DOCSLIB.ORG

Homotopy Equivalences of 3-Manifolds and Deformation Theory of Kleinian Groups

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Homotopy Equivalences of 3-Manifolds and Deformation Theory of Kleinian Groups Homotopy Equivalences of 3-Manifolds and Deformation Theory of Kleinian Groups Richard D. Canary Darryl McCullough Author address: Department of Mathematics University of Michigan Ann Arbor, MI 48109 E-mail address: [email protected] URL: www.math.lsa.umich.edu/ canary/ e Department of Mathematics University of Oklahoma Norman, OK 73019 E-mail address: [email protected] URL: www.math.ou.edu/ dmccullough/ e Contents Preface ix Chapter 1. Introduction 1 1.1. Motivation 1 1.2. The main theorems for Haken 3-manifolds 3 1.3. The main theorems for reducible 3-manifolds 8 1.4. Examples 9 Chapter 2. Johannson's Characteristic Submanifold Theory 15 2.1. Fibered 3-manifolds 16 2.2. Boundary patterns 20 2.3. Admissible maps and mapping class groups 23 2.4. Essential maps and useful boundary patterns 28 2.5. The classical theorems 35 2.6. Exceptional fibered 3-manifolds 38 2.7. Vertical and horizontal surfaces and maps 39 2.8. Fiber-preserving maps 41 2.9. The characteristic submanifold 48 2.10. Examples of characteristic submanifolds 51 2.11. The Classification Theorem 57 2.12. Miscellaneous topological results 59 Chapter 3. Relative Compression Bodies and Cores 65 3.1. Relative compression bodies 66 3.2. Minimally imbedded relative compression bodies 69 3.3. The maximal incompressible core 71 3.4. Normally imbedded relative compression bodies 73 3.5. The normal core and the useful core 74 Chapter 4. Homotopy Types 77 4.1. Homotopy equivalences preserve usefulness 77 4.2. Finiteness of homotopy types 83 Chapter 5. Pared 3-Manifolds 87 5.1. Definitions and basic properties 87 5.2. The topology of pared manifolds 89 5.3. The characteristic submanifold of a pared manifold 92 Chapter 6. Small 3-Manifolds 97 6.1. Small manifolds and small pared manifolds 97 6.2. Small pared homotopy types 101 v vi CONTENTS Chapter 7. Geometrically Finite Hyperbolic 3-Manifolds 105 7.1. Basic definitions 105 7.2. Quasiconformal deformation theory: a review 107 7.3. The Parameterization Theorem 114 Chapter 8. Statements of Main Theorems 117 8.1. Statements of Main Topological Theorems 117 8.2. Statements of Main Hyperbolic Theorem and Corollary 118 8.3. Derivation of hyperbolic results 119 Chapter 9. The Case When There Is a Compressible Free Side 121 9.1. Algebraic lemmas 122 9.2. The finite-index cases 124 9.3. The infinite-index cases 132 Chapter 10. The Case When the Boundary Pattern Is Useful 139 10.1. The homomorphism Ψ 141 10.2. Realizing homotopy equivalences of I-bundles 153 10.3. Realizing homotopy equivalences of Seifert-fibered manifolds 161 10.4. Proof of Main Topological Theorem 2 171 Chapter 11. Dehn Flips 175 Chapter 12. Finite Index Realization For Reducible 3-Manifolds 179 12.1. Homeomorphisms of connected sums 180 12.2. Reducible 3-manifolds with compressible boundary 190 12.3. Reducible 3-manifolds with incompressible boundary 191 Chapter 13. Epilogue 195 13.1. More topology 195 13.2. More geometry 197 Bibliography 207 Index 213 Abstract This text investigates a natural question arising in the topological theory of 3-manifolds, and applies the results to give new information about the deformation theory of hyperbolic 3-manifolds. It is well known that some compact 3-manifolds with boundary admit homotopy equivalences that are not homotopic to homeo- morphisms. We investigate when the subgroup (M) of outer automorphisms of R π1(M) which are induced by homeomorphisms of a compact 3-manifold M has fi- nite index in the group Out(π1(M)) of all outer automorphisms. This question is completely resolved for Haken 3-manifolds. It is also resolved for many classes of reducible 3-manifolds and 3-manifolds with boundary patterns, including all pared 3-manifolds. The components of the interior GF(π1(M)) of the space AH(π1(M)) of all (marked) hyperbolic 3-manifolds homotopy equivalent to M are enumerated by the marked homeomorphism types of manifolds homotopy equivalent to M, so one may apply the topological results above to study the topology of this deformation space. We show that GF(π1(M)) has finitely many components if and only if either M has incompressible boundary, but no \double trouble," or M has compressible boundary and is \small." (A hyperbolizable 3-manifold with incompressible boundary has double trouble if and only if there is a thickened torus component of its characteristic submanifold which intersects the boundary in at least two annuli.) More generally, the deformation theory of hyperbolic structures on pared manifolds is analyzed. Some expository sections detail Johannson's formulation of the Jaco-Shalen- Johannson characteristic submanifold theory, the topology of pared 3-manifolds, and the deformation theory of hyperbolic 3-manifolds. An epilogue discusses re- lated open problems and recent progress in the deformation theory of hyperbolic 3-manifolds. Received by the editor February 6, 2001. 2000 Mathematics Subject Classification. Primary 57M99; Secondary 20F34, 30F40, 57M50. Key words and phrases. 3-manifold, hyperbolic, pared, reducible, convex-cocompact, geo- metrically finite, boundary pattern, Haken, reducible, characteristic submanifold, compression body, homotopy equivalence, homeomorphism, diffeomorphism, automorphism, double trouble. Supported in part by the National Science Foundation and the Sloan Foundation. Supported in part by the National Science Foundation. vii Preface This work addresses a question about homotopy equivalences and homeomor- phisms of 3-manifolds, and gives an application to the topology of deformation spaces of hyperbolic 3-manifolds. Although the topological question is quite natu- ral, it did not receive much attention until motivated by the geometric application. It is simply this: in the group of homotopy classes of self-homotopy-equivalences of a 3-manifold, when does the subgroup consisting of the classes that contain a homeomorphism have finite index? Most of our results will apply only to Haken 3-manifolds, although in chap- ter 12 we develop rather general versions of our theorems for reducible 3-manifolds. For closed Haken manifolds, Waldhausen proved that all homotopy classes con- tain homeomorphisms, so we need only consider Haken manifolds with nonempty boundary. This case breaks into two very different subcases, the manifolds with incompressible boundary and those with compressible boundary. We resolve the question completely, giving exact conditions for the subgroup of classes realizable by homeomorphisms to have finite index. In the case when the boundary is incompressible, our principal means to ex- amine homotopy equivalences and homeomorphisms of Haken 3-manifolds is the characteristic submanifold theory due independently to Jaco-Shalen and Johann- son. For our geometric application, we need to consider a more general version of the question, in which we work with maps that preserve certain submanifolds of the boundary. The formulation of the characteristic submanifold given by Johannson, with its elegant theory of boundary patterns, provides exactly the kind of control that we need. For this reason, almost all of our work is carried out in the context of manifolds with boundary patterns. A complete resolution of the realization ques- tion for Haken manifolds with boundary patterns seems out of reach, and our main results are restricted to the case when the submanifolds forming the boundary pat- tern are disjoint. But these do include as a very special case the boundary patterns which arise in the study of the deformation theory of hyperbolic 3-manifolds. The statements of the main results require a number of preliminary concepts. Consequently, they cannot conveniently be given until rather late in the develop- ment of our exposition, indeed not until chapter 8. To motivate so much preliminary work, we state and discuss restricted versions of the main results in chapter 1. There we lay out the general program and provide several examples illustrating some of the phenomena that can occur. Chapter 2 contains an exposition of Johannson's version of the characteristic submanifold theory. We review the concepts which underlie his main results, give a variety of examples of boundary patterns and characteristic submanifolds, and develop some technical results which we need for our later work. All of Johannson's theory takes place under the assumption that the 3-manifold satisfies a certain ix x PREFACE incompressibility condition on its boundary (namely, that the boundary pattern has the property of being useful). To work with manifolds that do not satisfy this condition (that is, to allow some boundary patterns that are not useful, in the technical sense), we use a second kind of characteristic structure, the characteristic compression body invented by Bonahon. For manifolds with boundary patterns, we need a relative version of his theory, which we develop in chapter 3. (Both the characteristic compression body and our relative version are generalizations of the Abikoff-Maskit decomposition which was developed earlier in the context of Kleinian groups.) With these topological concepts in place, we can prove in chapter 4 some results about homotopy equivalences and homotopy types of 3- manifolds with boundary patterns, which generalize theorems of Johannson and Swarup. Geometry enters, at least implicitly, in chapter 5, where we introduce the boundary patterns naturally associated to manifolds with hyperbolic structures. These are called pared structures. We review the definitions and some known results, including the strong restrictions on the structure of the characteristic sub- manifold associated to a pared structure. In chapter 6, we define small 3-manifolds. These arise as exceptional cases for our main topological results, in the case where the manifold has compressible boundary. We determine which of the small mani- folds have pared structures, and prove a technical result which lists the homotopy types all of whose homeomorphism classes are either compression bodies or small manifolds. The actual geometric theory which we use draws on work of many mathemati- cians, including Ahlfors, Bers, Kra, Marden, Maskit, Sullivan, and Thurston.
Load more

Recommended publications
  • Nieuw Archief Voor Wiskunde
    Nieuw Archief voor Wiskunde Boekbespreking Kevin Broughan Equivalents of the Riemann Hypothesis Volume 1: Arithmetic Equivalents Cambridge University Press, 2017 xx + 325 p., prijs £ 99.99 ISBN 9781107197046 Kevin Broughan Equivalents of the Riemann Hypothesis Volume 2: Analytic Equivalents Cambridge University Press, 2017 xix + 491 p., prijs £ 120.00 ISBN 9781107197121 Reviewed by Pieter Moree These two volumes give a survey of conjectures equivalent to the ber theorem says that r()x asymptotically behaves as xx/log . That Riemann Hypothesis (RH). The first volume deals largely with state- is a much weaker statement and is equivalent with there being no ments of an arithmetic nature, while the second part considers zeta zeros on the line v = 1. That there are no zeros with v > 1 is more analytic equivalents. a consequence of the prime product identity for g()s . The Riemann zeta function, is defined by It would go too far here to discuss all chapters and I will limit 3 myself to some chapters that are either close to my mathematical 1 (1) g()s = / s , expertise or those discussing some of the most famous RH equiv- n = 1 n alences. Most of the criteria have their own chapter devoted to with si=+v t a complex number having real part v > 1 . It is easily them, Chapter 10 has various criteria that are discussed more brief- seen to converge for such s. By analytic continuation the Riemann ly. A nice example is Redheffer’s criterion. It states that RH holds zeta function can be uniquely defined for all s ! 1.
    [Show full text]
  • Sheaves and Homotopy Theory
    SHEAVES AND HOMOTOPY THEORY DANIEL DUGGER The purpose of this note is to describe the homotopy-theoretic version of sheaf theory developed in the work of Thomason [14] and Jardine [7, 8, 9]; a few enhancements are provided here and there, but the bulk of the material should be credited to them. Their work is the foundation from which Morel and Voevodsky build their homotopy theory for schemes [12], and it is our hope that this exposition will be useful to those striving to understand that material. Our motivating examples will center on these applications to algebraic geometry. Some history: The machinery in question was invented by Thomason as the main tool in his proof of the Lichtenbaum-Quillen conjecture for Bott-periodic algebraic K-theory. He termed his constructions `hypercohomology spectra', and a detailed examination of their basic properties can be found in the first section of [14]. Jardine later showed how these ideas can be elegantly rephrased in terms of model categories (cf. [8], [9]). In this setting the hypercohomology construction is just a certain fibrant replacement functor. His papers convincingly demonstrate how many questions concerning algebraic K-theory or ´etale homotopy theory can be most naturally understood using the model category language. In this paper we set ourselves the specific task of developing some kind of homotopy theory for schemes. The hope is to demonstrate how Thomason's and Jardine's machinery can be built, step-by-step, so that it is precisely what is needed to solve the problems we encounter. The papers mentioned above all assume a familiarity with Grothendieck topologies and sheaf theory, and proceed to develop the homotopy-theoretic situation as a generalization of the classical case.
    [Show full text]
  • The Modal Logic of Potential Infinity, with an Application to Free Choice
    The Modal Logic of Potential Infinity, With an Application to Free Choice Sequences Dissertation Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University By Ethan Brauer, B.A. ∼6 6 Graduate Program in Philosophy The Ohio State University 2020 Dissertation Committee: Professor Stewart Shapiro, Co-adviser Professor Neil Tennant, Co-adviser Professor Chris Miller Professor Chris Pincock c Ethan Brauer, 2020 Abstract This dissertation is a study of potential infinity in mathematics and its contrast with actual infinity. Roughly, an actual infinity is a completed infinite totality. By contrast, a collection is potentially infinite when it is possible to expand it beyond any finite limit, despite not being a completed, actual infinite totality. The concept of potential infinity thus involves a notion of possibility. On this basis, recent progress has been made in giving an account of potential infinity using the resources of modal logic. Part I of this dissertation studies what the right modal logic is for reasoning about potential infinity. I begin Part I by rehearsing an argument|which is due to Linnebo and which I partially endorse|that the right modal logic is S4.2. Under this assumption, Linnebo has shown that a natural translation of non-modal first-order logic into modal first- order logic is sound and faithful. I argue that for the philosophical purposes at stake, the modal logic in question should be free and extend Linnebo's result to this setting. I then identify a limitation to the argument for S4.2 being the right modal logic for potential infinity.
    [Show full text]
  • The Boundary Manifold of a Complex Line Arrangement
    Geometry & Topology Monographs 13 (2008) 105–146 105 The boundary manifold of a complex line arrangement DANIEL CCOHEN ALEXANDER ISUCIU We study the topology of the boundary manifold of a line arrangement in CP2 , with emphasis on the fundamental group G and associated invariants. We determine the Alexander polynomial .G/, and more generally, the twisted Alexander polynomial associated to the abelianization of G and an arbitrary complex representation. We give an explicit description of the unit ball in the Alexander norm, and use it to analyze certain Bieri–Neumann–Strebel invariants of G . From the Alexander polynomial, we also obtain a complete description of the first characteristic variety of G . Comparing this with the corresponding resonance variety of the cohomology ring of G enables us to characterize those arrangements for which the boundary manifold is formal. 32S22; 57M27 For Fred Cohen on the occasion of his sixtieth birthday 1 Introduction 1.1 The boundary manifold Let A be an arrangement of hyperplanes in the complex projective space CPm , m > 1. Denote by V S H the corresponding hypersurface, and by X CPm V its D H A D n complement. Among2 the origins of the topological study of arrangements are seminal results of Arnol’d[2] and Cohen[8], who independently computed the cohomology of the configuration space of n ordered points in C, the complement of the braid arrangement. The cohomology ring of the complement of an arbitrary arrangement A is by now well known. It is isomorphic to the Orlik–Solomon algebra of A, see Orlik and Terao[34] as a general reference.
    [Show full text]
  • 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.
    [Show full text]
  • Graph Manifolds and Taut Foliations Mark Brittenham1, Ramin Naimi, and Rachel Roberts2 Introduction
    Graph manifolds and taut foliations 1 2 Mark Brittenham , Ramin Naimi, and Rachel Roberts UniversityofTexas at Austin Abstract. We examine the existence of foliations without Reeb comp onents, taut 1 1 foliations, and foliations with no S S -leaves, among graph manifolds. We show that each condition is strictly stronger than its predecessors, in the strongest p os- sible sense; there are manifolds admitting foliations of eachtyp e which do not admit foliations of the succeeding typ es. x0 Introduction Taut foliations have b een increasingly useful in understanding the top ology of 3-manifolds, thanks largely to the work of David Gabai [Ga1]. Many 3-manifolds admit taut foliations [Ro1],[De],[Na1], although some do not [Br1],[Cl]. To date, however, there are no adequate necessary or sucient conditions for a manifold to admit a taut foliation. This pap er seeks to add to this confusion. In this pap er we study the existence of taut foliations and various re nements, among graph manifolds. What we show is that there are many graph manifolds which admit foliations that are as re ned as wecho ose, but which do not admit foliations admitting any further re nements. For example, we nd manifolds which admit foliations without Reeb comp onents, but no taut foliations. We also nd 0 manifolds admitting C foliations with no compact leaves, but which do not admit 2 any C such foliations. These results p oint to the subtle nature b ehind b oth top ological and analytical assumptions when dealing with foliations. A principal motivation for this work came from a particularly interesting exam- ple; the manifold M obtained by 37/2 Dehn surgery on the 2,3,7 pretzel knot K .
    [Show full text]
  • 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.
    [Show full text]
  • 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..........................
    [Show full text]
  • A TEXTBOOK of TOPOLOGY Lltld
    SEIFERT AND THRELFALL: A TEXTBOOK OF TOPOLOGY lltld SEI FER T: 7'0PO 1.OG 1' 0 I.' 3- Dl M E N SI 0 N A I. FIRERED SPACES This is a volume in PURE AND APPLIED MATHEMATICS A Series of Monographs and Textbooks Editors: SAMUELEILENBERG AND HYMANBASS A list of recent titles in this series appears at the end of this volunie. SEIFERT AND THRELFALL: A TEXTBOOK OF TOPOLOGY H. SEIFERT and W. THRELFALL Translated by Michael A. Goldman und S E I FE R T: TOPOLOGY OF 3-DIMENSIONAL FIBERED SPACES H. SEIFERT Translated by Wolfgang Heil Edited by Joan S. Birman and Julian Eisner @ 1980 ACADEMIC PRESS A Subsidiary of Harcourr Brace Jovanovich, Publishers NEW YORK LONDON TORONTO SYDNEY SAN FRANCISCO COPYRIGHT@ 1980, BY ACADEMICPRESS, INC. ALL RIGHTS RESERVED. NO PART OF THIS PUBLICATION MAY BE REPRODUCED OR TRANSMITTED IN ANY FORM OR BY ANY MEANS, ELECTRONIC OR MECHANICAL, INCLUDING PHOTOCOPY, RECORDING, OR ANY INFORMATION STORAGE AND RETRIEVAL SYSTEM, WITHOUT PERMISSION IN WRITING FROM THE PUBLISHER. ACADEMIC PRESS, INC. 11 1 Fifth Avenue, New York. New York 10003 United Kingdom Edition published by ACADEMIC PRESS, INC. (LONDON) LTD. 24/28 Oval Road, London NWI 7DX Mit Genehmigung des Verlager B. G. Teubner, Stuttgart, veranstaltete, akin autorisierte englische Ubersetzung, der deutschen Originalausgdbe. Library of Congress Cataloging in Publication Data Seifert, Herbert, 1897- Seifert and Threlfall: A textbook of topology. Seifert: Topology of 3-dimensional fibered spaces. (Pure and applied mathematics, a series of mono- graphs and textbooks ; ) Translation of Lehrbuch der Topologic. Bibliography: p. Includes index. 1.
    [Show full text]
  • 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).
    [Show full text]
  • Floer Homology, Gauge Theory, and Low-Dimensional Topology
    Floer Homology, Gauge Theory, and Low-Dimensional Topology Clay Mathematics Proceedings Volume 5 Floer Homology, Gauge Theory, and Low-Dimensional Topology Proceedings of the Clay Mathematics Institute 2004 Summer School Alfréd Rényi Institute of Mathematics Budapest, Hungary June 5–26, 2004 David A. Ellwood Peter S. Ozsváth András I. Stipsicz Zoltán Szabó Editors American Mathematical Society Clay Mathematics Institute 2000 Mathematics Subject Classification. Primary 57R17, 57R55, 57R57, 57R58, 53D05, 53D40, 57M27, 14J26. The cover illustrates a Kinoshita-Terasaka knot (a knot with trivial Alexander polyno- mial), and two Kauffman states. These states represent the two generators of the Heegaard Floer homology of the knot in its topmost filtration level. The fact that these elements are homologically non-trivial can be used to show that the Seifert genus of this knot is two, a result first proved by David Gabai. Library of Congress Cataloging-in-Publication Data Clay Mathematics Institute. Summer School (2004 : Budapest, Hungary) Floer homology, gauge theory, and low-dimensional topology : proceedings of the Clay Mathe- matics Institute 2004 Summer School, Alfr´ed R´enyi Institute of Mathematics, Budapest, Hungary, June 5–26, 2004 / David A. Ellwood ...[et al.], editors. p. cm. — (Clay mathematics proceedings, ISSN 1534-6455 ; v. 5) ISBN 0-8218-3845-8 (alk. paper) 1. Low-dimensional topology—Congresses. 2. Symplectic geometry—Congresses. 3. Homol- ogy theory—Congresses. 4. Gauge fields (Physics)—Congresses. I. Ellwood, D. (David), 1966– II. Title. III. Series. QA612.14.C55 2004 514.22—dc22 2006042815 Copying and reprinting. Material in this book may be reproduced by any means for educa- tional and scientific purposes without fee or permission with the exception of reproduction by ser- vices that collect fees for delivery of documents and provided that the customary acknowledgment of the source is given.
    [Show full text]
  • Enumeration of Non-Orientable 3-Manifolds Using Face Pairing Graphs and Union-Find∗
    Enumeration of non-orientable 3-manifolds using face pairing graphs and union-find∗ Benjamin A. Burton Author’s self-archived version Available from http://www.maths.uq.edu.au/~bab/papers/ Abstract Drawing together techniques from combinatorics and computer science, we improve the census algorithm for enumerating closed minimal P2-irreducible 3-manifold triangulations. In particular, new constraints are proven for face pairing graphs, and pruning techniques are improved using a modification of the union-find algorithm. Using these results we catalogue all 136 closed non-orientable P2-irreducible 3-manifolds that can be formed from at most ten tetrahedra. 1 Introduction With recent advances in computing power, topologists have been able to construct exhaustive tables of small 3-manifold triangulations, much like knot theorists have constructed exhaustive tables of simple knot projections. Such tables are valuable sources of data, but they suffer from the fact that enormous amounts of computer time are required to generate them. Where knot tables are often limited by bounding the number of crossings in a knot pro- jection, tables of 3-manifolds generally limit the number of tetrahedra used in a 3-manifold triangulation. A typical table (or census) of 3-manifolds lists all 3-manifolds of a particular type that can be formed from n tetrahedra or fewer. Beyond their generic role as a rich source of examples, tables of this form have a number of specific uses. They still offer the only general means for proving that a triangulation is minimal (i.e., uses as few tetrahedra as possible), much in the same way as knot tables are used to calculate crossing number.
    [Show full text]