DOCSLIB.ORG

A Hilbert Manifold Model for Mapping Spaces

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
A Hilbert Manifold Model for Mapping Spaces A Hilbert Manifold Model for Mapping Spaces Lennart Meier January 26, 2010 Let M be an n-dimensional compact manifold and N be an arbitrary mani- fold. The mapping space Map(M; N) can be equipped with the compact-open topology. This can be seen as an innite-dimensional manifold modelled on Banach spaces. For many geometric purposes, such Banach manifolds are in- convenient since several of the theorems one knows from the nite-dimensional cases fail in this context. Hilbert manifolds, i.e. manifolds modelled on separa- ble Hilbert spaces, behave much better (see e.g. [Lan]). Transversality theorems for Hilbert manifolds are used in work by Chataur and also by the author to provide geometric descriptions of string topology. Since for algebraic topology applications nally only the homotopy type of Map(M; N) matters, we are interested in getting a Hilbert manifold homotopy equivalent to Map(M; N). To that end, we use the theory of Sobolev spaces to construct a Hilbert manifold Hn(M; N) which serves our purposes. It should be noted that our main results are known for a long time. It has been stated which a sketched proof, for example, in [E-M]. Furthermore, a great part of our section 2 was proved in greater generality in [Pal2]. In spite of that, the author has the opinion that it is good to have a complete and concrete proof of this fundamental construction written down. Our constructions and argumentation are modelled on the situation of the free loop space as discussed in [Kli], 1.2. There is no claim for originality. 1 A Hilbert Manifold Model We dene Hn(M; N) ⊂ Map(M; N) as the space of all continuous maps f : M ! N such that there is for every p 2 M a chart (U; φ) around p and a chart (V; ) around f(p) such that fφ−1 : φ(U) ! (V ) is of Sobolev class Hn1. Note that every L2-map φ(U) ! (V ) of class Hn has a unique continuous representative (see [Alt], 8.13). To construct an atlas for Hn(M; N), we choose Riemannian metrics on M and N. As a preliminary notion, we dene the following: Denition 1.1. Dene for the energy R 2 f 2 Map(M; N) E(f) := M jjTpfjj and the length R of . Here . L(f) := M jjTpfjj f jjTpfjj = maxv2TpM;jvj=1 jTpf(v)j 1Some authors denote this also by 2 or 2;n. Ln W 1 Let g : O ! M be a map from a Riemannian manifold O. Then jjTq(fg)jj ≤ jjTg(q)fjj · jjTqgjj for q 2 O. Therefore, by Cauchy-Schwarz we have L(fg) ≤ pE(f) · E(g). Consider some f 2 C1(M; N). Since the image of f is compact, there exists an " > 0 such that exp ×π : TN ! N × N induces a dieomorphism from the "-neighbourhood O" ⊂ TN of the zero ∗ section onto an open neighbourhood of the diagonal in N ×N. Let f∗ : f TN ! −1 TN be the pullback and Of = (f∗) (O"). We dene n n expf : H (Of ) ! H (M; N); ξ 7! (p 7! exp(f∗ξ(p))) n ∗ Here H (Of ) denotes the following: all continuous sections ξ of π : f TN ! M with image in Of such that there is a chart (U; φ) around every point of M and a local trivialization −1 k such that −1 k is of π (U) ! U × R pr2 ξφ : φ(U) ! R n n n Sobolev class H . We see in the next section that H (Of ) ⊂ H (E) is an open subset. Lemma 1.2. The map is injective and its image is the open set expf n Uf = fg 2 H (M; N): g(p) 2 exp(O" \ Tf(p)N)g Proof. The injectivity and that are clear. Now we want to show im expf ⊂ Uf that is of Sobolev class n if is. If is of Sobolev class n, then ξ H expf (ξ) expf (ξ) H also is Sobolev class n. The map id × expf (ξ): M ! M × N H id × expf ◦f∗ denes a dieomorphism from Of onto an open set in M × N. Reversing this dieomorphism we get ξ which is therefore also Hn. 0 To show that Uf is open, it is enough to show Uf = fg 2 C (M; N): g(p) 2 0 exp(O" \ Tf(p)N)g is open in C (M; N). But Uf is just the "-ball around f in the maximum metric. Theorem 1.3. The space Hn(M; N) is a (smooth) Hilbert manifold. n Proof. Note that H (Of ) is an open subset of the separable Hilbert space n ∗ . Therefore, −1 n is a chart for every 1 . H (f TN) expf : Uf ! H (Of ) f 2 C (M; N) n Now we want to show that for every g 2 H (Of ) there is a Uf with g 2 Uf . Approximate g by smooth functions fk in the maximum metric. This can be done locally and then be globalized by a partition of unity. Then choose some δ > 0 such that the closed δ-neighbourhood of im(g) is compact. Let " be the minimal injectivity radius on the closed δ-neighbourhood of im(g). Choose k with . Then . The collection of all −1 d1(fk; g) < min(δ; ") g 2 Ufk (Uf ; expf ) denes now an atlas A on Hn(M; N). We will deal with the smoothness of A in the next section. 2 We still need to show that Hn(M; N) has a countable base of topology. To that end, it suces to show that our atlas A has a countable subatlas. So let k; c 2 N and be natural numbers. Triangulate M with a nite trian- gulation ner than 1p . Let be the vertices of this triangulation and 3k c e1; : : : es denote by a countable set of points in such that the 1 -balls around Pk pi N 3k the pi cover N. Choose for every sequence S = (p1; : : : ps) in Pk of cardinality 1 s a function fk;c;S 2 C (M; N) of energy less than c with fk;c;S(ej) = pj for every pj 2 S. These are countably many functions. Now let g 2 Hn(M; N) be a function of energy less than c. Choose some compact δ-neighbourhood K of im(g). Choose furthermore some k 2 N such that the injectivity radius on K is bounded below by 1=k and 1=k < δ. Trian- gulate M as above. Then we have for p and q contained in the same simplex: d(g(p); g(q)) ≤ L(gγ) ≤ pE(g)pE(γ) = pE(g)L(γ) = pE(g)d(p; q) p 1 1 < c p = 3k c 3k Here γ : [0; 1] ! M denotes a minimal geodesic connecting p and q. There- fore, every simplex is contained in a 1 -ball. Choose as above such 3k pj 2 N that g(ej) 2 B1=3k(pj). Let S = (p1; : : : ps) be the sequence dened by these points. Then we claim that jjfk;c;S − gjj1 < 1=k where jj • jj1 denotes the supremum norm. Clearly we have 1 . Let be a simplex d(fk;c;S(ej); g(ej)) < 3k ∆ containing . Then we have by the inequality above 1 ej d(fk;c;S(x); f(ej)) < 3k and 1 for every by the inequality above. Then the d(g(x); g(ej)) < 3k x 2 ∆ claim follows from the triangle inequality. We see that . g 2 Ufk;c;S 2 The Smoothness of the Chosen Atlas Before we come to the smoothness, we have to study Sobolev spaces of sections of a vector bundle. A good source for this topic is the section III.2 of [L-M]. Although they consider only complex vector bundles, all results carry over to the real case. So let π : E ! M be a vector bundle equipped with a connection r and a euclidean metric j·j over a compact base manifold of dimension n. Let the space Ck(E) be equipped with the norm k 2 X i 2 jjξjjk = jjr ξjj0: i=0 2 Here the euclidean metric is extended to all forms and jj·jj0 denotes the L -norm. We call the completion of k with respect to this norm 2 . According C (E) Lk(E) to [L-M], we get an equivalent norm for other choices of metric and connection. Proposition 2.1 ([L-M], III.2.14+2.15). For each integer k > n=2, there is a continuous inclusion 2 0 extending the inclusion of k . In Lk(E) !C (E) C (E) particular, there is a constant K(k) with jjujjk ≤ K(k)jjujj1. 3 We see that we can identify 2 n . Ln(E) = H (E) Lemma 2.2. Let O be an open subset of E. Then Hn(O) ⊂ Hn(E) is open. Proof. Let ξ 2 Hn(O). Since M is compact, there exists an " > 0 such that n for η 2 H (E), jjη − ξjj1 < " implies that η(p) 2 O for all p 2 M. Therefore, n jjη − ξjjn < "=K(n) implies η 2 H (O). Lemma 2.3. Let O be again an open subset of E and Φ: F ! M a second vector bundle. Furthermore, let f : O! F be a smooth bre map, i.e. Φ◦f = π. Then the induced map f~: Hn(O) ! Hn(F ); ξ 7! f ◦ ξ is continuous.
Load more

Recommended publications
  • Infinite-Dimensional Manifolds Are Open Subsets of Hilbert Space
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector INFINITE-DIMENSIONAL MANIFOLDS ARE OPEN SUBSETS OF HILBERT SPACE DAVID W. HENDERSON+ (Receketl z I Jflmrr,v 1969) IN THIS paper We prove, using Hilbert space microbundles, the THEOREM. If M is n separable metric manifold modeled on the separable injnite- dimensional Hilbert space, H, then M can be embedded as an open subset of H. Each infinite-dimensional separable Frechet space (and therefore each infinite- dimensional separable Banach space) is homeomorphic to H. (See [I].) We shall use “F-manifold ” to denote “ metric manifold modeled on a separable infinite-dimensional Frenchet space “. Thus we have COROLLARY 1. Each separable F-matzifold ccl/zbe embedded as an open subset of H. Recent results of Eells and Elworthy [6] and Kuiper and Burghelea [lo] and Moulis [14] combine to show (see [5]) that every homotopy equivalence between C”-Hilbert manifolds is homotopic to a C” diffeomorphism. Since open subsets of H have an induced C” structure, we have COROLLARY 2. Each F-manifokd has u unique C” Hilbert manifold structure. an d COROLLARY 3. Ecery homotopy eyuicalence betbreen F-manifolds is homotopic to a homeomorphism. Results about open subsets of H in [7] apply to give us COROLLARY 4. For each F-manifold Xl there is a coutltable locally-finite simplicial complex K, such that 1cI is homeomorphic to \l<j x H. an d COROLLARY 5. Each F-manifold is homeomorphic to an opett set U c H, such that H - U is homeomorphic to H and bd(U) is homeomorphic to U and to cl(U).
    [Show full text]
  • Appendix an Overview of Morse Theory
    Appendix An Overview of Morse Theory Morse theory is a beautiful subject that sits between differential geometry, topol- ogy and calculus of variations. It was first developed by Morse [Mor25] in the middle of the 1920s and further extended, among many others, by Bott, Milnor, Palais, Smale, Gromoll and Meyer. The general philosophy of the theory is that the topology of a smooth manifold is intimately related to the number and “type” of critical points that a smooth function defined on it can have. In this brief ap- pendix we would like to give an overview of the topic, from the classical point of view of Morse, but with the more recent extensions that allow the theory to deal with so-called degenerate functions on infinite-dimensional manifolds. A compre- hensive treatment of the subject can be found in the first chapter of the book of Chang [Cha93]. There is also another, more recent, approach to the theory that we are not going to touch on in this brief note. It is based on the so-called Morse complex. This approach was pioneered by Thom [Tho49] and, later, by Smale [Sma61] in his proof of the generalized Poincar´e conjecture in dimensions greater than 4 (see the beautiful book of Milnor [Mil56] for an account of that stage of the theory). The definition of Morse complex appeared in 1982 in a paper by Witten [Wit82]. See the book of Schwarz [Sch93], the one of Banyaga and Hurtubise [BH04] or the survey of Abbondandolo and Majer [AM06] for a modern treatment.
    [Show full text]
  • Remarks on the Geometry and the Topology of the Loop Spaces Hs(S1, N)
    Remarks on the geometry and the topology of the loop spaces Hs(S1;N); for s ≤ 1/2: Jean-Pierre Magnot To cite this version: Jean-Pierre Magnot. Remarks on the geometry and the topology of the loop spaces Hs(S1;N); for s ≤ 1/2:. International Journal of Maps in Mathematics, 2019, pp.14 - 37. hal-02285964 HAL Id: hal-02285964 https://hal.archives-ouvertes.fr/hal-02285964 Submitted on 13 Sep 2019 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. International Journal of Maps in Mathematics Volume (2), Issue (1), (2019), Pages:(14-37) ISSN: 2636-7467 (Online) www.journalmim.com REMARKS ON THE GEOMETRY AND THE TOPOLOGY OF THE LOOP SPACES Hs(S1;N); FOR s ≤ 1=2: JEAN-PIERRE MAGNOT∗ Abstract. We first show that, for a fixed locally compact manifold N; the space L2(S1;N) has not the homotopy type of the classical loop space C1(S1;N); by two theorems: - the inclusion C1(S1;N) ⊂ L2(S1;N) is null homotopic if N is connected, - the space L2(S1;N) is contractible if N is compact and connected.
    [Show full text]
  • Homotopy Negligible Subsets Compositio Mathematica, Tome 21, No 2 (1969), P
    COMPOSITIO MATHEMATICA JAMES EELLS,JR. NICOLAAS H. KUIPER Homotopy negligible subsets Compositio Mathematica, tome 21, no 2 (1969), p. 155-161 <http://www.numdam.org/item?id=CM_1969__21_2_155_0> © Foundation Compositio Mathematica, 1969, tous droits réservés. L’accès aux archives de la revue « Compositio Mathematica » (http: //http://www.compositio.nl/) implique l’accord avec les conditions géné- rales d’utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d’une infrac- tion pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ COMPOSITIO MATHEMATICA, Vol. 21, Fasc. 2, 1969, pag. 155-161 Wolters-Noordhoff Publishing Printed in the Netherlands Homotopy negligible subsets by James Eells Jr. 1 and Nicolaas H. Kuiper 1. Introduction to homotopy negligibility We shall say that a subset A of a topological space X is homo- topy negligible if the inclusion map X-A - X is a homotopy equivalence. The main result in this note is that under certain conditions this global property of A in X can be deduced from an analogous local property. In § 2, 3 examples are given where X is an infinite dimensional manifold. For a Palais-stable separable Hilbert manifold X the homotopy negligibility of A in X implies that X and X -A are even diffeomorphic, by recent results of Kuiper, Burghelea [5] and Nicole Moulis [6]. These results are based on Bessaga [2] and Kuiper [4]. THEOREM 1. Let X be an absolute neighborhood retract and A a closed subset.
    [Show full text]
  • Topological Complexity of Manifolds of Preferences
    MPRA Munich Personal RePEc Archive Topological complexity of manifolds of preferences Graciela Chichilnisky 1986 Online at http://mpra.ub.uni-muenchen.de/8119/ MPRA Paper No. 8119, posted 7. April 2008 01:37 UTC REPRINTED FROM : CONTRIBUTIONS TO MATHEMATICAL ECONOMICS In Honor of Gerard Debreu Edited by WERNER HILDENBRAND, University of Bonn ANDREU MAS-COLELL, Harvard University 1986 NORTH-HOLLAND Amsterdam - New York - Oxford - Tokyo Chapter 8 TOPOLOGICAL COMPLEXITY OF MANIFOLDS OF PREFERENCES* GRACIELA CHICHILNISKY Columbia University, New York, NY, USA 1. Introduction The problem of endowing preferences with manifold structures emerged from discussions with Gerard Debreu in 1975 . Time has shown that such structures can be useful in understanding the behavior of economic systems . In Chichilnisky (1976) spaces of smooth preferences were endowed with a Hilbert manifold structure, and this was used to study the existence and structural stability of competitive equilibria in economies where preferences might be non-monotonic and non-convex. This paper constructs manifolds of preferences and applies this construction to the aggregation of preferences. We examine the topological complexity of manifolds of smooth preferences and use this to determine when appropriate aggregation rules exist and when they do not. In mathematical terms, a smooth preference is an oriented foliation of the choice space. General spaces of foliations are still poorly understood. Our approach is to consider subspaces of foliations which are of special interest in economic theory and which can be endowed with manifold structures. These spaces are large enough to be infinite-dimensional, and to include non-convex and non-monotonic preferences. Spaces of convex and monotonic preferences are shown to be submanifolds.
    [Show full text]
  • Lecture Notes in Mathematics
    Lecture Notes in Mathematics Edited by A. Dold and B. Eckmann Subseries: Nankai Institute of Mathematics, (Tianjin, P. R. China) vol. 7 Advisers: S.S. Chern, B.-j. Jiang 1353 Richard S. Palais Chuu-lian Terng Critical Point Theory and Submanifold Geometry Springer-Verlag Berlin Heidelberg New York London Paris Tokyo Copyright c 1988 by Richard S. Palais and Chuu-lian Terng. All rights reserved. To Shiing-shen Chern Scholar, Teacher, Friend Preface This book is divided into two parts. Part I is a modern introduction to the very classical theory of submanifold geometry. We go beyond the classical theory in at least one important respect; we study submanifolds of Hilbert space as well as of Euclidean spaces. Part II is devoted to critical point theory, and here again the theory is developed in the setting of Hilbert manifolds. The two parts are inter-related through the Morse Index Theorem, that is, the fact that the structure of the set of critical points of the distance function from a point to a submanifold can be described completely in terms of the local geometric invariants of the submanifold. Now it is perfectly standard and natural to study critical point theory in infinite dimensions; one of the major applications of critical point theory is to the Calculus of Variations, where an infinite dimensional setting is essential. But what is the rationale for extending the classical theory of submanifolds to Hilbert space? The elementary theory of Riemannian Hilbert manifolds was developed in the 1960’s, including for example the existence of Levi-Civita connections, geodesic coordinates, and some local theory of submanifolds.
    [Show full text]
  • Arxiv:Math/0403552V1 [Math.DS] 31 Mar 2004 Nt Os Index Morse finite Sdnt Ycrit( by Denote Us Let Introduction Iesoa Eaal Elhletspace
    When the Morse index is infinite Alberto Abbondandolo and Pietro Majer March 31, 2004 Abstract Let f be a smooth Morse function on an infinite dimensional separable Hilbert manifold, all of whose critical points have infinite Morse index and co-index. For any critical point x choose an integer a(x) arbitrarily. Then there exists a Riemannian structure on M such that the corresponding gradient flow of f has the following property: for any pair of critical points x,y, the unstable manifold of x and the stable manifold of y have a transverse intersection of dimension a(x) − a(y). Introduction Let f be a smooth Morse function on an infinite dimensional separable Hilbert manifold1 M. Let us denote by crit(f) the set of its critical points, and let us assume that each x ∈ crit(f) has finite Morse index i(x). A Riemannian structure g on M determines the vector field −grad f, whose local flow φt : M → M has the critical points of f as rest points. The Morse condition is translated into the fact that these rest points are hyperbolic. Their unstable and stable manifolds, u u W (x; f,g)= W (x)= p ∈ M | lim φt(p)= x , t→−∞ s s W (x; f,g)= W (x)= p ∈ M | lim φt(p)= x , t→+∞ are submanifolds with dim W u(x) = codim W s(x) = i(x). Moreover for a generic choice of the Riemannian structure g, W u(x) and W s(y) intersect transversally for every pair of critical points x, y.
    [Show full text]
  • A BORDISM APPROACH to STRING TOPOLOGY 1. Introduction The
    A BORDISM APPROACH TO STRING TOPOLOGY DAVID CHATAUR LABORATOIRE PAINLEVE´ UNIVERSITE´ DE LILLE1 FRANCE Abstract. Using intersection theory in the context of Hilbert mani- folds and geometric homology we show how to recover the main oper- ations of string topology built by M. Chas and D. Sullivan. We also study and build an action of the homology of reduced Sullivan’s chord diagrams on the singular homology of free loop spaces, extending pre- vious results of R. Cohen and V. Godin and unifying part of the rich algebraic structure of string topology as an algebra over the partial Prop of these reduced chord diagrams. 1. Introduction The study of spaces of maps is an important and difficult task of algebraic topology. Our aim is to study the algebraic structure of the homology of free loop spaces. The discovery by M. Chas and D. Sullivan of a Batalin Vilkovisky structure on the singular homology of free loop spaces [4] had a deep impact on the subject and has revealed a part of a very rich algebraic structure [5], [7]. The BV -structure consists of: - A loop product − • −, which is commutative and associative; it can be understood as an attempt to perform intersection theory of families of closed curves, - A loop bracket {−, −}, which comes from a symmetrization of the homo- topy controling the graded commutativity of the loop product. - An operator ∆ coming from the action of S1 on the free loop space (S1 acts by reparametrization of the loops). M. Chas and D. Sullivan use (in [4]) ”classical intersection theory of chains in a manifold”.
    [Show full text]
  • Complex Cobordism of Hilbert Manifolds with Some Applications to flag Varieties of Loop Groups
    Contemporary Mathematics Complex cobordism of Hilbert manifolds with some applications to flag varieties of loop groups A. Baker & C. Ozel¨ Abstract. We develop a version of Quillen’s geometric cobordism theory for infinite dimensional separable Hilbert manifolds. This cobordism theory has a graded group structure under the topological union operation and has push- forward maps for Fredholm maps. We discuss transverse approximations and products, and the contravariant property of this cobordism theory. We define Euler classes for finite dimensional complex vector bundles and describe some applications to the complex cobordism of flag varieties of loop groups. Introduction In [17], Quillen gave a geometric interpretation of complex cobordism groups which suggests a way of defining the cobordism of separable Hilbert manifolds. In order that such an extension be reasonable, it ought to reduce to his construction for finite dimensional manifolds and also be capable of supporting calculations for important types of infinite dimensional manifolds such as homogeneous spaces of free loop groups of finite dimensional Lie groups and Grassmannians. In this paper, we outline an extension of Quillen’s work to separable Hilbert manifolds and discuss its main properties. Although we are able to verify some expected features, there appears to be a serious gap in the literature on infinite dimensional transversality and without appropriate transverse approximations of Fredholm and smooth maps we are unable to obtain contravariance or product structure. However, covariance along Fredholm maps does hold as does contravari- ance along submersions. If the relevant infinite dimensional transversality results are indeed true then our version of Quillen’s theory may be of wider interest.
    [Show full text]
  • Open Subsets of Hilbert Space Compositio Mathematica, Tome 21, No 3 (1969), P
    COMPOSITIO MATHEMATICA DAVID W. HENDERSON Open subsets of Hilbert space Compositio Mathematica, tome 21, no 3 (1969), p. 312-318 <http://www.numdam.org/item?id=CM_1969__21_3_312_0> © Foundation Compositio Mathematica, 1969, tous droits réservés. L’accès aux archives de la revue « Compositio Mathematica » (http: //http://www.compositio.nl/) implique l’accord avec les conditions géné- rales d’utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d’une infrac- tion pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ COMPOSITIO MATHEMATICA, Vol. 21, Fasc. 3, 1969, pag. 312-318 Wolters-Noordhoff Publishing Printed in the Netherlands Open subsets of Hilbert space by David W. Henderson 1 In this paper we prove the following THEOREM. Il M is an open subset of the separable Hilbert space. l2’ which we shall call H, then M is homeomorphic to an open set N C H where ("~" denotes "is homeomorphic to" and R ~ Reals) a countable locally-finte simplicial complex (clfsc), and (c) there is an embedding h: bd(N) X R ~ H such that (i) h (bd (N ) X R ) is open in H, (ii) h 1 bd (N) X {0} is the inclusion o f bd(N) into H, and (iii) h(bd(N) X ( - ~, 0)) = N. Recent results of Eells and Elworthy [7] show that each sepa- rable C°° Hilbert manifold is C°°-diffeomorphic to an open subset of H, and, since Hilbert C~-structures are unique [7], we may assert, in this case, that M is C°°-diffeomorphic to N.
    [Show full text]
  • Topological Spaces and Manifolds
    Appendix A Topological Spaces and Manifolds A.1 Topology and Metric A subset of R is called open if it is the union of open intervals (a, b) ⊆ R.This definition makes R into a topological space: A.1 Definition 1. A topological space is a pair (M, O), consisting of a set M and a set O of subsets of M (these subsets will be called open sets) such that the following properties are satisfied: 1. ∅ and M are open, 2. arbitrary unions of open sets are open, 3. the intersection of any two open sets is open. 2. O is called a topology on M. 3. If O1 and O2 are topologies on M and O1 ⊆ O2, then we call O1 coarser or weaker than O2, and O2 finer or stronger than O1. A.2 Example 1. The discrete topology 2M (the power set) is the finest topology onasetM, and the trivial topology {M, ∅} is the coarsest topology on M. Topological spaces (M, 2M ) are called discrete. 2. If N ⊆ M is a subset of the topological space (M, O), then {U ∩ N | U ∈ O}⊆2N (A.1.1) defines a topology on N, called the subspace topology, relative topology, induced topology,ortrace topology. For example, for the subset N := [0, ∞) of R,all subintervals of the form [0, b) ⊂ N with b ∈ N are open in N, but not open in R. 3. If (M, OM ) and (N, ON ) are disjoint topological spaces (i.e., M ∩ N =∅), then M∪N the union M ∪ N with {U ∪ V | U ∈ OM , V ∈ ON }⊆2 is a topological space, called the sum space.
    [Show full text]
  • On the Classification of the Coadjoint Orbits of the Sobolev Bott–Virasoro
    On the classification of the coadjoint orbits of the Sobolev Bott–Virasoro group Fran¸coisGay-Balmaz1 AMS Classification: 81R10, 22E65, 58D05, 17B68 Keywords: Bott-Virasoro group, Virasoro algebra, coadjoint orbit, Hilbert manifold, diffeomorphism group. Abstract The purpose of this paper is to give the classification of the Bott-Virasoro coadjoint orbits, with nonzero central charge, in the functional analytic setting of smooth Hilbert manifolds. The central object of the paper is thus the completion of the Bott-Virasoro group with respect to a Sobolev topology, giving rise to a smooth Hilbert manifold and topological group, called the Sobolev Bott–Virasoro group. As a consequence of this approach, analytic and geometric properties of the coadjoint orbits are studied. 1 Introduction The classification of the coadjoint orbits of the Virasoro algebra in the smooth category can be found in different formulations in Lazutkin and Pankratova [1975], Segal [1981], Kirillov [1982], Witten [1988] and others. Balog, Feh´erand Palla [1998] reformulate and complete these results; see also Dai [2000] and Dai and Pickrell [2003] for another approach. In this paper the classification problem of the coadjoint orbits of the Vira- soro algebra is studied in the setting of smooth infinite dimensional Hilbert manifolds. The techniques are extensions of those in Balog, Feh´erand Palla [1998] to the Sobolev manifold category. The classical approach to the Bott–Virasoro group consists in seeing it as a one- C∞ 1 dimensional central extension of the group D+ (S ) of smooth orientation preserving diffeomorphisms of the circle. This group can be endowed with the structure of a smooth infinite dimensional Fr´echet manifold, making it into a regular Lie group.
    [Show full text]