In infinite dimensional Banach spaces, many analogies of classical sets are topologically trivial ( even contractible). E.g., infinite dimensional spheres are contractible by  Y. Benyamini, Y. Sternfeld "Spheres in Infinite-Dimensional Normed Spaces are Lipschitz Contractible"

It is not difficult to construct infinite dimensional sets that are topologically non trivial. E.g., infinite dimensional torus $T^\infty = \prod^\infty S^1$ has non trivial fundamental group.

It is possible to construct embeddings of such sets into Banach spaces. E.g., one can embed the above torus into a complex $l_p(\mathbb{C})$ space. But such embeddings are usually incomplete in Banach norm. After completion, they become contractible, as is the case with the above torus.

Question: is there a systematic way to construct topologically non trivial complete sets in Banach spaces?

Q2: the same for sets that are smooth Banach manifolds?

Q3: the same two questions in the case the ambient Banach space is one of the classical sequence spaces $l_p$ or the classical function spaces $C(R^n), L_p(R^n, W^{p,q}(R^n)$.


Note: I do not know if the analogies of general linear groups are topologically trivial. Is it true that the space of linear operators between two spaces X,Y (with some norm restriction , to make it analogous to SO(n)) is topologically trivial? I think it is interesting for both classical sequence spaces and for function spaces.

For the last remark, I believe relevant discussion is provided in 

Daniel Freeman, Thomas Schlumprecht, Andras Zsak  "Banach spaces for which the space of operators has 2𝔠 closed ideals"

Spiros A Argyros, Richard G Haydon "A hereditarily indecomposable L \infty-space that solves the scalar-plus-compact problem"