Group-Groupoids and Monodromy Groupoids ∗ O
Total Page:16
File Type:pdf, Size:1020Kb

Load more
Recommended publications
-
Monodromy Representations Associated to a Continuous Map
Monodromy representations associated to a continuous map Salvador Borrós Cullell Supervisor: Dr. Pietro Polesello ALGANT Master Program Universität Duisburg-Essen; Università degli studi di Padova This dissertation is submitted for the degree of Master in Science July 2017 Acknowledgements First and foremost, I would like to thank my advisor Dr. Polesello for his infinite patience and his guidance while writing the thesis. His emphasis on writing mathematics in a precise and concise manner will forever remain a valuable lesson. I would also like to thank my parents for their constant support, specially during these two years I have been out of home. Despite the distance, they have kept me going when things weren’t going my way. A very special mention must be made of Prof. Marc Levine who introduced me to many of the notions that now appear in this thesis and showed unparalleled patience and kindness while helping me understand them in depth. Finally, I would like to thank my friends Shehzad Hathi and Jan Willem Frederik van Beck for taking up the role of a supportive family while living abroad. Abstract The aim of this thesis is to give a geometrical meaning to the induced monodromy rep- resentation. More precisely, given f : X ! Y a continuous map, the associated functor ind f f : P1(X) ! P1(Y) induces a functor Repk(P1(X)) ! Repk(P1(Y)) of the corresponding LCSH categories of representations. We will define a functor f∗ : LCSH(kX ) ! LCSH(kY ) from the category of locally constant sheaves on X to that of locally constant sheaves on Y in a way that the monodromy representation m LCSH is given by ind (m ), where m denotes f∗ F f F F the monodromy representation of a locally constant sheaf F on X. -
Arxiv:Math/0507171V1 [Math.AG] 8 Jul 2005 Monodromy
Monodromy Wolfgang Ebeling Dedicated to Gert-Martin Greuel on the occasion of his 60th birthday. Abstract Let (X,x) be an isolated complete intersection singularity and let f : (X,x) → (C, 0) be the germ of an analytic function with an isolated singularity at x. An important topological invariant in this situation is the Picard-Lefschetz monodromy operator associated to f. We give a survey on what is known about this operator. In particular, we re- view methods of computation of the monodromy and its eigenvalues (zeta function), results on the Jordan normal form of it, definition and properties of the spectrum, and the relation between the monodromy and the topology of the singularity. Introduction The word ’monodromy’ comes from the greek word µoνo − δρoµψ and means something like ’uniformly running’ or ’uniquely running’. According to [99, 3.4.4], it was first used by B. Riemann [135]. It arose in keeping track of the solutions of the hypergeometric differential equation going once around arXiv:math/0507171v1 [math.AG] 8 Jul 2005 a singular point on a closed path (cf. [30]). The group of linear substitutions which the solutions are subject to after this process is called the monodromy group. Since then, monodromy groups have played a substantial rˆole in many areas of mathematics. As is indicated on the webside ’www.monodromy.com’ of N. M. Katz, there are several incarnations, classical and l-adic, local and global, arithmetic and geometric. Here we concentrate on the classical lo- cal geometric monodromy in singularity theory. More precisely we focus on the monodromy operator of an isolated hypersurface or complete intersection singularity. -
Arxiv:2010.15657V1 [Math.NT] 29 Oct 2020
LOW-DEGREE PERMUTATION RATIONAL FUNCTIONS OVER FINITE FIELDS ZHIGUO DING AND MICHAEL E. ZIEVE Abstract. We determine all degree-4 rational functions f(X) 2 Fq(X) 1 which permute P (Fq), and answer two questions of Ferraguti and Micheli about the number of such functions and the number of equivalence classes of such functions up to composing with degree-one rational func- tions. We also determine all degree-8 rational functions f(X) 2 Fq(X) 1 which permute P (Fq) in case q is sufficiently large, and do the same for degree 32 in case either q is odd or f(X) is a nonsquare. Further, for several other positive integers n < 4096, for each sufficiently large q we determine all degree-n rational functions f(X) 2 Fq(X) which permute 1 P (Fq) but which are not compositions of lower-degree rational func- tions in Fq(X). Some of these results are proved by using a new Galois- theoretic characterization of additive (linearized) polynomials among all rational functions, which is of independent interest. 1. Introduction Let q be a power of a prime p.A permutation polynomial is a polyno- mial f(X) 2 Fq[X] for which the map α 7! f(α) is a permutation of Fq. Such polynomials have been studied both for their own sake and for use in various applications. Much less work has been done on permutation ra- tional functions, namely rational functions f(X) 2 Fq(X) which permute 1 P (Fq) := Fq [ f1g. However, the topic of permutation rational functions seems worthy of study, both because permutation rational functions have the same applications as permutation polynomials, and because of the con- struction in [24] which shows how to use permutation rational functions over Fq to produce permutation polynomials over Fq2 . -
The Monodromy Groups of Schwarzian Equations on Closed
Annals of Mathematics The Monodromy Groups of Schwarzian Equations on Closed Riemann Surfaces Author(s): Daniel Gallo, Michael Kapovich and Albert Marden Reviewed work(s): Source: Annals of Mathematics, Second Series, Vol. 151, No. 2 (Mar., 2000), pp. 625-704 Published by: Annals of Mathematics Stable URL: http://www.jstor.org/stable/121044 . Accessed: 15/02/2013 18:57 Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at . http://www.jstor.org/page/info/about/policies/terms.jsp . JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact [email protected]. Annals of Mathematics is collaborating with JSTOR to digitize, preserve and extend access to Annals of Mathematics. http://www.jstor.org This content downloaded on Fri, 15 Feb 2013 18:57:11 PM All use subject to JSTOR Terms and Conditions Annals of Mathematics, 151 (2000), 625-704 The monodromy groups of Schwarzian equations on closed Riemann surfaces By DANIEL GALLO, MICHAEL KAPOVICH, and ALBERT MARDEN To the memory of Lars V. Ahlfors Abstract Let 0: 7 (R) -* PSL(2, C) be a homomorphism of the fundamental group of an oriented, closed surface R of genus exceeding one. We will establish the following theorem. THEOREM. Necessary and sufficient for 0 to be the monodromy represen- tation associated with a complex projective stucture on R, either unbranched or with a single branch point of order 2, is that 0(7ri(R)) be nonelementary. -
Hausdorff Morita Equivalence of Singular Foliations
Hausdorff Morita Equivalence of singular foliations Alfonso Garmendia∗ Marco Zambony Abstract We introduce a notion of equivalence for singular foliations - understood as suitable families of vector fields - that preserves their transverse geometry. Associated to every singular foliation there is a holonomy groupoid, by the work of Androulidakis-Skandalis. We show that our notion of equivalence is compatible with this assignment, and as a consequence we obtain several invariants. Further, we show that it unifies some of the notions of transverse equivalence for regular foliations that appeared in the 1980's. Contents Introduction 2 1 Background on singular foliations and pullbacks 4 1.1 Singular foliations and their pullbacks . .4 1.2 Relation with pullbacks of Lie groupoids and Lie algebroids . .6 2 Hausdorff Morita equivalence of singular foliations 7 2.1 Definition of Hausdorff Morita equivalence . .7 2.2 First invariants . .9 2.3 Elementary examples . 11 2.4 Examples obtained by pushing forward foliations . 12 2.5 Examples obtained from Morita equivalence of related objects . 15 3 Morita equivalent holonomy groupoids 17 3.1 Holonomy groupoids . 17 arXiv:1803.00896v1 [math.DG] 2 Mar 2018 3.2 Morita equivalent holonomy groupoids: the case of projective foliations . 21 3.3 Pullbacks of foliations and their holonomy groupoids . 21 3.4 Morita equivalence for open topological groupoids . 27 3.5 Holonomy transformations . 29 3.6 Further invariants . 30 3.7 A second look at Hausdorff Morita equivalence of singular foliations . 31 4 Further developments 33 4.1 An extended equivalence for singular foliations . 33 ∗KU Leuven, Department of Mathematics, Celestijnenlaan 200B box 2400, BE-3001 Leuven, Belgium. -
Galois Theory Towards Dessins D'enfants
Galois Theory towards Dessins d'Enfants Marco Ant´onioDelgado Robalo Disserta¸c~aopara obten¸c~aodo Grau de Mestre em Matem´aticae Aplica¸c~oes J´uri Presidente: Prof. Doutor Rui Ant´onioLoja Fernandes Orientador: Prof. Doutor Jos´eManuel Vergueiro Monteiro Cidade Mour~ao Vogal: Prof. Doutor Carlos Armindo Arango Florentino Outubro de 2009 Agradecimentos As` Obsess~oes Em primeiro lugar e acima de tudo, agrade¸co`aminha M~aeFernanda. Por todos os anos de amor e de dedica¸c~ao.Obrigado. Agrade¸coao meu Irm~aoRui. Obrigado. As` minhas Av´os,Diamantina e Maria, e ao meu Av^o,Jo~ao.Obrigado. Este trabalho foi-me bastante dif´ıcilde executar. Em primeiro lugar pela vastid~aodo tema. Gosto de odisseias, da proposi¸c~aoaos objectivos inalcan¸c´aveis e de vis~oespanor^amicas.Seria dif´ıcilter-me dedicado a uma odisseia matem´aticamaior que esta, no tempo dispon´ıvel. Agrade¸coprofundamente ao meu orientador, Professor Jos´eMour~ao: pela sua enorme paci^enciaperante a minha constante mudan¸cade direc¸c~aono assunto deste trabalho, permitindo que eu pr´oprio encontrasse o meu caminho nunca deixando de me apoiar; por me ajudar a manter os p´esnos ch~aoe a n~aome perder na enorme vastid~aodo tema; pelos in´umeros encontros regulares ao longo de mais de um ano. A finaliza¸c~aodeste trabalho em tempo ´utils´ofoi poss´ıvel com a ajuda da sua sensatez e contagiante capacidade de concretiza¸c~ao. Agrade¸cotamb´emao Professor Paulo Almeida, que talvez sem o saber, me ajudou a recuperar o entu- siasmo pela procura da ess^enciadas coisas. -
Groupoids and Local Cartesian Closure
Groupoids and local cartesian closure Erik Palmgren∗ Department of Mathematics, Uppsala University July 2, 2003 Abstract The 2-category of groupoids, functors and natural isomorphisms is shown to be locally cartesian closed in a weak sense of a pseudo- adjunction. Key words: groupoids, 2-categories, local cartesian closure. Mathematics Subject Classification 2000: 18B40, 18D05, 18A40. 1 Introduction A groupoid is a category in which each morphism is invertible. The notion may be considered as a common generalisation of the notions of group and equivalence relation. A group is a one-object groupoid. Every equivalence relation on a set, viewed as a directed graph, is a groupoid. A recent indepen- dence proof in logic used groupoids. Hofmann and Streicher [8] showed that groupoids arise from a construction in Martin-Löf type theory. This is the so-called identity type construction, which applied to a type gives a groupoid turning the type into a projective object, in the category of types with equiv- alence relations. As a consequence there are plenty of choice objects in this category: every object has a projective cover [12] — a property which seems essential for doing constructive mathematics according to Bishop [1] inter- nally to the category. For some time the groupoids associated with types were believed to be discrete. However, in [8] a model of type theory was given using fibrations over groupoids, showing that this need not be the case. The purpose of this article is to draw some further conclusions for group- oids from the idea of Hofmann and Streicher. Since type theory has a Π- construction, one could expect that some kind of (weak) local cartesian clo- sure should hold for groupoids. -
Arxiv:1705.00466V2 [Math.DG] 1 Nov 2017 Bet,I Atclrqoins Si Hywr Moh Hsi Bec Is This Smooth
ORBISPACES AS DIFFERENTIABLE STRATIFIED SPACES MARIUS CRAINIC AND JOAO˜ NUNO MESTRE Abstract. We present some features of the smooth structure, and of the canonical stratification on the orbit space of a proper Lie groupoid. One of the main features is that of Morita invariance of these structures - it allows us to talk about the canonical structure of differentiable stratified space on the orbispace (an object analogous to a separated stack in algebraic geometry) presented by the proper Lie groupoid. The canonical smooth structure on an orbispace is studied mainly via Spallek’s framework of differentiable spaces, and two alternative frameworks are then presented. For the canonical stratification on an orbispace, we ex- tend the similar theory coming from proper Lie group actions. We make no claim to originality. The goal of these notes is simply to give a complemen- tary exposition to those available, and to clarify some subtle points where the literature can sometimes be confusing, even in the classical case of proper Lie group actions. Contents 1. Introduction 1 2. Background 5 3. Orbispaces as differentiable spaces 13 4. Orbispaces as stratified spaces 27 5. Orbispaces as differentiable stratified spaces 43 References 47 1. Introduction Lie groupoids are geometric objects that generalize Lie groups and smooth man- arXiv:1705.00466v2 [math.DG] 1 Nov 2017 ifolds, and permit a unified approach to the study of several objects of interest in differential geometry, such as Lie group actions, foliations and principal bundles (see for example [9, 37, 44] and the references therein). They have found wide use in Poisson and Dirac geometry (e.g. -
Integrability of Lie Brackets
Annals of Mathematics, 157 (2003), 575–620 Integrability of Lie brackets By Marius Crainic and Rui Loja Fernandes* Abstract In this paper we present the solution to a longstanding problem of dif- ferential geometry: Lie’s third theorem for Lie algebroids. We show that the integrability problem is controlled by two computable obstructions. As ap- plications we derive, explain and improve the known integrability results, we establish integrability by local Lie groupoids, we clarify the smoothness of the Poisson sigma-model for Poisson manifolds, and we describe other geometrical applications. Contents 0. Introduction 1. A-paths and homotopy 1.1. A-paths 1.2. A-paths and connections 1.3. Homotopy of A-paths 1.4. Representations and A-paths 2. The Weinstein groupoid 2.1. The groupoid G(A) 2.2. Homomorphisms 2.3. The exponential map 3. Monodromy 3.1. Monodromy groups 3.2. A second-order monodromy map 3.3. Computing the monodromy 3.4. Measuring the monodromy ∗ The first author was supported in part by NWO and a Miller Research Fellowship. The second author was supported in part by FCT through program POCTI and grant POCTI/1999/MAT/33081. Key words and phrases. Lie algebroid, Lie groupoid. 576 MARIUS CRAINIC AND RUI LOJA FERNANDES 4. Obstructions to integrability 4.1. The main theorem 4.2. The Weinstein groupoid as a leaf space 4.3. Proof of the main theorem 5. Examples and applications 5.1. Local integrability 5.2. Integrability criteria 5.3. Tranversally parallelizable foliations Appendix A. Flows A.1. Flows and infinitesimal flows A.2. -
Integrability of Lie Algebroids by Proper Groupoids
Integrability of Lie algebroids by proper Lie groupoids Lu´ıs Alexandre Meira Fernandes Alves Pereira Disserta¸c˜ao para obten¸c˜aodo Grau de Mestre em Matem´atica e Aplica¸c˜oes J´uri Presidente: Prof. Doutor Miguel Tribolet de Abreu Orientador: Prof. Doutor Rui Ant´onio Loja Fernandes Vogais: Prof. Doutor Gustavo Rui de Oliveira Granja Julho de 2008 Agradecimentos Este trabalho foi apoiado por uma Bolsa de Inicia¸c˜aoCient´ıfica da Funda¸c˜ao para a Ciˆenciae a Tecnologia. 1 Resumo Um crit´eriocl´assico diz-nos que uma ´algebra de Lie (real) g ´ea ´algebra de Lie de um grupo de Lie G compacto se e s´ose existe um produto interno em g que ´einvariante para a ac¸c˜aoadjunta de g em si mesma. O objectivo deste trabalho foi o de investigar se este resultado poderia ser extendido para algebr´oides de Lie, obtendo-se um crit´erioan´alogo caracterizando quando ´eque um algebr´oide de Lie A ´eo algebr´oidede Lie de um grup´oidede Lie G pr´oprio. Teve-se como base a formula¸c˜aode uma conjectura de trabalho afirmando que a existˆenciade um produto interno em A satisfazendo uma certa propriedade de invariˆancia seria uma condi¸c˜aonecess´ariae suficiente para que A fosse in- tegrado por um grup´oide pr´oprio. O trabalho consistiu ent˜aoem decidir se a conjectura era v´alida, e caso contr´ario, porquˆee como falhava. Numa direc¸c˜ao,prov´amosque a existˆenciade um grup´oidede Lie pr´oprio integrando A e satisfazendo algumas condi¸c˜oes razo´aveis implica a existˆencia de um produto interno em A nas condi¸c˜oesda nossa conjectura. -
Correspondences and Groupoids 1 Introduction
Proceedings of the IX Fall Workshop on Geometry and Physics, Vilanova i la Geltr´u, 2000 Publicaciones de la RSME, vol. X, pp. 1–6. Correspondences and groupoids 1Marta Macho-Stadler and 2Moto O’uchi 1Departamento de Matem´aticas, Universidad del Pa´ıs Vasco-Euskal Herriko Unibertsitatea 2Department of Applied Mathematics, Osaka Women’s University emails: [email protected], [email protected] Abstract Our definition of correspondence between groupoids (which generali- zes the notion of homomorphism) is obtained by weakening the conditions in the definition of equivalence of groupoids in [5]. We prove that such a correspondence induces another one between the associated C*-algebras, and in some cases besides a Kasparov element. We wish to apply the results obtained in the particular case of K-oriented maps of leaf spaces in the sense of [3]. Key words: Groupoid, C*-algebra, correspondence, KK-group. MSC 2000: 46L80, 46L89 1Introduction A. Connes introduced in [1] the notion of correspondence in the theory of Von Neumann algebras. It is a concept of morphism, which gives the well known notion of correspondence of C*-algebras. In [5], P.S. Mulhy, J.N. Re- nault and D. Williams defined the notion of equivalence of groupoids and showed that if two groupoids are equivalent, then the associated C*-algebras are Morita-equivalent. But, this notion is too strong: here we give a definition of correspondence of groupoids,byweakening the conditions of equivalence in [5]. We prove that such a correspondence induces another one (in the sense of A. Connes) between the associated reduced C*-algebras. -
The Monodromy Groupoid of a Lie Groupoid Cahiers De Topologie Et Géométrie Différentielle Catégoriques, Tome 36, No 4 (1995), P
CAHIERS DE TOPOLOGIE ET GÉOMÉTRIE DIFFÉRENTIELLE CATÉGORIQUES RONALD BROWN OSMAN MUCUK The monodromy groupoid of a Lie groupoid Cahiers de topologie et géométrie différentielle catégoriques, tome 36, no 4 (1995), p. 345-369 <http://www.numdam.org/item?id=CTGDC_1995__36_4_345_0> © Andrée C. Ehresmann et les auteurs, 1995, tous droits réservés. L’accès aux archives de la revue « Cahiers de topologie et géométrie différentielle catégoriques » implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction 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/ CAHIERS DE TOPOLOGIE ET Volume XXXVI-4 (1995) GEOMETRIE DIFFERENTIELLE CATEGORIQUES THE MONODROMY GROUPOID OF A LIE GROUPOID by Ronald BROWN and Osman MUCUK Resume: Dans cet article, on montre que, sous des condi- tions générales, Punion disjointe des recouvrements universels des étoiles d’un groupoide de Lie a la structure d’un groupoide de Lie dans lequel la projection possède une propriete de mon- odromie pour les extensions des morphismes locaux reguliers. Ceci complete un rapport 46ta,iU6 de r6sultats announces par J. Pradines. Introduction The notion of monodromy groupoid which we describe here a.rose from the grand scheme of J. Pradines in the ea.rly 1960s to generalise the standard construction of a simply connected Lie group from a Lie al- gebra to a corresponding construction of a. Lie grOllpOld from a Lie algebroid, a notion first defined by Pradines.