I'm looking for the total space. I know that $F_{oo}\left(S^{n}\right)=SO(n+1)$. I'm guessing that it's not as simple as:
$$F_{oo}\left(S^{2}\times S^{1}\right)=F_{oo}\left(S^{2}\right)\times F_{oo}\left(S^{1}\right)=SO(3)\times SO(2)$$
I would definitely expect it to have more than 4 dimensions. It's a closed space, but not simply connected. From another perspective:
$$S^{2}\times S^{1}=\frac{SU(2)}{U(1)}\times U(1)=\frac{S^{3}}{U(1)}\times U(1)$$
So maybe it's something very similar to $F_{oo}(S^{3})=SO(4)$? Because the products are simple, low dimensional manifolds I thought this would be easy, but honestly I'm not sure where to go from here lol!