In [this recent question][1], I learned that any two separable Banach spaces are homeomorphic. Based on some readings, I'm guessing that $L^2(\mathbb R)$ is homeomorphic to $\prod_{n=1}^{\infty} (0,1)$ (infinite product of the open interval, with the product topology).

<b>Warm-up question:</b> Is it true that $L^2(\mathbb R)$ is homeomorphic to $\prod_{n=1}^{\infty} (0,1)$?

(The space $\prod_{n=1}^{\infty} (0,1)$ is known as the pseudo-interior of the Hilbert cube $I^\infty:=\prod_{n=1}^{\infty} [0,1]$.)

Spaces locally homeomorphic to $I^\infty$ are known as Hilbert cube manifolds; spaces locally homeomorphic to $L^2(\mathbb R)$ are known as $\ell_2$-manifolds.

I seem to recall that the classification of compact Hilbert cube manifolds is very interesting: compact Hilbert cube manifolds are in bijective correspondence with compact CW-complexes, up to simple homotopy equivalence.
But I'm not completely sure of the above statement, so I'll repeat it in my question below:

<b>Questions:</b><br>
• What is the classification of compact Hilbert cube manifolds?<br>
• What is the classification of non-compact Hilbert cube manifolds?<br>
• What is the classification of $\ell_2$-manifolds?<br>

PS: Let's assume that all spaces are separable (or whatever implies Polish), to avoid pathologies.

  [1]: https://mathoverflow.net/questions/269294/is-l2-mathbb-r-homeomorphic-to-l1-mathbb-r/269297#269297