DOCSLIB.ORG

A Differential Form Approach to the Genus of Open Riemann Surfaces 3

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
A Differential Form Approach to the Genus of Open Riemann Surfaces 3 A DIFFERENTIAL FORM APPROACH TO THE GENUS OF OPEN RIEMANN SURFACES FRANCO VARGAS PALLETE, JESUS´ ZAPATA SAMANEZ Abstract. We will show that any open Riemann surface M of finite genus is biholomorphic to an open set of a compact Riemann surface. Moreover, we will introduce a quotient space of forms in M that determines if M has finite genus and also the minimal genus where M can be holomorphically embedded. 1. Introduction Even when de Rham cohomology of an open surface is infinite in dimension 1 (and hence a bit complicated), there is a sufficient and necessary condition in the language of differential 1-forms for the problem of embedding the surface into the Riemann sphere (Koebe’s Gen- eralized Uniformization Theorem). In this article we generalize this sufficient and necessary condition for surfaces with non-zero genus in terms of the dimension of certain quotient space of 1-forms. This is also in general a necessary condition for the problem of embedding a open n-manifold into a compact n-manifold. The article is organized as follows. In Section 2 we set some classical notation and facts about differential forms, and we also define a quotient space using either closed differential forms that are either exact outside a compact set or with compact support. We observe that these spaces are canonically isomorphic. In Section 3 we study ck, the dimension of our quotient space of k-forms. We show that ck is a topological invariant, is non-decreasing under inclusion and additive under connected sum (except for k = 0, n where n is the dimension of the total space). In Section 4 we study the relationship between c1(X) and the genus of a surface X, namely that c1(X) is twice the genus. We also prove that for any given Riemann surface structure on X there is a holomorphic embedding into some compact Riemann surface of the same genus. We finalize with some comments and applications. arXiv:1903.06131v1 [math.CV] 14 Mar 2019 Acknowledgements The authors would like to thank the Institute for Advanced Study for hosting them while completing this article. 2. Background Given a differentiable manifold M let us define the many space of k-forms that will appear through the article. Denote by Ωk(M) the space of real-valued skew-symmetric differential k-forms of M, where we define the exterior derivative operator d : Ωk(M) → Ωk+1(M) that Vargas Pallete was partially funded by the Proyecto de Apoyo a la Iniciaci´on Cient´ıfica (PAIN) from the Pontificia Universidad Cat´olica del Per´u(PUCP) and by the Scholarship for the IMCA Project of the Brazil- Per´uCovenant, and is also supported by the Minerva Research Foundation. Zapata Samanez is supported by the Departamento Acad´emico de Ciencias of the Pontificia Universidad Cat´olica del Per´u(PUCP). 1 2 FRANCO VARGAS PALLETE, JESUS´ ZAPATA SAMANEZ satisfies d ◦ d = 0. Hence we can define the spaces of closed and exact k-forms Zk(M) = d d Ker(Ωk(M) → Ωk+1(M)), Bk(M) = Im(Ωk−1(M) → Ωk(M)) where Bk(M) ⊆ Zk(M). k Definition 2.1. Denote by Z (M) the space of k-closed form of M such that are exact outside a compact set. That is k k k−1 Z (M)= {λ ∈ Z (M) | ∃K ⊆ M compact, ν ∈ Ω (M \ K), dν = λ|M\K} k Note that we have the inclusion Bk(M) ⊆Z (M), so we can define as well k k k k Definition 2.2. H (M)=Z (M)/B (M), ck(M) = dimRH (M) k We can obtain a quotient real vector space canonically isomorphic to H (M) by taking k k differentials with compact support. Let us denote then by Ωc (M) ⊂ Ω (M) the space of differential k-forms of M with compact support. We have the following commutative diagram k i k Ωc (M) Ω (M) (1) d d k+1 i k+1 Ωc (M) Ω (M) k where d is again the exterior derivative and the horizontal maps are the inclusions Ωc (M) ⊂ Ωk(M). In case that M is compact these inclusions are identities. We can then define the k k d k+1 closed forms with compact support Zc (M) = Ker(Ωc (M) → Ωc (M)) and the exact forms k k−1 d k with compact support Bc (M) = Im(Ωc (M) → Ωc (M)). If we were going to take the k k quotient Zc (M)/Bc (M) we will rescue the de Rham cohomology with compact support. Instead, we will look at the space of forms with compact support that are exact. k Definition 2.3. Denote by Bc (M) the space of forms with compact support that are the k exterior derivative of a form that does not need to have compact support. That is, Bc (M)= −1 k k k i (B (M)) = B (M) ∩ Ωc (M) Given the commutativity of Diagram 1 and that d◦d ≡ 0, we have the following inclusions k k Bc (M) B (M) (2) k k k Bc (M) Zc (M) Z (M) k k which in particular allows us to define the quotient Zc (M)/Bc (M). Let us see that this quotient is canonically isomorphic to the quotient in Definition 2.2 via the inclusion map k k .(Zc (M) ֒→Z (M A DIFFERENTIAL FORM APPROACH TO THE GENUS OF OPEN RIEMANN SURFACES 3 ∼ k k k k = → (Lemma 2.1. The inclusion map Zc (M) ֒→Z (M) induces an isomorphism Zc (M)/Bc (M k Z (M)/Bk(M). k k Proof. Notice first that the preimage of B (M) under the inclusion map is Bc (M), so the k k k k quotient map Zc (M)/Bc (M) → Z (M)/B (M) is well defined and injective. To show k surjectivity, take any form ω ∈ Z (M). Since ω is exact outside a compact set, by using a k−1 k bump function we can write ω = dα + ω0, where α ∈ Ω (M) and ω0 ∈ Zc (M). But this exactly says that the image of [ω0] is [ω]. k Under this natural isomorphism we will represent the vector spaceH (M) as a quotient of the space of forms more appropriate to our goals. Forms with compact support are convenient for their relation with the restriction and extension maps of forms, that we describe now. Differentiable forms have the natural map restriction(r) for U ⊂ M a open set that makes the following diagram commute Ωk(M)r Ωk(U) (3) d d Ωk+1(M)r Ωk+1(U) Likewise, compactly supported forms have the natural map extension(e) which is injective at all ranks and makes the following diagram commute k e k Ωc (U) Ωc (M) (4) d d k+1 e k+1 Ωc (U) Ωc (M) Given Diagrams 3 and 4 we know that restriction and extension preserve closed and exact forms. We can describe the interaction between restriction and extension in the following com- mutative diagram k e k Ωc (U) Ωc (M) (5) i i Ωk(U)r Ωk(M) Zk(M) R We can quickly compare the dimension ck with the Betti numbers bk = dim Bk(M) . Lemma 2.2. For any differentiable manifold M and non-negative integer k, ck(M) ≤ bk(M). Moreover, if M is compact, then ck(M)= bk(M). 4 FRANCO VARGAS PALLETE, JESUS´ ZAPATA SAMANEZ k Z (M) Zk(M) Proof. The inequality follows from the injective map Bk(M) ֒→ Bk(M) induced by the inclusion k Bk(M) ⊆Z (M) ⊆ Zk(M), where the last inclusion is an equality if M is compact. Forms behave well with respect to pullbacks of smooth functions. Since at times we will be dealing with forms with compact support, we will restrict ourselves to proper function between manifold. Let them f : M → N be a smooth proper function, the pullback of f is the natural map of chains f ∗ that makes the following diagrams commute. ∗ ∗ k f k k f k Ω (N) Ω (M) Ωc (N) Ωc (M) (6) d d d d ∗ ∗ k+1 f k+1 k+1 f k+1 Ω (N) Ω (M) Ωc (N) Ωc (M) k ∗ Z (N) which in particular concludes that f can be defined in cohomology as map from Bk(N) to Zk(M) Bk(M) . Moreover, if f,g : M → N are homotopic, we know that there is a algebraic homotopy between f ∗ and g∗ given by a linear map P : Ωk+1(N) → Ωk(M) that commutes with the inclusion of (1) and satisfies (7) f ∗(ω) − g∗(ω)= P (dω)+ dP (ω) k k k ∗ ∗ Z (N) Z (M) for all ω ∈ Ω (M). In particular the maps f ,g : Bk(N) → Bk(M) are equal. The analogue statement is true if we switch to de Rham cohomology with compact support. For the topological and geometric significance of the ck invariants (particularly c1) we will focus on Riemann surfaces. We will adopt the notation of Σg,n for the surface with genus g and n boundary components. A connected Riemann surface X is said to be planar or schlichtartig if every closed 1- form on X with compact support is exact. Under our notation this is the same as c1(X) vanishing, for which we have the Generalized Uniformization Theorem of Koebe (see for instance [Sim89]). Theorem 2.1 (Generalized Uniformization Theorem). Every planar Riemann surface is biholomorphic (i.e. conformally equivalent) to an open subset of the Riemann sphere C.
Load more

Recommended publications
  • Uniformization of Riemann Surfaces
    CHAPTER 3 Uniformization of Riemann surfaces 3.1 The Dirichlet Problem on Riemann surfaces 128 3.2 Uniformization of simply connected Riemann surfaces 141 3.3 Uniformization of Riemann surfaces and Kleinian groups 148 3.4 Hyperbolic Geometry, Fuchsian Groups and Hurwitz’s Theorem 162 3.5 Moduli of Riemann surfaces 178 127 128 3. UNIFORMIZATION OF RIEMANN SURFACES One of the most important results in the area of Riemann surfaces is the Uni- formization theorem, which classifies all simply connected surfaces up to biholomor- phisms. In this chapter, after a technical section on the Dirichlet problem (solutions of equations involving the Laplacian operator), we prove that theorem. It turns out that there are very few simply connected surfaces: the Riemann sphere, the complex plane and the unit disc. We use this result in 3.2 to give a general formulation of the Uniformization theorem and obtain some consequences, like the classification of all surfaces with abelian fundamental group. We will see that most surfaces have the unit disc as their universal covering space, these surfaces are the object of our study in 3.3 and 3.5; we cover some basic properties of the Riemaniann geometry, §§ automorphisms, Kleinian groups and the problem of moduli. 3.1. The Dirichlet Problem on Riemann surfaces In this section we recall some result from Complex Analysis that some readers might not be familiar with. More precisely, we solve the Dirichlet problem; that is, to find a harmonic function on a domain with given boundary values. This will be used in the next section when we classify all simply connected Riemann surfaces.
    [Show full text]
  • Holomorphic K-Differentials and Holomorphic Approximation on Open Riemann Surfaces
    Western University Scholarship@Western Electronic Thesis and Dissertation Repository 8-19-2010 12:00 AM Holomorphic k-differentials and holomorphic approximation on open Riemann surfaces Nadya Askaripour The University of Western Ontario Supervisor A. Boivin The University of Western Ontario Joint Supervisor T. Foth The University of Western Ontario Graduate Program in Mathematics A thesis submitted in partial fulfillment of the equirr ements for the degree in Doctor of Philosophy © Nadya Askaripour 2010 Follow this and additional works at: https://ir.lib.uwo.ca/etd Part of the Analysis Commons Recommended Citation Askaripour, Nadya, "Holomorphic k-differentials and holomorphic approximation on open Riemann surfaces" (2010). Electronic Thesis and Dissertation Repository. 9. https://ir.lib.uwo.ca/etd/9 This Dissertation/Thesis is brought to you for free and open access by Scholarship@Western. It has been accepted for inclusion in Electronic Thesis and Dissertation Repository by an authorized administrator of Scholarship@Western. For more information, please contact [email protected]. Holomorphic k-differentials and holomorphic approximation on open Riemann surfaces (Spine title: k-differentials and approximation on open Riemann surfaces) (Thesis format: Monograph) by Nadya Askaripour Graduate Program in Mathematics A thesis submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy School of Graduate and Postdoctoral Studies The University of Western Ontario London, Ontario, Canada c Nadya Askaripour 2010 Certificate
    [Show full text]
  • © in This Web Service Cambridge University
    Cambridge University Press 978-0-521-80937-5 - Complex analysis Kunihiko Kodaira Index More information Index 0-chain, 191 boundary, 65, 73, 191, 261 1-chain, 191 boundary point, 261, 285 1-cycle, 316 branch, 178 1-form, 247 1-form, harmonic, 258, 295 canonical divisor, 386–388 1-form, holomorphic, 257, 258 Cauchy–Hadamard formula, 18 1-form, real, 247 Cauchy–Riemann equations, 10, 258 2-form, 248 Cauchy’s criterion, 4, 6 Cauchy’s integral formula, 36, 104 a-point, 119 Cauchy’s Theorem, 101, 103, 194, 396, 399 Abelian differential, 380, 386 Cauchy’s Theorem, homotopy variant, 190 Abelian differential of the first kind, 376, 379, cell, 74, 77 387, 390 cellular decomposition, 80, 400 Abelian differential of the second kind, 380 cellularly decomposable, 80 Abelian differential of the third kind, 380, center of a power series, 16 389, 390, 392 chain group, 0-dimensional, 191 Abelian differential, residue of an, 380 chain group, 1-dimensional, 191 Abelian differential, zero of the nth order, 380 chain, 0-, 191 Abelian integral, 380, 392 chain, 1- , 191, 360 Abel’s Theorem for meromorphic functions, circle of convergence, 18 391 closed differential form, 256, 318 absolute value, 1 closed region, 6 absolutely convergent, 4, 53 closed set, 261 absolutely convergent, uniformly, 43 closure, 261 absolutely convergent integral, 251 cocycle, 1-, 342 accumulation point, 272, 273 cohomologous, 342 analytic continuation, 153, 162 cohomology class, 342 analytic continuation, direct, 154 cohomology class, d-, 341 analytic continuation, result of, 164 cohomology
    [Show full text]
  • An Introduction to Riemann Surfaces
    Cornerstones Series Editors Charles L. Epstein, University of Pennsylvania, Philadelphia, PA, USA Steven G. Krantz, Washington University, St. Louis, MO, USA Advisory Board Anthony W. Knapp, Emeritus, State University of New York at Stony Brook, Stony Brook, NY, USA For further volumes: www.springer.com/series/7161 Terrence Napier Mohan Ramachandran An Introduction to Riemann Surfaces Terrence Napier Mohan Ramachandran Department of Mathematics Department of Mathematics Lehigh University SUNY at Buffalo Bethlehem, PA 18015 Buffalo, NY 14260 USA USA [email protected] [email protected] ISBN 978-0-8176-4692-9 e-ISBN 978-0-8176-4693-6 DOI 10.1007/978-0-8176-4693-6 Springer New York Dordrecht Heidelberg London Library of Congress Control Number: 2011936871 Mathematics Subject Classification (2010): 14H55, 30Fxx, 32-01 © Springer Science+Business Media, LLC 2011 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.birkhauser-science.com) To Raghavan Narasimhan Preface A Riemann surface X is a connected 1-dimensional complex manifold, that is, a connected Hausdorff space that is locally homeomorphic to open subsets of C with complex analytic coordinate transformations (this also makes X a real 2-dimensional smooth manifold).
    [Show full text]
  • Uniform Approximation on Riemann Surfaces
    Western University Scholarship@Western Electronic Thesis and Dissertation Repository 7-22-2016 12:00 AM Uniform Approximation on Riemann Surfaces Fatemeh Sharifi The University of Western Ontario Supervisor Paul. M. Gauthier The University of Western Ontario Joint Supervisor Gordon J Sinnamon The University of Western Ontario Graduate Program in Mathematics A thesis submitted in partial fulfillment of the equirr ements for the degree in Doctor of Philosophy © Fatemeh Sharifi 2016 Follow this and additional works at: https://ir.lib.uwo.ca/etd Part of the Analysis Commons Recommended Citation Sharifi, atemeh,F "Uniform Approximation on Riemann Surfaces" (2016). Electronic Thesis and Dissertation Repository. 3954. https://ir.lib.uwo.ca/etd/3954 This Dissertation/Thesis is brought to you for free and open access by Scholarship@Western. It has been accepted for inclusion in Electronic Thesis and Dissertation Repository by an authorized administrator of Scholarship@Western. For more information, please contact [email protected]. Abstract This thesis consists of three contributions to the theory of complex approximation on Riemann surfaces. It is known that if E is a closed subset of an open Riemann surface R and f is a holomorphic function on a neighbourhood of E; then it is \usually" not possible to approximate f uniformly by functions holomorphic on all of R: In Chapter 2, we show, however, that for every open Riemann surface R and every closed subset E ⊂ R; there is a closed subset F ⊂ E; which approximates E extremely well, and has the following property. Every function holomorphic on F can be approximated tangentially (much better than uniformly) by functions holomorphic on R: In Chapter 3, given a function f : E ! C from a closed subset of a Riemann surface R to the Riemann sphere C; we seek to approximate f in the spherical distance by functions meromorphic on R: As a consequence we generalize a recent extension of Mergelyan's theorem, due to Fragoulopoulou, Nestoridis and Papadoperakis [3.13].
    [Show full text]
  • 1 Period Matrices
    Seminar Harvard University — Spring 1999 Course Notes 1 Period matrices 1. The case of an annulus A. Using the Riemann mapping theorem, A is isomorphic to A(R) = 1 < z < R . The natural invariant is the { | | } modulus, mod(A) = log R. This is both the ratio of height to radius, and a measurement of the height with respect to the unique invariant holomorphic 1-form with period 2π, namely dz/z. Thus we have one motivation for the use of 1-forms as moduli. 2. Holomorphic 1-forms. On a compact Riemann surface there are no holomorphic functions. However there may be functions well-defined up to translation, i.e. holomorphic 1-forms. 3. Let Ω(X) denote the complex vector space of holomorphic 1-forms ω = ω(z) dz. Every such 1-form can be thought of as a local map X C, well-defined up to an additive constant. → 4. Periods. By integrating along loops any 1-form gives a map π1(X) C called the period homomorphism. Since C is abelian, ω determines→ an element of H1(X, C). Thus we have a map Ω(X) H1(X, C). This map is injective when X is compact; since a form with→ no periods can be integrated. We can also take periods of anti-holomorphic forms, Ω(X) H1(X, C). These (1, 0) and (0, 1) give all harmonic 1-forms (the exterior→ differen- tials of harmonic functions). Since the period map is injective on both, we find that for X compact: dim Ω(X) g(X).
    [Show full text]
  • Riemai{N Surfaces and Small Point Sets
    Annales Academiae Scientiarum Fennicr Commentationes in memoriam Series A. I. Mathematica Rolf Nevanlinna Volumer 7, 1982, 49-57 RIEMAI{N SURFACES AND SMALL POINT SETS LARS V. ATILFORS I have been asked to talk about Nevanlinna's contributions to Riemann sur- faces and the related subject of small point sets in function theory. What I shall try to do is to make an effort to understand how Nevanlinna came to Riemann surfaces in the first place and to make a factual assessment of what he did and of its impact on contemporary and later work in the same field. I want to prepare you that I shall not shy away from a certain amount of criticism when I think it furthers my purpose, but please keep in mind that I am speaking from a safe distance of hindsight, and that my critique is not intended to detract from the value of Nevan- linna's pioneering work. It is not quite easy to draw the right line, for there were many persons involved and it is difficult to tell who was influenced by whom. Riemann surfaces were not Nevanlinna's first love. In his younger years he was exclusively interested in functions on a plane region. Nevanlinna's fame came early and it was principally based on his theory of meromorphic functions and clever use of potential theory which he had started together with his brother Fri- thiof and then developed further to the powerful method of harmonic measure. Even for these topics Riemann surfaces were needed, above all in the context of uniformization, but he did not yet study Riemann surfaces for their own sake.
    [Show full text]
  • Hyperbolic Geometry of Superstring Perturbation Theory
    Prepared for submission to JHEP Hyperbolic Geometry of Superstring Perturbation Theory Seyed Faroogh Moosavian, Roji Pius Perimeter Institute for Theoretical Physics, Waterloo, ON N2L 2Y5, Canada E-mail: [email protected] [email protected] Abstract: The hyperbolic structure of the RNS formulation of perturbative su- perstring theory is explored, with the goal of providing a systematic method for explicitly evaluating both the on-shell and the off-shell amplitudes in closed super- string theory with arbitrary number of external states and loops. A set of local coordinates around the punctures and a distribution of picture-changing operators on the worldsheet satisfying the off-shell factorization requirement needed for defining the off-shell amplitudes are specified using the hyperbolic metric on the worldsheet with constant curvature −1. The superstring amplitudes are expressed as sum of real integrals over certain covering spaces of the corresponding moduli space. These covering spaces are identified as simple domains inside the Teichm¨ullerspace of the worldsheet parameterized using the Fenchel-Nielsen coordinates. arXiv:1703.10563v2 [hep-th] 3 Feb 2019 Contents 1 Introduction1 1.1 Objectives1 1.1.1 Bosonic String Amplitudes and Moduli Space of Riemann Sur- faces2 1.1.2 Superstring Amplitudes and Picture-Changing Operators4 1.1.3 Off-Shell Superstring Amplitudes and 1PI Approach6 1.2 Summary of the Results9 1.2.1 Bosonic-String Measure and the Fenchel-Nielsen Coordinates 10 1.2.2 Gluing-Compatible Local Coordinates
    [Show full text]
  • Report 7/2004
    Mathematisches Forschungsinstitut Oberwolfach Report No. 7/2004 Funktionentheorie Organised by Walter Bergweiler (Kiel) Stephan Ruscheweyh (W¨urzburg) Ed Saff (Vanderbilt) February 8th – February 14th, 2004 Introduction by the Organisers The present conference was organized by Walter Bergweiler (Kiel), Stephan Ruscheweyh (W¨urzburg) and Ed Saff (Vanderbilt). The 24 talks gave an overview of recent results and current trends in function theory. 349 Funktionentheorie 351 Workshop on Funktionentheorie Table of Contents Kenneth Stephenson (joint with Charles Collins and Tobin Driscoll) Curvature flow in conformal mapping . 353 Igor Pritsker Julia polynomials and the Szeg˝o kernel method .......................... 354 Daniela Kraus Conformal Pseudo–metrics and a free boundary value problem for analytic functions .............................................................. 356 J.K. Langley (joint with J. Rossi) Critical points of discrete potentials in space ............................ 358 Marcus Stiemer Efficient Discretization of Green Energy and Grunsky-Type Development of Functions Univalent in an Annulus .................................. 359 Gunther Semmler Boundary Interpolation in the Theory of Nonlinear Riemann-Hilbert Problems .............................................................. 360 Mihai Putinar (joint with B. Gustafsson and H.S. Shapiro) Restriction operators on Bergman space ................................ 361 Jan-Martin Hemke Measurable dynamics of transcendental entire functions on their Julia sets 363 Lasse Rempe On
    [Show full text]
  • The Hausdorff Dimension of the Limit Sets of Infinitely Generated Kleinian
    Math. Proc. Camb. Phil. Soc. (2000), 128, 123 123 Printed in the United Kingdom c 2000 Cambridge Philosophical Society The Hausdorff dimension of the limit sets of infinitely generated Kleinian groups By KATSUHIKO MATSUZAKI Department of Mathematics, University of Michigan Ann Arbor, MI 48109-1109, U.S.A. and Department of Mathematics, Ochanomizu University Otsuka 2-1-1, Bunkyo-ku, Tokyo 112, Japan e-mail: [email protected] (Received 15 September 1997; revised 26 November 1998) Abstract In this paper we investigate the Hausdorff dimension of the limit set of an infinitely generated discrete subgroup of hyperbolic isometries and obtain conditions for the limit set to have full dimension. 1. Introduction Let Γ be a Kleinian group acting on the upper half space model HD+1 of (D + 1)- dimensional hyperbolic space and let Λ(Γ) and Λc(Γ) denote the limit set and the conical limit set of Γ respectively. A limit point x lies in Λc(Γ) if the orbit under Γ of any point in HD+1 approaches x inside a (Euclidean) non-tangential cone with vertex at x. The Hausdorff dimension of Λc(Γ) has geometric interpretations via the exponent of convergence of Γ; for example as the bottom of the spectrum of the hyperbolic Laplacian and the hyperbolic isoperimetric constant on the associated hy- D+1 perbolic manifold NΓ ÷ H =Γ (cf. [1]). In the case of geometrically finite Kleinian groups the limit set consists of Λc(Γ) and at most a countable set (cf. [8]). Thus the dimensions of Λ(Γ) and Λc(Γ) coincide.
    [Show full text]