The *framed* little disks operad $ f{\cal D}_n$ can be described as the semidirect product $ {\cal D}_n \rtimes SO(n)$ , where $ {\cal D}_n$ is the little disks operad and $ SO(n)$ is the special orthogonal group, or rotation group.

As a topological space, forgetting the operad structure, this simply means that, in each arity $ k$ , we have

$ $ f{\cal D}_n (k) = ({\cal D}_n \rtimes SO(n))(k) = {\cal D}_n (k) \times SO(n)^k \ . $ $

See for instance, Salvatore-Wahl.

So, in arity one, we get

$ $ f{\cal D}_n(1) = {\cal D}_n(1) \times SO(n) \ , $ $

which is easily seen to be homotopically equivalent to $ SO(n)$ , since $ {\cal D}_n(1)$ is contractible:

$ $ f{\cal D}_n(1) \simeq SO(n) \ . $ $

This is pretty exciting, because it’s an example of a *non*-unitary topological operad; that is, $ P(1) \neq *$ , and Fresse had to do a nice effort to adapt his Sullivan rational homotopy theory for topological operads Homotopy of operads to encompass it: see Extended

**Now my question is the following**: is there any other topological model $ P$ of $ f{\cal D}_n$ for which $ P(1)$ *is* actually $ P(1) = SO(n)$ —not just homotopically equivalent?

By “topological model” I mean, any topological operad $ P$ that can be joined with $ f{\cal D}_n$ through a chain of quasi-isomorphisms (topological operad morphisms inducing isomorphisms in homology) like

$ $ P \stackrel{\sim}{\longleftarrow} \cdot \stackrel{\sim}{\longrightarrow} \cdot \dots \stackrel{\sim}{\longleftarrow}\cdot \stackrel{\sim}{\longrightarrow} f{\cal D}_n \ . $ $

And when I say “homology”, I mean homology with coefficients in a zero characteristic field: over the real numbers, for instance, would be fine.

I guess a brute-force approach—just delete $ {\cal D}_n(1)$ from the definition, leave $ SO(n)$ and compose with the homotopy equivalence—can not work because you would destroy the operad structure.