Uniqueness of loop spaces - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T04:03:26Z http://mathoverflow.net/feeds/question/65103 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/65103/uniqueness-of-loop-spaces Uniqueness of loop spaces Dr Shello 2011-05-16T02:50:58Z 2011-05-18T14:55:55Z <p>Suppose X is a loop space; by this we mean there is some space $Y$ with $\Omega Y \simeq X$. </p> <p>Under what assumptions is (the homotopy type of) $Y$ unique? </p> <p>As has been pointed out below, the homotopy type of $Y$ being determined uniquely is far from true in general. But for connected $Y$, are there conditions we can impose that make it so?</p> http://mathoverflow.net/questions/65103/uniqueness-of-loop-spaces/65104#65104 Answer by Mike Shulman for Uniqueness of loop spaces Mike Shulman 2011-05-16T03:27:25Z 2011-05-16T03:27:25Z <p>As Ryan points out, if Y is allowed to be disconnected, then there is no hope, since the loop-space construction sees only the connected component of the basepoint. But even if Y is assumed to be connected, it is not unique. For instance, let G and H be two discrete groups whose underlying sets are bijective, but which are not isomorphic. Then as (discrete) topological spaces, we have $G\simeq H$, and so both $K(G,1)$ and $K(H,1)$ are spaces Y such that $\Omega Y \simeq G \simeq H$. But $K(G,1)$ and $K(H,1)$ are not homotopy equivalent unless $G\cong H$ as groups.</p> <p>What is true, however, is that if we remember the "up-to-coherent-homotopy" multiplication (i.e. "$A_\infty$-structure") on a loop space $\Omega Y$, then the connected space Y is characterized up to homotopy equivalence by $\Omega Y$ and this additional data. For there is a delooping functor "B" from $A_\infty$-spaces to connected spaces, which preserves homotopy equivalence, and such that $B\Omega Y \simeq Y$.</p> http://mathoverflow.net/questions/65103/uniqueness-of-loop-spaces/65117#65117 Answer by Tilman for Uniqueness of loop spaces Tilman 2011-05-16T07:47:38Z 2011-05-16T07:47:38Z <p>As others have pointed out, the generic case (whatever that should mean in this case) is that the loop structure on a loop space is not unique. However, things get quite interesting whenever we have a space that actually does have a unique loop structure. I highly recommend looking at:</p> <p>Dwyer, Miller, Wilkerson: The homotopic uniqueness of $BS^3$, LNM 1298</p> <p>and</p> <p>Dwyer, Miller, Wilkerson: Homotopical uniqueness of classifying spaces. Topology 31 (1992), no. 1, 29–45.</p>