Man and Machine Thinking About the Smooth 4-Dimensional Poincar\'E Conjecture
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Topics in Geometric Group Theory. Part I
Topics in Geometric Group Theory. Part I Daniele Ettore Otera∗ and Valentin Po´enaru† April 17, 2018 Abstract This survey paper concerns mainly with some asymptotic topological properties of finitely presented discrete groups: quasi-simple filtration (qsf), geometric simple connec- tivity (gsc), topological inverse-representations, and the notion of easy groups. As we will explain, these properties are central in the theory of discrete groups seen from a topo- logical viewpoint at infinity. Also, we shall outline the main steps of the achievements obtained in the last years around the very general question whether or not all finitely presented groups satisfy these conditions. Keywords: Geometric simple connectivity (gsc), quasi-simple filtration (qsf), inverse- representations, finitely presented groups, universal covers, singularities, Whitehead manifold, handlebody decomposition, almost-convex groups, combings, easy groups. MSC Subject: 57M05; 57M10; 57N35; 57R65; 57S25; 20F65; 20F69. Contents 1 Introduction 2 1.1 Acknowledgments................................... 2 arXiv:1804.05216v1 [math.GT] 14 Apr 2018 2 Topology at infinity, universal covers and discrete groups 2 2.1 The Φ/Ψ-theoryandzippings ............................ 3 2.2 Geometric simple connectivity . 4 2.3 TheWhiteheadmanifold............................... 6 2.4 Dehn-exhaustibility and the QSF property . .... 8 2.5 OnDehn’slemma................................... 10 2.6 Presentations and REPRESENTATIONS of groups . ..... 13 2.7 Good-presentations of finitely presented groups . ........ 16 2.8 Somehistoricalremarks .............................. 17 2.9 Universalcovers................................... 18 2.10 EasyREPRESENTATIONS . 21 ∗Vilnius University, Institute of Data Science and Digital Technologies, Akademijos st. 4, LT-08663, Vilnius, Lithuania. e-mail: [email protected] - [email protected] †Professor Emeritus at Universit´eParis-Sud 11, D´epartement de Math´ematiques, Bˆatiment 425, 91405 Orsay, France. -
Part III 3-Manifolds Lecture Notes C Sarah Rasmussen, 2019
Part III 3-manifolds Lecture Notes c Sarah Rasmussen, 2019 Contents Lecture 0 (not lectured): Preliminaries2 Lecture 1: Why not ≥ 5?9 Lecture 2: Why 3-manifolds? + Intro to Knots and Embeddings 13 Lecture 3: Link Diagrams and Alexander Polynomial Skein Relations 17 Lecture 4: Handle Decompositions from Morse critical points 20 Lecture 5: Handles as Cells; Handle-bodies and Heegard splittings 24 Lecture 6: Handle-bodies and Heegaard Diagrams 28 Lecture 7: Fundamental group presentations from handles and Heegaard Diagrams 36 Lecture 8: Alexander Polynomials from Fundamental Groups 39 Lecture 9: Fox Calculus 43 Lecture 10: Dehn presentations and Kauffman states 48 Lecture 11: Mapping tori and Mapping Class Groups 54 Lecture 12: Nielsen-Thurston Classification for Mapping class groups 58 Lecture 13: Dehn filling 61 Lecture 14: Dehn Surgery 64 Lecture 15: 3-manifolds from Dehn Surgery 68 Lecture 16: Seifert fibered spaces 69 Lecture 17: Hyperbolic 3-manifolds and Mostow Rigidity 70 Lecture 18: Dehn's Lemma and Essential/Incompressible Surfaces 71 Lecture 19: Sphere Decompositions and Knot Connected Sum 74 Lecture 20: JSJ Decomposition, Geometrization, Splice Maps, and Satellites 78 Lecture 21: Turaev torsion and Alexander polynomial of unions 81 Lecture 22: Foliations 84 Lecture 23: The Thurston Norm 88 Lecture 24: Taut foliations on Seifert fibered spaces 89 References 92 1 2 Lecture 0 (not lectured): Preliminaries 0. Notation and conventions. Notation. @X { (the manifold given by) the boundary of X, for X a manifold with boundary. th @iX { the i connected component of @X. ν(X) { a tubular (or collared) neighborhood of X in Y , for an embedding X ⊂ Y . -
Arxiv:1303.6028V2 [Math.DG] 29 Dec 2014 B Ahrglrlvlhprufc Scle an Called Is Hypersurface Level Regular Each E Scle the Called Is Set E.[T3 )
ISOPARAMETRIC FUNCTIONS ON EXOTIC SPHERES CHAO QIAN AND ZIZHOU TANG Abstract. This paper extends widely the work in [GT13]. Existence and non-existence results of isoparametric functions on exotic spheres and Eells-Kuiper projective planes are established. In particular, every homotopy n-sphere (n > 4) carries an isoparametric function (with certain metric) with 2 points as the focal set, in strong contrast to the classification of cohomogeneity one actions on homotopy spheres [St96] ( only exotic Kervaire spheres admit cohomogeneity one actions besides the standard spheres ). As an application, we improve a beautiful result of B´erard-Bergery [BB77] ( see also pp.234-235 of [Be78] ). 1. Introduction Let N be a connected complete Riemannian manifold. A non-constant smooth function f on N is called transnormal, if there exists a smooth function b : R R such that f 2 = → |∇ | b( f ), where f is the gradient of f . If in addition, there exists a continuous function a : ∇ R R so that f = a( f ), where f is the Laplacian of f , then f is called isoparametric. → △ △ Each regular level hypersurface is called an isoparametric hypersurface and the singular level set is called the focal set. The two equations of the function f mean that the regular level hypersurfaces of f are parallel and have constant mean curvatures, which may be regarded as a geometric generalization of cohomogeneity one actions in the theory of transformation groups ( ref. [GT13] ). Owing to E. Cartan and H. F. M¨unzner [M¨u80], the classification of isoparametric hy- persurfaces in a unit sphere has been one of the most challenging problems in submanifold geometry. -
EXOTIC SPHERES and CURVATURE 1. Introduction Exotic
BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY Volume 45, Number 4, October 2008, Pages 595–616 S 0273-0979(08)01213-5 Article electronically published on July 1, 2008 EXOTIC SPHERES AND CURVATURE M. JOACHIM AND D. J. WRAITH Abstract. Since their discovery by Milnor in 1956, exotic spheres have pro- vided a fascinating object of study for geometers. In this article we survey what is known about the curvature of exotic spheres. 1. Introduction Exotic spheres are manifolds which are homeomorphic but not diffeomorphic to a standard sphere. In this introduction our aims are twofold: First, to give a brief account of the discovery of exotic spheres and to make some general remarks about the structure of these objects as smooth manifolds. Second, to outline the basics of curvature for Riemannian manifolds which we will need later on. In subsequent sections, we will explore the interaction between topology and geometry for exotic spheres. We will use the term differentiable to mean differentiable of class C∞,and all diffeomorphisms will be assumed to be smooth. As every graduate student knows, a smooth manifold is a topological manifold that is equipped with a smooth (differentiable) structure, that is, a smooth maximal atlas. Recall that an atlas is a collection of charts (homeomorphisms from open neighbourhoods in the manifold onto open subsets of some Euclidean space), the domains of which cover the manifold. Where the chart domains overlap, we impose a smooth compatibility condition for the charts [doC, chapter 0] if we wish our manifold to be smooth. Such an atlas can then be extended to a maximal smooth atlas by including all possible charts which satisfy the compatibility condition with the original maps. -
Higher Dimensional Conundra
Higher Dimensional Conundra Steven G. Krantz1 Abstract: In recent years, especially in the subject of harmonic analysis, there has been interest in geometric phenomena of RN as N → +∞. In the present paper we examine several spe- cific geometric phenomena in Euclidean space and calculate the asymptotics as the dimension gets large. 0 Introduction Typically when we do geometry we concentrate on a specific venue in a particular space. Often the context is Euclidean space, and often the work is done in R2 or R3. But in modern work there are many aspects of analysis that are linked to concrete aspects of geometry. And there is often interest in rendering the ideas in Hilbert space or some other infinite dimensional setting. Thus we want to see how the finite-dimensional result in RN changes as N → +∞. In the present paper we study some particular aspects of the geometry of RN and their asymptotic behavior as N →∞. We choose these particular examples because the results are surprising or especially interesting. We may hope that they will lead to further studies. It is a pleasure to thank Richard W. Cottle for a careful reading of an early draft of this paper and for useful comments. 1 Volume in RN Let us begin by calculating the volume of the unit ball in RN and the surface area of its bounding unit sphere. We let ΩN denote the former and ωN−1 denote the latter. In addition, we let Γ(x) be the celebrated Gamma function of L. Euler. It is a helpful intuition (which is literally true when x is an 1We are happy to thank the American Institute of Mathematics for its hospitality and support during this work. -
MTH 304: General Topology Semester 2, 2017-2018
MTH 304: General Topology Semester 2, 2017-2018 Dr. Prahlad Vaidyanathan Contents I. Continuous Functions3 1. First Definitions................................3 2. Open Sets...................................4 3. Continuity by Open Sets...........................6 II. Topological Spaces8 1. Definition and Examples...........................8 2. Metric Spaces................................. 11 3. Basis for a topology.............................. 16 4. The Product Topology on X × Y ...................... 18 Q 5. The Product Topology on Xα ....................... 20 6. Closed Sets.................................. 22 7. Continuous Functions............................. 27 8. The Quotient Topology............................ 30 III.Properties of Topological Spaces 36 1. The Hausdorff property............................ 36 2. Connectedness................................. 37 3. Path Connectedness............................. 41 4. Local Connectedness............................. 44 5. Compactness................................. 46 6. Compact Subsets of Rn ............................ 50 7. Continuous Functions on Compact Sets................... 52 8. Compactness in Metric Spaces........................ 56 9. Local Compactness.............................. 59 IV.Separation Axioms 62 1. Regular Spaces................................ 62 2. Normal Spaces................................ 64 3. Tietze's extension Theorem......................... 67 4. Urysohn Metrization Theorem........................ 71 5. Imbedding of Manifolds.......................... -
General Topology
General Topology Tom Leinster 2014{15 Contents A Topological spaces2 A1 Review of metric spaces.......................2 A2 The definition of topological space.................8 A3 Metrics versus topologies....................... 13 A4 Continuous maps........................... 17 A5 When are two spaces homeomorphic?................ 22 A6 Topological properties........................ 26 A7 Bases................................. 28 A8 Closure and interior......................... 31 A9 Subspaces (new spaces from old, 1)................. 35 A10 Products (new spaces from old, 2)................. 39 A11 Quotients (new spaces from old, 3)................. 43 A12 Review of ChapterA......................... 48 B Compactness 51 B1 The definition of compactness.................... 51 B2 Closed bounded intervals are compact............... 55 B3 Compactness and subspaces..................... 56 B4 Compactness and products..................... 58 B5 The compact subsets of Rn ..................... 59 B6 Compactness and quotients (and images)............. 61 B7 Compact metric spaces........................ 64 C Connectedness 68 C1 The definition of connectedness................... 68 C2 Connected subsets of the real line.................. 72 C3 Path-connectedness.......................... 76 C4 Connected-components and path-components........... 80 1 Chapter A Topological spaces A1 Review of metric spaces For the lecture of Thursday, 18 September 2014 Almost everything in this section should have been covered in Honours Analysis, with the possible exception of some of the examples. For that reason, this lecture is longer than usual. Definition A1.1 Let X be a set. A metric on X is a function d: X × X ! [0; 1) with the following three properties: • d(x; y) = 0 () x = y, for x; y 2 X; • d(x; y) + d(y; z) ≥ d(x; z) for all x; y; z 2 X (triangle inequality); • d(x; y) = d(y; x) for all x; y 2 X (symmetry). -
And Free Cyclic Group Actions on Homotopy Spheres
TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 220, 1976 DECOMPOSABILITYOF HOMOTOPYLENS SPACES ANDFREE CYCLICGROUP ACTIONS ON HOMOTOPYSPHERES BY KAI WANG ABSTRACT. Let p be a linear Zn action on C and let p also denote the induced Z„ action on S2p~l x D2q, D2p x S2q~l and S2p~l x S2q~l " 1m_1 where p = [m/2] and q = m —p. A free differentiable Zn action (£ , ju) on a homotopy sphere is p-decomposable if there is an equivariant diffeomor- phism <t>of (S2p~l x S2q~l, p) such that (S2m_1, ju) is equivalent to (£(*), ¿(*)) where S(*) = S2p_1 x D2q U^, D2p x S2q~l and A(<P) is a uniquely determined action on S(*) such that i4(*)IS p~l XD q = p and A(Q)\D p X S = p. A homotopy lens space is p-decomposable if it is the orbit space of a p-decomposable free Zn action on a homotopy sphere. In this paper, we will study the decomposabilities of homotopy lens spaces. We will also prove that for each lens space L , there exist infinitely many inequivalent free Zn actions on S m such that the orbit spaces are simple homotopy equiva- lent to L 0. Introduction. Let A be the antipodal map and let $ be an equivariant diffeomorphism of (Sp x Sp, A) where A(x, y) = (-x, -y). Then there is a uniquely determined free involution A($) on 2(4>) where 2(4») = Sp x Dp+1 U<¡,Dp+l x Sp such that the inclusions S" x Dp+l —+ 2(d>), Dp+1 x Sp —*■2(4>) are equi- variant. -
Ball, Deborah Loewenberg Developing Mathematics
DOCUMENT RESUME ED 399 262 SP 036 932 AUTHOR Ball, Deborah Loewenberg TITLE Developing Mathematics Reform: What Don't We Know about Teacher Learning--But Would Make Good Working Hypotheses? NCRTL Craft Paper 95-4. INSTITUTION National Center for Research on Teacher Learning, East Lansing, MI. SPONS AGENCY Office of Educational Research and Improvement (ED), Washington, DC. PUB DATE Oct 95 NOTE 53p.; Paper presented at a conference on Teacher Enhancement in Mathematics K-6 (Arlington, VA, November 18-20, 1994). AVAILABLE FROMNational Center for Research on Teacher Learning, 116 Erickson Hall, Michigan State University, East Lansing, MI 48824-1034. PUB TYPE Reports Research/Technical (143) Speeches /Conference Papers (150) EDRS PRICE MF01/PC03 Plus Postage. DESCRIPTORS *Educational Change; *Educational Policy; Elementary School Mathematics; Elementary Secondary Education; *Faculty Development; *Knowledge Base for Teaching; *Mathematics Instruction; Mathematics Teachers; Secondary School Mathematics; *Teacher Improvement ABSTRACT This paper examines what teacher educators, policymakers, and teachers think they know about the current mathematics reforms and what it takes to help teachers engage with these reforms. The analysis is organized around three issues:(1) the "it" envisioned by the reforms;(2) what teachers (and others) bring to learning "it";(3) what is known and believed about teacher learning. The second part of the paper deals with what is not known about the reforms and how to help a larger number of teachers engage productively with the reforms. This section of the paper confronts the current pressures to "scale up" reform efforts to "reach" more teachers. Arguing that what is known is not sufficient to meet the demand for scaling up, three potential hypotheses about teacher learning and professional development are proposed that might serve to meet the demand to offer more teachers opportunities for learning. -
Simplices in the Euclidean Ball Matthieu Fradelizi, Grigoris Paouris, Carsten Schütt
Simplices in the Euclidean ball Matthieu Fradelizi, Grigoris Paouris, Carsten Schütt To cite this version: Matthieu Fradelizi, Grigoris Paouris, Carsten Schütt. Simplices in the Euclidean ball. Canadian Mathematical Bulletin, 2011, 55 (3), pp.498-508. 10.4153/CMB-2011-142-1. hal-00731269 HAL Id: hal-00731269 https://hal-upec-upem.archives-ouvertes.fr/hal-00731269 Submitted on 12 Sep 2012 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. Simplices in the Euclidean ball Matthieu Fradelizi, Grigoris Paouris,∗ Carsten Sch¨utt To appear in Canad. Math. Bull. Abstract We establish some inequalities for the second moment 1 2 |x|2dx |K| ZK of a convex body K under various assumptions on the position of K. 1 Introduction The starting point of this paper is the article [2], where it was shown that if all the extreme points of a convex body K in Rn have Euclidean norm greater than r > 0, then 1 r2 x 2dx > (1) K | |2 9n | | ZK where x 2 stands for the Euclidean norm of x and K for the volume of K. | | | | r2 We improve here this inequality showing that the optimal constant is , n + 2 with equality for the regular simplex, with vertices on the Euclidean sphere of radius r. -
Lecture Notes C Sarah Rasmussen, 2019
Part III 3-manifolds Lecture Notes c Sarah Rasmussen, 2019 Contents Lecture 0 (not lectured): Preliminaries2 Lecture 1: Why not ≥ 5?9 Lecture 2: Why 3-manifolds? + Introduction to knots and embeddings 13 Lecture 3: Link diagrams and Alexander polynomial skein relations 17 Lecture 4: Handle decompositions from Morse critical points 20 Lecture 5: Handles as Cells; Morse functions from handle decompositions 24 Lecture 6: Handle-bodies and Heegaard diagrams 28 Lecture 7: Fundamental group presentations from Heegaard diagrams 36 Lecture 8: Alexander polynomials from fundamental groups 39 Lecture 9: Fox calculus 43 Lecture 10: Dehn presentations and Kauffman states 48 Lecture 11: Mapping tori and Mapping Class Groups 54 Lecture 12: Nielsen-Thurston classification for mapping class groups 58 Lecture 13: Dehn filling 61 Lecture 14: Dehn surgery 64 Lecture 15: 3-manifolds from Dehn surgery 68 Lecture 16: Seifert fibered spaces 72 Lecture 17: Hyperbolic manifolds 76 Lecture 18: Embedded surface representatives 80 Lecture 19: Incompressible and essential surfaces 83 Lecture 20: Connected sum 86 Lecture 21: JSJ decomposition and geometrization 89 Lecture 22: Turaev torsion and knot decompositions 92 Lecture 23: Foliations 96 Lecture 24. Taut Foliations 98 Errata: Catalogue of errors/changes/addenda 102 References 106 1 2 Lecture 0 (not lectured): Preliminaries 0. Notation and conventions. Notation. @X { (the manifold given by) the boundary of X, for X a manifold with boundary. th @iX { the i connected component of @X. ν(X) { a tubular (or collared) neighborhood of X in Y , for an embedding X ⊂ Y . ◦ ν(X) { the interior of ν(X). This notation is somewhat redundant, but emphasises openness. -
01. Review of Metric Spaces and Point-Set Topology 1. Euclidean
(September 28, 2018) 01. Review of metric spaces and point-set topology Paul Garrett [email protected] http:=/www.math.umn.edu/egarrett/ [This document is http:=/www.math.umn.edu/egarrett/m/real/notes 2018-19/01 metric spaces and topology.pdf] 1. Euclidean spaces 2. Metric spaces 3. Completions of metric spaces 4. Topologies of metric spaces 5. General topological spaces 6. Compactness and sequential compactness 7. Total-boundedness criterion for compact closure 8. Baire's theorem 9. Appendix: mapping-property characterization of completions 1. Euclidean spaces n Let R be the usual Euclidean n-space, that is, ordered n-tuples x = (x1; : : : ; xn) of real numbers. In addition to vector addition (termwise) and scalar multiplication, we have the usual distance function on Rn, in coordinates x = (x1; : : : ; xn) and y = (y1; : : : ; yn), defined by p 2 2 d(x; y) = (x1 − y1) + ::: + (xn − yn) Of course there is visible symmetry d(x; y) = d(y; x), and positivity: d(x; y) = 0 only for x = y. The triangle inequality d(x; z) ≤ d(x; y) + d(y; z) is not trivial to prove. In the one-dimensional case, the triangle inequality is an inequality on absolute values, and can be proven case-by-case. In Rn, it is best to use the following set-up. The usual inner product (or dot-product) on Rn is x · y = hx; yi = h(x1; : : : ; xn); (y1; : : : ; yn)i = x1y1 + ::: + xnyn and jxj2 = hx; xi. Context distinguishes the norm jxj of x 2 Rn from the usual absolute value jcj on real or complex numbers c.