It is well-known (due to [this](https://www.sciencedirect.com/science/article/pii/0040938366900024) work of Palais, I believe) that Banach manifolds are dominated by countable CW complexes. It then follows (due Whitehead, as indicated by Milnor in [this](https://www.ams.org/journals/tran/1959-090-02/S0002-9947-1959-0100267-4/) work) that they have the homotopy type of CW complexes. 


1. Are there conditions ensuring that a Banach or Fréchet manifold admits a genuine CW complex structure?
2. What are the main examples of that?


Actually, I'm working with Banach fiber bundles, which are Serre fibrations. I would like to regard them as Hurewicz fibrations to use HLP (homotopy lifting property). Due to [this](https://mathoverflow.net/q/241611) answer of Peter May (and the references cited by him), I know that it is enough to get a CW structure.


Any help is welcome.



CW structure on infinite-dimensional manifolds
Let $M$ be a Hilbert manifold. Then there exists a Riemannian metric $g$ on $M$ such that the induced metric $d$ is complete (See <https://arxiv.org/pdf/1610.01527.pdf> by Biliotti and Mercuri). Thus $M$ has the topology of a complete metric space. By the Baire category theorem $(M,d)$ is a Baire space: It cannot be written as a countable union of closed sets with empty interior. 


If $M$ had the structure of a CW complex, it does. It must clearly be an infinite dimensional $CW$-complex, but such a thing is the countable union of its $n$-skeleta, which have empty interior.


With this argument you can treat also other infinite dimensional manifolds: using the fact that seperable Frechet spaces are homeomorphic to seperable Hilbert space, one can show that a Frechet manifold is homeomorphic to a Hilbert manifold. 


