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It is known that any space of loops is an H-space. So my question has two parts:

  1. What are the obstacles for a complex to be an H-space? Is there any hope to somehow reasonably classify/characterize all H-spaces?
  2. What are the obstacles for an H-space to be a loop space?
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    $\begingroup$ Every loop space is an H-space, so Adams's resolution of the Hopf invariant 1 problem shows that $S^0, S^1, S^3$ and $S^7$ are the only spheres that are loop spaces. And is it really true that all H-spaces are loop spaces? $\endgroup$
    – Lennart Meier
    Commented Mar 21, 2022 at 11:41
  • $\begingroup$ Oh, really, I mixed up the direction of inclusion and asked a stupid question. $\endgroup$
    – Arshak Aivazian
    Commented Mar 21, 2022 at 11:44
  • $\begingroup$ I changed the question, thanks $\endgroup$
    – Arshak Aivazian
    Commented Mar 21, 2022 at 11:49
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    $\begingroup$ All H-spaces have abelian $\pi_1$ and vanishing Whitehead products. These are your primary obstructions. $\endgroup$
    – Tyrone
    Commented Mar 21, 2022 at 13:22
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    $\begingroup$ The multiplication on a loopspace is always homotopy associative, while on an H-space it need not be, e.g. $S^7$. $\endgroup$
    – Mark Grant
    Commented Mar 21, 2022 at 18:19

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