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"
EDIT: It seems that one can try to make analogy of polynomially defined sets as follows. Consider a sequence space X ( any one likes - $c_0,l_p$, James, Tsirelson,...). Consider "polynomial equations" \begin{equation} 0= p_i(x) = \sum_{j_1,...,j_n} p_{i;j_1,...,j_n} x_{j_1}...x_{j_n} \end{equation} for the case when the derivative $\frac{\partial p_i}{\partial x_j}$ has infinite dimensional kernel generically. Then this defines a "algebraic Banach manifold". ( There are various subtleties involved with definition of $p_i$ as these involve tensor products). One may try to study the question on the topological type of the resulting set, for generic $p_i$. Is this done somewhere? Does it lead to something interesting?
To make analogy with algebraic geometry, it may be useful to try to define $\mathbb{P}(X)$ - corresponding projective space. To do so, one can consider limits $x_i=\xi_i/\xi_0$ $\xi_0 \rightarrow 0$. I.e. one adds codimension 1 subspaces of $X$. It seems there is no problem with this construction, is there?
