Recall that an H-group is a space $X$ (in the sense of homotopy theory, so say CW complex) that is a group object in the homotopy category. I.e., there's a multiplication map $X \times X \rightarrow X$ which is associative up to homotopy and with an inverse map again up to homotopy. The standard example of an H-group is a loop space $\Omega X$. What is a simple example of an H-group that is not a loop space?

If I understand the language right, I'm asking for a group-like $A_3$ algebra that's not $A_\infty$. If I were asking for $A_2$ but not $A_3$ then I know $S^7$ is a good example.

My motivation is illustrating some of the subtleties in defining $\infty$-groups for a HoTT seminar.

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    $\begingroup$ mathoverflow.net/questions/81721/… Have a look at Dylan Wilson's answer. $\endgroup$
    – David C
    Aug 27 '16 at 15:38
  • $\begingroup$ @DavidC: I even ran across that question when googling, but failed to scroll down to the last answer. Thanks! $\endgroup$ Aug 27 '16 at 15:50
  • $\begingroup$ That said, that answer is a bit advanced and terse for me, I'd love to see it fleshed out in one of the simplest cases. Does $RP^2$ satisfy the assumptions of that answer? $\endgroup$ Aug 27 '16 at 15:55
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    $\begingroup$ Dylan's answer quotes Stasheff; Stasheff's paper jstor.org/stable/1993608 includes the examples $M(Q(p),2n+1)$ where $Q(p)$ is the $p$-adic subgroup of the rationals --- this is $A_{p-1}$ but not $A_p$. $\endgroup$ Aug 31 '16 at 2:22
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    $\begingroup$ $RP^2$ cannot be an $H$-space; Exercises there are four homotopy classes of map $RP^2\to RP^2$, and there are four homotopy classes of map $\Sigma RP^2 \to \Sigma RP^2$; but the suspension $RP^2\to \Omega\Sigma RP^2$ misses the degree-2 map. (otherwise the cofiber of something would have a nontrivial $Sq^2$ acting on $H^1$! ) Consequently, some of the four maps below get confused in the suspension, too. Upshot is that no map $\Omega\Sigma RP^2 \to RP^2 $ can be a retraction. $\endgroup$ Aug 31 '16 at 2:39

The first example appears in the paper "Homotopy associativity and finite CW complexes" Topology vol.9 (1970) 121-128 by Alexander Zabrodsky. With present knowledge it is possible to construct many examples.
Here are some examples. Consider Sp(n) with n 3 or more, or G_2. Construct X from a pullback diagram of p-local spaces, where X localized at 2 or 3 is the Lie group and localized at the remaining primes is the product of local sphere having the same type as the Lie group. Then X is homotopy associative because each of its localizations is homotopy associative. The space X is not a mod p group for p between 5 and 2n if Sp(n) is used and not a mod 5 group if G_2 is used in the construction. The latter claim is based of a theorem proved by Clarence Wilkerson, "K-theory Operations in mod-p Loop Spaces" Math.Z. 132, 29-44, (1973)

  • $\begingroup$ Welcome to mathoverflow John! $\endgroup$ Feb 10 '17 at 9:58

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