Uniqueness of loop spaces - MathOverflow

Uniqueness of loop spaces

2011-05-16T02:50:58Z

Suppose X is a loop space; by this we mean there is some space $Y$ with $\Omega Y \simeq X$.

Under what assumptions is (the homotopy type of) $Y$ unique?

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?

Answer by Mike Shulman for Uniqueness of loop spaces

2011-05-16T03:27:25Z

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.

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 $B\Omega Y \simeq Y$.

Answer by Tilman for Uniqueness of loop spaces

2011-05-16T07:47:38Z

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:

Dwyer, Miller, Wilkerson: The homotopic uniqueness of $BS^3$, LNM 1298

and

Dwyer, Miller, Wilkerson: Homotopical uniqueness of classifying spaces. Topology 31 (1992), no. 1, 29–45.