This is essentially a (cw) reference request, because it seems like the sort of thing that should have been looked at already.

Setting aside, for now, how to think what the localization of a general space is really; what's the right way to think about localization of $SO(n)$ --- or, if that doesn't make sense, of $Spin(n)$ --- as a space? (whether Sulivan's construction or Bousfield-Kan or ...)

some points of curiosity: Is it still a homotopy group? Is it better known as something else? Is there any good interaction between the algebraic side of localization and the group structure?

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    $\begingroup$ I am very naive about such things, but I find the formalism of Bousfield classes to be helpful. I think from that perspective you do get that it is a group up to homotopy (which is what I assume you mean by homotopy group). $\endgroup$ – Sean Tilson May 24 '11 at 15:41
  • $\begingroup$ er... yes... "group object in the homotopy category", not "$\pi_n$ of something"... I'm not quite awake yet; for instance why I'm being parenthetical about (simply connected cover) when mentioning "spin" again, after... $\endgroup$ – some guy on the street May 24 '11 at 15:50
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    $\begingroup$ My impulse (for any connected group $G$) would be to localize the classifying space of $G$ (which is simply connected and therefore not at all problematic) and then loop the result. $\endgroup$ – Tom Goodwillie May 24 '11 at 20:09
  • $\begingroup$ Oh, that sounds good. Thanks, Tom! $\endgroup$ – some guy on the street May 25 '11 at 5:25

I assume you mean the $p$-localization at a prime $p$? Any localization preserves loop spaces, so yes, the localization of a Lie group will be a loop space. Whether it's actually a group again depends on the model of localization you choose -- it's usually something defined in the homotopy category, so you might get something that is just homotopy equivalent to a group. But you can always choose a group model by, for example, applying the Kan loop group functor to the classifying space of the loop space. Localization at $\mathbf Z/p$, or $p$-completion, has turned out to be the more powerful concept in that context. Have a look at Dwyer's survey paper "Lie groups and p-compact groups".


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