Page "Hilbert manifold" Paragraph 4

* Every homotopy equivalence between two Hilbert manifolds is homotopic to a diffeomorphism.

In particular every two homotopy equivalent Hilbert manifolds are already diffeomorphic.

This stands in contrast to lens spaces and exotic spheres, which demonstrate that in the finite-dimensional situation, homotopy equivalence, homeomorphism, and diffeomorphism of manifolds are distinct properties.