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Scalar Curvature and Geometrization Conjectures for 3-Manifolds
Comparison Geometry MSRI Publications Volume 30, 1997 Scalar Curvature and Geometrization Conjectures for 3-Manifolds MICHAEL T. ANDERSON Abstract. We first summarize very briefly the topology of 3-manifolds and the approach of Thurston towards their geometrization. After dis- cussing some general properties of curvature functionals on the space of metrics, we formulate and discuss three conjectures that imply Thurston’s Geometrization Conjecture for closed oriented 3-manifolds. The final two sections present evidence for the validity of these conjectures and outline an approach toward their proof. Introduction In the late seventies and early eighties Thurston proved a number of very re- markable results on the existence of geometric structures on 3-manifolds. These results provide strong support for the profound conjecture, formulated by Thur- ston, that every compact 3-manifold admits a canonical decomposition into do- mains, each of which has a canonical geometric structure. For simplicity, we state the conjecture only for closed, oriented 3-manifolds. Geometrization Conjecture [Thurston 1982]. Let M be a closed , oriented, 2 prime 3-manifold. Then there is a finite collection of disjoint, embedded tori Ti 2 in M, such that each component of the complement M r Ti admits a geometric structure, i.e., a complete, locally homogeneous RiemannianS metric. A more detailed description of the conjecture and the terminology will be given in Section 1. A complete Riemannian manifold N is locally homogeneous if the universal cover N˜ is a complete homogenous manifold, that is, if the isometry group Isom N˜ acts transitively on N˜. It follows that N is isometric to N=˜ Γ, where Γ is a discrete subgroup of Isom N˜, which acts freely and properly discontinuously on N˜. -
Generalized Riemann Hypothesis Léo Agélas
Generalized Riemann Hypothesis Léo Agélas To cite this version: Léo Agélas. Generalized Riemann Hypothesis. 2019. hal-00747680v3 HAL Id: hal-00747680 https://hal.archives-ouvertes.fr/hal-00747680v3 Preprint submitted on 29 May 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. Generalized Riemann Hypothesis L´eoAg´elas Department of Mathematics, IFP Energies nouvelles, 1-4, avenue de Bois-Pr´eau,F-92852 Rueil-Malmaison, France Abstract (Generalized) Riemann Hypothesis (that all non-trivial zeros of the (Dirichlet L-function) zeta function have real part one-half) is arguably the most impor- tant unsolved problem in contemporary mathematics due to its deep relation to the fundamental building blocks of the integers, the primes. The proof of the Riemann hypothesis will immediately verify a slew of dependent theorems (Borwien et al.(2008), Sabbagh(2002)). In this paper, we give a proof of Gen- eralized Riemann Hypothesis which implies the proof of Riemann Hypothesis and Goldbach's weak conjecture (also known as the odd Goldbach conjecture) one of the oldest and best-known unsolved problems in number theory. 1. Introduction The Riemann hypothesis is one of the most important conjectures in math- ematics. -
Towards the Poincaré Conjecture and the Classification of 3-Manifolds John Milnor
Towards the Poincaré Conjecture and the Classification of 3-Manifolds John Milnor he Poincaré Conjecture was posed ninety- space SO(3)/I60 . Here SO(3) is the group of rota- nine years ago and may possibly have tions of Euclidean 3-space, and I60 is the subgroup been proved in the last few months. This consisting of those rotations which carry a regu- note will be an account of some of the lar icosahedron or dodecahedron onto itself (the Tmajor results over the past hundred unique simple group of order 60). This manifold years which have paved the way towards a proof has the homology of the 3-sphere, but its funda- and towards the even more ambitious project of mental group π1(SO(3)/I60) is a perfect group of classifying all compact 3-dimensional manifolds. order 120. He concluded the discussion by asking, The final paragraph provides a brief description of again translated into modern language: the latest developments, due to Grigory Perelman. A more serious discussion of Perelman’s work If a closed 3-dimensional manifold has will be provided in a subsequent note by Michael trivial fundamental group, must it be Anderson. homeomorphic to the 3-sphere? The conjecture that this is indeed the case has Poincaré’s Question come to be known as the Poincaré Conjecture. It At the very beginning of the twentieth century, has turned out to be an extraordinarily difficult Henri Poincaré (1854–1912) made an unwise claim, question, much harder than the corresponding which can be stated in modern language as follows: question in dimension five or more,2 and is a If a closed 3-dimensional manifold has key stumbling block in the effort to classify the homology of the sphere S3 , then it 3-dimensional manifolds. -
PIECEWISE LINEAR TOPOLOGY Contents 1. Introduction 2 2. Basic
PIECEWISE LINEAR TOPOLOGY J. L. BRYANT Contents 1. Introduction 2 2. Basic Definitions and Terminology. 2 3. Regular Neighborhoods 9 4. General Position 16 5. Embeddings, Engulfing 19 6. Handle Theory 24 7. Isotopies, Unknotting 30 8. Approximations, Controlled Isotopies 31 9. Triangulations of Manifolds 33 References 35 1 2 J. L. BRYANT 1. Introduction The piecewise linear category offers a rich structural setting in which to study many of the problems that arise in geometric topology. The first systematic ac- counts of the subject may be found in [2] and [63]. Whitehead’s important paper [63] contains the foundation of the geometric and algebraic theory of simplicial com- plexes that we use today. More recent sources, such as [30], [50], and [66], together with [17] and [37], provide a fairly complete development of PL theory up through the early 1970’s. This chapter will present an overview of the subject, drawing heavily upon these sources as well as others with the goal of unifying various topics found there as well as in other parts of the literature. We shall try to give enough in the way of proofs to provide the reader with a flavor of some of the techniques of the subject, while deferring the more intricate details to the literature. Our discussion will generally avoid problems associated with embedding and isotopy in codimension 2. The reader is referred to [12] for a survey of results in this very important area. 2. Basic Definitions and Terminology. Simplexes. A simplex of dimension p (a p-simplex) σ is the convex closure of a n set of (p+1) geometrically independent points {v0, . -
Prospects in Topology
Annals of Mathematics Studies Number 138 Prospects in Topology PROCEEDINGS OF A CONFERENCE IN HONOR OF WILLIAM BROWDER edited by Frank Quinn PRINCETON UNIVERSITY PRESS PRINCETON, NEW JERSEY 1995 Copyright © 1995 by Princeton University Press ALL RIGHTS RESERVED The Annals of Mathematics Studies are edited by Luis A. Caffarelli, John N. Mather, and Elias M. Stein Princeton University Press books are printed on acid-free paper and meet the guidelines for permanence and durability of the Committee on Production Guidelines for Book Longevity of the Council on Library Resources Printed in the United States of America by Princeton Academic Press 10 987654321 Library of Congress Cataloging-in-Publication Data Prospects in topology : proceedings of a conference in honor of W illiam Browder / Edited by Frank Quinn. p. cm. — (Annals of mathematics studies ; no. 138) Conference held Mar. 1994, at Princeton University. Includes bibliographical references. ISB N 0-691-02729-3 (alk. paper). — ISBN 0-691-02728-5 (pbk. : alk. paper) 1. Topology— Congresses. I. Browder, William. II. Quinn, F. (Frank), 1946- . III. Series. QA611.A1P76 1996 514— dc20 95-25751 The publisher would like to acknowledge the editor of this volume for providing the camera-ready copy from which this book was printed PROSPECTS IN TOPOLOGY F r a n k Q u in n , E d it o r Proceedings of a conference in honor of William Browder Princeton, March 1994 Contents Foreword..........................................................................................................vii Program of the conference ................................................................................ix Mathematical descendants of William Browder...............................................xi A. Adem and R. J. Milgram, The mod 2 cohomology rings of rank 3 simple groups are Cohen-Macaulay........................................................................3 A. -
The Process of Student Cognition in Constructing Mathematical Conjecture
ISSN 2087-8885 E-ISSN 2407-0610 Journal on Mathematics Education Volume 9, No. 1, January 2018, pp. 15-26 THE PROCESS OF STUDENT COGNITION IN CONSTRUCTING MATHEMATICAL CONJECTURE I Wayan Puja Astawa1, I Ketut Budayasa2, Dwi Juniati2 1Ganesha University of Education, Jalan Udayana 11, Singaraja, Indonesia 2State University of Surabaya, Ketintang, Surabaya, Indonesia Email: [email protected] Abstract This research aims to describe the process of student cognition in constructing mathematical conjecture. Many researchers have studied this process but without giving a detailed explanation of how students understand the information to construct a mathematical conjecture. The researchers focus their analysis on how to construct and prove the conjecture. This article discusses the process of student cognition in constructing mathematical conjecture from the very beginning of the process. The process is studied through qualitative research involving six students from the Mathematics Education Department in the Ganesha University of Education. The process of student cognition in constructing mathematical conjecture is grouped into five different stages. The stages consist of understanding the problem, exploring the problem, formulating conjecture, justifying conjecture, and proving conjecture. In addition, details of the process of the students’ cognition in each stage are also discussed. Keywords: Process of Student Cognition, Mathematical Conjecture, qualitative research Abstrak Penelitian ini bertujuan untuk mendeskripsikan proses kognisi mahasiswa dalam mengonstruksi konjektur matematika. Beberapa peneliti mengkaji proses-proses ini tanpa menjelaskan bagaimana siswa memahami informasi untuk mengonstruksi konjektur. Para peneliti dalam analisisnya menekankan bagaimana mengonstruksi dan membuktikan konjektur. Artikel ini membahas proses kognisi mahasiswa dalam mengonstruksi konjektur matematika dari tahap paling awal. -
History of Mathematics
History of Mathematics James Tattersall, Providence College (Chair) Janet Beery, University of Redlands Robert E. Bradley, Adelphi University V. Frederick Rickey, United States Military Academy Lawrence Shirley, Towson University Introduction. There are many excellent reasons to study the history of mathematics. It helps students develop a deeper understanding of the mathematics they have already studied by seeing how it was developed over time and in various places. It encourages creative and flexible thinking by allowing students to see historical evidence that there are different and perfectly valid ways to view concepts and to carry out computations. Ideally, a History of Mathematics course should be a part of every mathematics major program. A course taught at the sophomore-level allows mathematics students to see the great wealth of mathematics that lies before them and encourages them to continue studying the subject. A one- or two-semester course taught at the senior level can dig deeper into the history of mathematics, incorporating many ideas from the 19th and 20th centuries that could only be approached with difficulty by less prepared students. Such a senior-level course might be a capstone experience taught in a seminar format. It would be wonderful for students, especially those planning to become middle school or high school mathematics teachers, to have the opportunity to take advantage of both options. We also encourage History of Mathematics courses taught to entering students interested in mathematics, perhaps as First Year or Honors Seminars; to general education students at any level; and to junior and senior mathematics majors and minors. Ideally, mathematics history would be incorporated seamlessly into all courses in the undergraduate mathematics curriculum in addition to being addressed in a few courses of the type we have listed. -
Triangulations of 3-Manifolds with Essential Edges
TRIANGULATIONS OF 3–MANIFOLDS WITH ESSENTIAL EDGES CRAIG D. HODGSON, J. HYAM RUBINSTEIN, HENRY SEGERMAN AND STEPHAN TILLMANN Dedicated to Michel Boileau for his many contributions and leadership in low dimensional topology. ABSTRACT. We define essential and strongly essential triangulations of 3–manifolds, and give four constructions using different tools (Heegaard splittings, hierarchies of Haken 3–manifolds, Epstein-Penner decompositions, and cut loci of Riemannian manifolds) to obtain triangulations with these properties under various hypotheses on the topology or geometry of the manifold. We also show that a semi-angle structure is a sufficient condition for a triangulation of a 3–manifold to be essential, and a strict angle structure is a sufficient condition for a triangulation to be strongly essential. Moreover, algorithms to test whether a triangulation of a 3–manifold is essential or strongly essential are given. 1. INTRODUCTION Finding combinatorial and topological properties of geometric triangulations of hyper- bolic 3–manifolds gives information which is useful for the reverse process—for exam- ple, solving Thurston’s gluing equations to construct an explicit hyperbolic metric from an ideal triangulation. Moreover, suitable properties of a triangulation can be used to de- duce facts about the variety of representations of the fundamental group of the underlying 3-manifold into PSL(2;C). See for example [32]. In this paper, we focus on one-vertex triangulations of closed manifolds and ideal trian- gulations of the interiors of compact manifolds with boundary. Two important properties of these triangulations are introduced. For one-vertex triangulations in the closed case, the first property is that no edge loop is null-homotopic and the second is that no two edge loops are homotopic, keeping the vertex fixed. -
Experimental Statistics of Veering Triangulations
EXPERIMENTAL STATISTICS OF VEERING TRIANGULATIONS WILLIAM WORDEN Abstract. Certain fibered hyperbolic 3-manifolds admit a layered veering triangulation, which can be constructed algorithmically given the stable lamination of the monodromy. These triangulations were introduced by Agol in 2011, and have been further studied by several others in the years since. We obtain experimental results which shed light on the combinatorial structure of veering triangulations, and its relation to certain topological invariants of the underlying manifold. 1. Introduction In 2011, Agol [Ago11] introduced the notion of a layered veering triangulation for certain fibered hyperbolic manifolds. In particular, given a surface Σ (possibly with punctures), and a pseudo-Anosov homeomorphism ' ∶ Σ → Σ, Agol's construction gives a triangulation ○ ○ ○ ○ T for the mapping torus M' with fiber Σ and monodromy ' = 'SΣ○ , where Σ is the surface resulting from puncturing Σ at the singularities of its '-invariant foliations. For Σ a once-punctured torus or 4-punctured sphere, the resulting triangulation is well understood: it is the monodromy (or Floyd{Hatcher) triangulation considered in [FH82] and [Jør03]. In this case T is a geometric triangulation, meaning that tetrahedra in T can be realized as ideal hyperbolic tetrahedra isometrically embedded in M'○ . In fact, these triangulations are geometrically canonical, i.e., dual to the Ford{Voronoi domain of the mapping torus (see [Aki99], [Lac04], and [Gu´e06]). Agol's definition of T was conceived as a generalization of these monodromy triangulations, and so a natural question is whether these generalized monodromy triangulations are also geometric. Hodgson{Issa{Segerman [HIS16] answered this question in the negative, by producing the first examples of non- geometric layered veering triangulations. -
Equivalences to the Triangulation Conjecture 1 Introduction
ISSN online printed Algebraic Geometric Topology Volume ATG Published Decemb er Equivalences to the triangulation conjecture Duane Randall Abstract We utilize the obstruction theory of GalewskiMatumotoStern to derive equivalent formulations of the Triangulation Conjecture For ex n ample every closed top ological manifold M with n can b e simplicially triangulated if and only if the two distinct combinatorial triangulations of 5 RP are simplicially concordant AMS Classication N S Q Keywords Triangulation KirbySieb enmann class Bo ckstein op erator top ological manifold Intro duction The Triangulation Conjecture TC arms that every closed top ological man n ifold M of dimension n admits a simplicial triangulation The vanishing of the KirbySieb enmann class KS M in H M Z is b oth necessary and n sucient for the existence of a combinatorial triangulation of M for n n by A combinatorial triangulation of a closed manifold M is a simplicial triangulation for which the link of every i simplex is a combinatorial sphere of dimension n i Galewski and Stern Theorem and Matumoto n indep endently proved that a closed connected top ological manifold M with n is simplicially triangulable if and only if KS M in H M ker where denotes the Bo ckstein op erator asso ciated to the exact sequence ker Z of ab elian groups Moreover the Triangulation Conjecture is true if and only if this exact sequence splits by or page The Ro chlin invariant morphism is dened on the homology b ordism group of oriented -
Some Decomposition Lemmas of Diffeomorphisms Compositio Mathematica, Tome 84, No 1 (1992), P
COMPOSITIO MATHEMATICA VICENTE CERVERA FRANCISCA MASCARO Some decomposition lemmas of diffeomorphisms Compositio Mathematica, tome 84, no 1 (1992), p. 101-113 <http://www.numdam.org/item?id=CM_1992__84_1_101_0> © Foundation Compositio Mathematica, 1992, tous droits réservés. L’accès aux archives de la revue « Compositio Mathematica » (http: //http://www.compositio.nl/) implique l’accord avec les conditions gé- nérales d’utilisation (http://www.numdam.org/conditions). Toute utilisa- tion commerciale ou impression systématique est constitutive d’une in- fraction pénale. Toute copie ou impression de ce fichier doit conte- nir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ Compositio Mathematica 84: 101-113,101 1992. cg 1992 Kluwer Academic Publishers. Printed in the Netherlands. Some decomposition lemmas of diffeomorphisms VICENTE CERVERA and FRANCISCA MASCARO Departamento de Geometria, y Topologia Facultad de Matemàticas, Universidad de Valencia, 46100-Burjassot (Valencia) Spain Received 9 April 1991; accepted 14 February 1992 Abstract. Let Q be a volume element on an open manifold, M, which is the interior of a compact manifold M. We will give conditions for a non-compact supported Q-preserving diffeomorphism to decompose as a finite product of S2-preserving diffeomorphisms with supports in locally finite families of disjoint cells. A widely used and very powerful technique in the study of some subgroups of the group of diffeomorphisms of a differentiable manifold, Diff(M), is the decomposition of its elements as a finite product of diffeomorphisms with support in cells (See for example [1], [5], [7]). It was in a paper of Palis and Smale [7] that first appeared one of those decompositions for the case of a compact manifold, M. -
Arxiv:1506.05408V1 [Math.AT]
NOVIKOV’S CONJECTURE JONATHAN ROSENBERG Abstract. We describe Novikov’s “higher signature conjecture,” which dates back to the late 1960’s, as well as many alternative formulations and related problems. The Novikov Conjecture is perhaps the most important unsolved problem in high-dimensional manifold topology, but more importantly, vari- ants and analogues permeate many other areas of mathematics, from geometry to operator algebras to representation theory. 1. Origins of the Original Conjecture The Novikov Conjecture is perhaps the most important unsolved problem in the topology of high-dimensional manifolds. It was first stated by Sergei Novikov, in various forms, in his lectures at the International Congresses of Mathematicians in Moscow in 1966 and in Nice in 1970, and in a few other papers [84, 87, 86, 85]. For an annotated version of the original formulation, in both Russian and English, we refer the reader to [37]. Here we will try instead to put the problem in context and explain why it might be of interest to the average mathematician. For a nice book- length exposition of this subject, we recommend [65]. Many treatments of various aspects of the problem can also be found in the many papers in the collections [38, 39]. For the typical mathematician, the most important topological spaces are smooth manifolds, which were introduced by Riemann in the 1850’s. However, it took about 100 years for the tools for classifying manifolds (except in dimension 1, which is trivial, and dimension 2, which is relatively easy) to be developed. The problem is that manifolds have no local invariants (except for the dimension); all manifolds of the same dimension look the same locally.