're satisfied, for instance, by any group homotopy equivalent to a finite CW complex (e.g. any Lie group, as you recognise). $\endgroup$
    – Tyrone
    Commented Mar 9 at 11:38
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    $\begingroup$ @ThorbenK the rational $A_\infty$-equivalence will push down to the classifying space. $\endgroup$
    – Tyrone
    Commented Mar 9 at 20:15

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