most recent 30 from http://mathoverflow.net 2013-05-26T06:01:43Z http://mathoverflow.net/feeds/question/47335 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/47335/recognition-principle Oblomov 2010-11-25T16:12:48Z 2010-11-25T16:12:48Z

<p>Hello,</p> <p>The classical statement of the recognition principle (after Boardman, Vogt, Milgram and May) that I know is: </p> <p>Let $X$ be a (group-like) topological space acted on by the little $n$-discs operad, then $X$ is (weakly) homotopy equivalent (as an algebra over the little discs operad) to an $n$-fold loop space.</p> <p>In particular, the theorem leaves me unsatisfied since there are some spaces which have the homotopy type of an $n$-fold loop space but which are not acted on by the little discs operad.</p> <p>The "right" statement of the recognition principle should be a statement inside the homotopy category (of spaces and operads), but I have never seen it stated properly.</p> <p>Does it appear somewhere? Also, is there a modern version of P. May's proof that appears in the geometry of iterated loop spaces?</p> <p>Thanks,</p> <p>K.</p>