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I was browsing MO and I have stumbled upon this answer which discusses why we should expect homotopy groups of spheres to be complicated. One heuristic argument given is that "the theory needs to have some complexity somewhere", which means that since spheres have simple cohomology, they have to have complicated homotopy, and dually for EM spaces, since they have simple homotopy, they have complicated cohomology.

Now, I understand that this might have been nothing more than a heuristic given to prove a point -- after all, they are spaces with both homotopy and cohomology simple, like the contractible ones. However, something tells me that there is something actually going on there. Hence the question:

Is there some sense in which a (sufficiently nontrivial) space cannot have both homotopy and cohomology simple?

If this is not a general phenomenon, what are some good (families of) examples for which both homotopy and cohomology are simple?

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    $\begingroup$ Examples include BU and BO. These spaces have both understandable cohomology and homotopy. More generally, you can look at Wilson spaces, which are p-local simply-connected H-spaces with homotopy and homology free and finitely generated over Z_(p). A theorem of Steve Wilson's says that every Wilson space is a product of indecomposable Wilson spaces. Each indecomposable Wilson space is the zeroth space of a spectrum $BP\langle n \rangle$, which can be constructed from the complex cobordism spectrum MU (the Thom spectrum of a spherical bundle over BU). $\endgroup$ – skd Sep 18 '18 at 20:16
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    $\begingroup$ If you restrict to simply connected spaces and interpret "simple" as "finite dimensional", then a dichotomy of this kind exists. Look up the McGibbon-Neisendorfer theorem. $\endgroup$ – Dan Petersen Sep 19 '18 at 0:02
  • $\begingroup$ I take the unpopular view, but which goes back to pre 1935, that we should seek nonabelian higher versions of the (strict) fundamental group, but which we now realise has to be a groupoid, and try to Model and compute homotopy types, of which homotopy groups form only a small part of the global structure, requiring more analysis. Background to this view is in the paper available at sciencedirect.com/science/article/pii/S0019357717300460 How cohomology fits into this picture is less clear. So this is more on relating the questions to more general issues in that paper. $\endgroup$ – Ronnie Brown Sep 20 '18 at 11:12