Hilbert manifold

Properties

Many basic constructions of manifold theory, such as the tangent space of a manifold and a tubular neighbourhood of a submanifold (of finite codimension) carry over from the finite dimensional situation to the Hilbert setting with little change. However, in statements involving maps between manifolds, one often has to restrict consideration to Fredholm maps, that is, maps whose differential at every point is Fredholm. The reason for this is that Sard's lemma holds for Fredholm maps, but not in general. Notwithstanding this difference, Hilbert manifolds have several very nice properties.

• Kuiper's theorem: If ${\displaystyle X}$ is a compact topological space or has the homotopy type of a CW complex then every (real or complex) Hilbert space bundle over ${\displaystyle X}$ is trivial. In particular, every Hilbert manifold is parallelizable.
• Every smooth Hilbert manifold can be smoothly embedded onto an open subset of the model Hilbert space.
• 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.
• Although Sard's Theorem does not hold in general, every continuous map ${\displaystyle f:X\to {\mathbb{R}}^n}$ from a Hilbert manifold can be arbitrary closely approximated by a smooth map ${\displaystyle g:X\to {\mathbb{R}}^n}$ which has no critical points.