Hilbert manifold

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===Homotopy Theory===
===Homotopy Theory===
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{{beginthm|Theorem|[{{cite|Eells1966}}, 4(A)]}}Every Hilbert manifold (indeed, every manifold modelled on a metrizable locally convex topological vector space) is an absolute neighborhood retract and has therefore the homotopy type of a countable, locally finite simplicial complex.{{endthm}}
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{{beginthm|Theorem|{{cite|Eells1966|4(A)}}}Every Hilbert manifold (indeed, every manifold modelled on a metrizable locally convex topological vector space) is an absolute neighborhood retract and has therefore the homotopy type of a countable, locally finite simplicial complex.{{endthm}}
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===Specialties of Infinite Dimension===
===Specialties of Infinite Dimension===
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Revision as of 10:01, 1 July 2013

An earlier version of this page was published in the Definitions section of the Bulletin of the Manifold Atlas: screen, print.

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Contents

1 Introduction and Definition

Even if one is only interested in finite-dimensional manifolds, the need for infinite-dimensional manifolds sometimes arises. For example, one approach to study closed geodesics on a manifold is to use Morse theory on its (free) loop space; while for some purposes it is enough to work with finite-dimensional approximations, it is necessary for some finer aspects of the theory to use models of the free loop space which are infinite-dimensional manifolds. While there is up to isomorphism only one vector space of every finite dimension, there are many different kinds of infinite-dimensional topological vector spaces one can choose. Modeling spaces on Fréchet spaces gives the notion of Fréchet manifolds, modelling on Banach spaces gives Banach manifolds, modelling on the Hilbert cube (the countably infinite product of intervals) gives Hilbert cube manifolds. We will stick to Hilbert manifolds (which are not directly related to Hilbert cube manifolds).

Definition 1.1. Let H be the (up to isomorphism unique) separable Hilbert space of infinite dimension. Then a Hilbert manifold is a separable metrizable space such that every point has a neighborhood that is homeomorphic to an open subset of H.

Some authors have slightly different definitions, leaving out the metrizability or the separability condition. Note that metrizability always implies paracompactness and here also the converse is true. Being metrizable and separable is in this context also equivalent to being second countable and Hausdorff by Uryson's metrization theorem (see also [Eells1966, 4(A)]). A \mathcal{C}^k-structure can be defined as an equivalence class of atlases whose chart transition maps are of class \mathcal{C}^k. The tangent bundle can be defined as usual and is a Hilbert space bundle with structure group GL(H) with the norm topology (see [Lang1995, II.1 and III.2]).

2 Properties

2.1 Basic Differential Topology

Many basic theorems of differential topology carry over from the finite dimensional situation to the Hilbert (and even Banach) setting with little change. For example, every closed submanifold of a smooth Hilbert manifold has a tubular neighborhood (unique up to isotopy) (see [Lang1995, IV.5-6]). Also, every Hilbert manifold can be embedded as a closed submanifold into the standard Hilbert space ([Kuiper&Terpstra-Keppler1970]). However, in statements involving maps between manifolds, one often has to restrict consideration to \textit{Fredholm maps}, i.e. maps whose differential at every point has closed image and finite-dimensional kernel and cokernel. The reason for this is that Sard's lemma holds for Fredholm maps, but not in general (see [Smale1965] and [Bonic1966]). The precise statement is:

Theorem 2.1. Let f: M \to N be a smooth Fredholm map between Hilbert manifolds. Then its set of regular values is the intersection of countably many sets with dense interior.

2.2 Homotopy Theory

{{beginthm|Theorem|[Eells1966, 4(A)]}Every Hilbert manifold (indeed, every manifold modelled on a metrizable locally convex topological vector space) is an absolute neighborhood retract and has therefore the homotopy type of a countable, locally finite simplicial complex.</div>

2.3 Specialties of Infinite Dimension

While proofs are often harder in infinite dimensions, some things are true for Hilbert manifolds that could not be hoped for in finite dimensions.

Theorem 2.2 [[Kuiper1965]]. The unitary group and the general linear group of the separable infinite-dimensional Hilbert space are contractible with the norm topology.
Corollary 2.3.If X is a paracompact space, then every (real or complex) Hilbert space vector bundle with these structure groups over X is trivial. In particular, every (smooth) Hilbert manifold is parallelizable.
Theorem 2.4 [[Henderson1970]]. Every Hilbert manifold X can be embedded onto an open subset of the model Hilbert space.

{{beginthm|Corollary|[Eells&Elworthy1970]}Every homotopy equivalence between two smooth Hilbert manifolds is homotopic to a diffeomorphism. In particular every two homotopy equivalent smooth Hilbert manifolds are already diffeomorphic. </div> Although Sard's Theorem does not hold in general, we have also the following theorem:

{{beginthm|Theorem|[Azagra&Cepedello Boiso2004]}Every continuous map f: X \to \mathbb{R}^n from a Hilbert manifold can be arbitrary closely approximated by a smooth map g: X \to \mathbb{R}^n which has no critical points.</div>

3 Examples

Example 3.1. Any open subset U of a separable Hilbert space H is a Hilbert manifold with a single global chart given by the inclusion into H.
Example 3.2. The unit sphere in a separable Hilbert space is a smooth Hilbert manifold.
Example 3.3. Mapping spaces between manifolds can often be viewed as Hilbert manifolds if one considers only maps of suitable Sobolev class. Set H^r = W^{2,n} to be the Sobolev class of
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-functions which are k-fold weakly differentiable in
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. Let now M be an n-dimensional compact smooth manifold, N be an arbitrary smooth finite-dimensional or Hilbert manifold and Map(M,N) be the space of continuous maps with the compact-open topology. Then the subspace Sob(M,N) \subset Map(M,N) of functions of Sobolev type H^r for r> n/2 can be given the structure of a smooth Hilbert manifold ([Eells1966, 6(D)]). This inclusion is a homotopy equivalence ([Eells1966, 6(E)]). In particular, Sob(M,N) is diffeomorphic to any other Hilbert manifold homotopy equivalent to Map(M,N) and therefore its diffeomorphism type depends only on the homotopy type of M and N. Hilbert manifold models for mapping spaces (in particular, free loop spaces) have been used, for example, in the study of closed geodesics ([Klingenberg1995], [Klingenberg1978]), string topology ([Chataur2005], [Meier2011]) and fluid dynamics ([Ebin&Marsden1970]).

4 References

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