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*-connected 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.

**EDIT** Idrissi kindly warns me that, in the presence of a non-simply connected $SO(n)$, this is the *wrong* notion of a topological model: I should have said *weak homotopy equivalences* instead of quasi-isomorphisms. (See the comments.)

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.

includingthe "deleted" framed little disks operad into the framed little disks operad, not projecting through the homotopy equivalence? -Silly me: it sounds great. Thank you. I'll check it. $\endgroup$ – Agustí Roig Jun 21 '19 at 19:34