Is there a version of homology theory for spaces for which explicitly infinite dimensional "cells" are allowed?
The spaces in question include e.g. \begin{equation} X = (x: x \in l_2: p_i(x) =0, i\in[1,..,n]) \end{equation} where \begin{gather} p_i(x) = \sum p_{i;\mu_1,...,\mu_d} x_{\mu_1}..x_{\mu_d} \end{gather} are polynomials on the Hilbert space $l_2$.
I believe these sets are not CW complexes ( although they may be homotopy equivalent to CW complexes). I would like to draw a parallel to algebraic geometry by "compactifying" $l_2$ to a Hilbert version of projective space. Although there is no such "compactification", because the space is not locally compact, but I think it is possible to add copies of $l_2$ "at infinity" i.e. by letting some coordinates $x_\mu \rightarrow \infty$. I vaguely remember such construction in the contect of "infinite dimensional Grassmannians" ( names that come to mind are Segal, Borodin, Olshanskii...). After this "compactification", we can consider the above equations in the modified space, defining an analogy of projective manifolds of finite codimension.
In the Banach setting, one may consider spaces \begin{equation} Y=(f: f \in C[0,1], F_i[f]=0 ) \end{equation} where \begin{equation} F_i[f] = \sum_{n=1}^N \int_0^1 ... \int_0^1 F_i(x_1,...,x_n) f(x_1)...f(x_n) dx_1....dx_n \end{equation} $F_i(x_1,...,x_n)$ are e.g $C^\infty([0,1]^n)$. (note that C[0,1] is quite "large" by Banach Mazur; maybe smaller spaces should work better (e.g. the ones from the papers of B. Bossard))
I believe more examples of topologically nontrivial polyhedral spaces can be constructed by quotienting convex polyhedra ( e.g. Paulsen polyhedron).
Question: is there a sensible "Hilbert homology theory" in which there are allowed generalizations of CW complexes with explicitly infinite dimensional cells and where homology groups are Hilbert spaces, and boundary maps are operators between Hilbert spaces?
In the literature, there is $l_2$ homology groups in the context of von Neumann algebras, but this seems to be still finite dimensional context. They show up in the study of finite dimensional manifolds with large fundamental group.
There is also classical paper by Palais on Morse theory for Hilbert manifolds, but it had very little explicit calculations , and there was no definition of homology, in the above sense.
I also realize that it is probably difficult to define such a homology theory because:
there are problems with tensor product. We know it is not unique, and it will be difficult to even define what we mean by the product of manifolds and their tangent spaces. (see papers on Connes embedding problem) And tensor product is necessary ingredient for homology theory.
There are problems with existence of projections. While projections are also necessary for homology theory. The projection theory in Hilbert space is also quite complicated. In the von Neumann algebra context involves the theory of factors.
In the Banach situation, existence of indecomposable spaces must be taken into account (Maurey Gowers).
But, still, I would like to know the opinion of the experts.
