Hello,
The so-called "recognition principle" of Boardman-Vogt and May leaves me unsatisfied.
My problem is the following:
The "recognition principle" says that every "group-like" algebra over the little $k$-disks operad is equivalent (as an algebra over the little $k$-disks operad) to a $k$-fold loop space. However, if I start with a space homotopy equivalent to a $k$-fold loop space, then it is not a priori equipped with an action of the little disks. So:
1) First, can someone give me an explicit example of a space homotopy equivalent, say, to a double loop space and such that it admits no (compatible) actions of the little $2$-disks operad?
2) Can you characterize among the spaces homotopy equivalent to double loop spaces those which are algebras over the little disks operad?
3)I heard that the problem was related to the fact that the little disks operads are not cofibrant (in the homotopy category of operads), and that the cofibrant replacements would be the so-called Fulton-McPherson operads. These are "compactifications" of configuration spaces of points in $\mathbf{R}^k$ (modulo the action of the affine group maybe) defined using a variant of a construction due to Fulton and McPherson.
Is it obvious from the definition that this operad indeed acts on iterated loop spaces?
Many thanks,
K.
Best Answer
I'll give an answer from the old days. The $W$ construction of Boardman and Vogt corresponds to the modern notion of cofibrant replacement of operads. If an operad $\mathcal{C}$ acts on $X$ and $Y$ is homotopy equivalent to $X$, then $W\mathcal{C}$ acts on $Y$. By the way, there is an inherent flaw in the little discs operads $\mathcal{D}_n$, namely there is no map of operads $\mathcal{D}_n\longrightarrow \mathcal{D}_{n+1}$ that is compatible with suspension (in the obvious sense: consider $\Omega^n \longrightarrow \Omega^{n+1}\Sigma$). The little $n$-cubes operads do not have this problem, but have others not shared by the $\mathcal{D}_n$. The Steiner operads have all the good properties of both the $\mathcal{C}_n$ and the $\mathcal{D}_n$. In practice, that is in actual applications, such geometric differences are far more important than the questions of cofibrancy and homotopy invariance.