DOCSLIB.ORG

A TEXTBOOK of TOPOLOGY Lltld

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read 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. Topology. 1. Threlfall, William, 1888- joint author. 11. Birman, Joan S., 1927- 111. Eisner, Julian. IV. Title. V. Title: Text- book of topology. VI. Series. QA3.P8 [QA611] 510s [514] 79-28163 ISBN 0-12-634850-2 PRINTED IN THE UNITED STATES OF AMERICA 80 81 82 83 9 8 7 6 5 4 3 2 1 CON TENTS ix Preface to English Edition ... Acknowledgments Xlll Preface to German Edition xv SEIFERT AND THRELFALL: A TEXTBOOK OF TOPOLOGY CHAITER ONE ILLUSTRATIVE MATERIAL 1. The Principal Problem of Topology i 2. Closed Surfaces 5 3. Isotopy, Homotopy, Homology 14 4. Higher Dimensional Manifolds 16 CHAF'TER TWO SlMPLIClAL COMPLEXES 5. Neighborhood Spaces 22 6. Mappings 25 7. Point Sets in Euclidean Spaces 30 8. Identification Spaces 33 9. n-Simplexes 31 10. Simplicia1 Complexes 43 11. The Schema of a Simplicial Complex 45 12. Finite, Pure, Homogeneous Complexes 48 13. Normal Subdivision 50 14. Examples of Complexes 52 V vi CON TENTS CHAPTER THREE HOMOLOGY GROUPS 15. Chains 60 16. Boundary, Closed Chains 61 17. Homologous Chains 63 18. Homology Groups 66 19. Computation of the Homology Groups in Simple Cases 68 20. Homologies with Division 71 21. Computation of Homology Groups from the Incidence Matrices 73 22. Block Chains 80 23. Chains mod 2, Connectivity Numbers, Euler’s Formula 83 24. Pseudomanifolds and Orientability 90 CHAPTER FOUR SIMPLICIAL APPROXIMATIONS 25. Singular Simplexes 95 26. Singular Chains 97 27. Singular Homology Groups 98 28. The Approximation Theorem, Invariance of Simplicia1 Homology Groups 102 29. Prisms in Euclidean Spaces 103 30. Proof of the Approximation Theorem 107 3 I. Deformation and Simplicial Approximation of Mappings I I5 CHAPTER FIVE LOCAL PROPERTIES 32. Homology Groups of a Complex at a Point 123 33. Invariance of Dimension 129 34. Invariance of the Purity of a Complex 129 35. Invariance of Boundary I3 1 36. Invariance of Pseudomanifolds and of Orientability 132 CHAPTER SIX SURFACE TOPOLOGY 37. Closed Surfaces 134 38. Transformation to Normal Form 139 39. Types of Normal Form: The Principal Theorem 145 40. Surfaces with Boundary 146 41. Homology Groups of Surfaces 149 CHAPTER SEVEN THE FUNDAMENTAL GROUP 42. The Fundamental Group 154 43. Examples 161 C'ONT'ENTS vii 44. The Edge Path Group of a Simplicia1 Complex i63 45. The Edge Path Group of a Surface Complex 167 46. Generators and Relations 171 47. Edge Complexes and Closed Surfaces 174 48. The Fundamental and Homology Groups 117 49. Free Deformation of Closed Paths 180 50. Fundamental Group and Deformation of Mappings 182 51. The Fundamental Group at a Point 182 52. The Fundamental Group of a Composite Complex 183 CHAITER EIGHT COVERING COMPLEXES 53. Unbranched Covering Complexes 188 54. Base Path and Covering Path 191 55. Coverings and Subgroups of the Fundamental Group 195 56. Universal Coverings 200 57. Regular Coverings 20 1 58. The Monodromy Group 205 CHAITER NINE 3-DIMENSIONAL MANIFOLDS 59. General Principles 21 I 60. Representation by a Polyhedron 213 61. Homology Groups 218 62. The Fundamental Group 22 1 63. The Heegaard Diagram 226 64. 3-Dimensional Manifolds with Boundary 229 65. Construction of 3-Dimensional Manifolds out of Knots 23 1 CHAITER TEN n-DIMENSIONAL MANIFOLDS 66. Star Complexes 235 67. Cell Complexes 24 1 68. Manifolds 244 69. The Poincare Duality Theorem 250 70. Intersection Numbers of Cell Chains 255 71. Dual Bases 258 72. Cellular Approximations 263 73. Intersection Numbers of Singular Chains 267 74. lnvariance of Intersection Numbers 269 75. Examples 279 76. Orientability and Two-Sidedness 283 77. Linking Numbers 288 viii CON TENTS CHAPTER ELEVEN CONTINUOUS MAPPINGS 78. The Degree of a Mappings 294 79. A Trace Formula 296 80. A Fixed Point Formula 300 8 1. Applications 301 CHAPTER TWELVE AUXILIARY THEOREMS FROM THE THEORY OF GROUPS 82. Generators and Relations 305 83. Homomorphic Mappings and Factor Groups 309 84. Abelianization of Groups 312 85. Free and Direct Products 313 86. Abelian Groups 316 87. The Normal Form of Integer Matrices 323 COMMENTS 328 BIBLIOGRAPHY 341 SEIFERT: TOPOLOGY OF 3-DIMENSIONAL FIBERED SPACES I. Fibered Spaces 360 2. Orbit Surface 364 3. Fiberings of S3 368 4. Triangulations of Fibered Spaces 370 5. Drilling and Filling (Surgery) 372 6. Classes of Fibered Spaces 378 7. The Orientable Fibered Spaces 383 8. The Nonorientable Fibered Spaces 389 9. Covering Spaces 395 10. Fundamental Groups of Fibered Spaces 398 I I. Fiberings of the 3-Sphere (Complete List) 40 1 12. The Fibered Poincare Spaces 402 13. Constructing Poincar; Spaces from Torus Knots 405 14. Translation Groups of Fibered Spaces 406 15. Spaces Which Cannot Be Fibered 413 Appendix. Branched Coverings 418 Index to "A Textbook of Topology" 423 PREFACE TO ENGLISH EDITION The first German edition of Seifert and Threlfall’s “Lehrbuch der Topologie” was published in 1934. The book very quickly became the leading introductory textbook for students of geometric-algebraic topology (as distinguished from point set or “general” topology), a position which it held for possibly 30 to 35 years, during which time it was translated into Russian, Chinese, and Spanish. An English language edition is, then, long overdue. The translation presented here is due to Michael A. Goldman. In spite of the fact that with the passage of time our understanding of the subject matter has changed enormously (this is particularly true with regard to homology theory) the book continues to be of interest for its geometric insight and leisurely, careful presentation with its many beautiful examples which convey so well to the student the flavor of the subject. In fact, a quick perusal of the more successful modern textbooks aimed at advanced under- graduates or beginning graduate students reveals, inevitably, large blocks of material which appear to have been inspired by if not directly modeled on sections of this book. For example: the introductory pages on the problems of topology, the classification of surfaces, the discussion of incidence matrices and of methods for bringing them to normal form, the chapter on 3-dimensional manifolds (in particular the discussion of lens spaces), the section on intersection theory, and especially the notes at the end of the text have withstood the test of time and are as useful and readable today as they no doubt were in 1934. This volume contains, in addition to Seifert and Threlfall’s book, a translation into English, by Wolfgang Heil, of Seifert’s foundational research paper “The topology of 3-dimensional fibered spaces” [“Topologie dreidi- mensionales gefaserter Raum,” Acta Mathematics 60, 147-288 (1933)l. The manuscript treats a simple and beautiful question: what kinds of 3-dimensional manifolds can be made up as unions of disjoint circles, put together nicely? Using practically no technical machinery, Seifert takes hold ix X PREFACE TO ENGLISH EDITION of the question, lays down a set of ground rules, and with exceptional clarity of thought and particular attention to detail weaves a theory which culminates in the complete classification of his “fibered spaces” up to fiber-preserving homeomorphism. The manuscript is a model of mathematical exposition, and on these grounds alone it is worthy of translation, preservation, and study by new generations of mathematicians. But the primary reason for publishing this translation is, in our minds, because the ideas in the paper have proved to be at the origins of so many different flourishing areas of current research. We give a brief review. The question that Seifert asks and answers is not an idle question. The first paragraph of the paper shows that Seifert understood well the crucial role that the manifolds which have since come to be known as Seifert fibered spaces were likely to play in the classification problem for compact 3- manifolds. In fact it could be said that Seifert anticipated the theory of geometric structures on 3-manifolds, as it is evolving at this writing. Although Seifert only classified his fibered spaces up to fiber-preserving homeomorphism, the classification has turned out (with some exceptions) to be topological (Orlik, Vogt, and Zieschang [lo]). This classification was generalized by Waldhausen [ 1 I] to a related classification of “graph manifolds” which are Seifert fiber spaces (now with nonempty torus boundaries) pasted together along their boundaries, much as one pastes together 3-manifolds along 2-spheres in the boundary to obtain connected sums.
Load more

Recommended publications
  • ON Hg-HOMEOMORPHISMS in TOPOLOGICAL SPACES
    Proyecciones Vol. 28, No 1, pp. 1—19, May 2009. Universidad Cat´olica del Norte Antofagasta - Chile ON g-HOMEOMORPHISMS IN TOPOLOGICAL SPACES e M. CALDAS UNIVERSIDADE FEDERAL FLUMINENSE, BRASIL S. JAFARI COLLEGE OF VESTSJAELLAND SOUTH, DENMARK N. RAJESH PONNAIYAH RAMAJAYAM COLLEGE, INDIA and M. L. THIVAGAR ARUL ANANDHAR COLLEGE Received : August 2006. Accepted : November 2008 Abstract In this paper, we first introduce a new class of closed map called g-closed map. Moreover, we introduce a new class of homeomor- phism called g-homeomorphism, which are weaker than homeomor- phism.e We prove that gc-homeomorphism and g-homeomorphism are independent.e We also introduce g*-homeomorphisms and prove that the set of all g*-homeomorphisms forms a groupe under the operation of composition of maps. e e 2000 Math. Subject Classification : 54A05, 54C08. Keywords and phrases : g-closed set, g-open set, g-continuous function, g-irresolute map. e e e e 2 M. Caldas, S. Jafari, N. Rajesh and M. L. Thivagar 1. Introduction The notion homeomorphism plays a very important role in topology. By definition, a homeomorphism between two topological spaces X and Y is a 1 bijective map f : X Y when both f and f − are continuous. It is well known that as J¨anich→ [[5], p.13] says correctly: homeomorphisms play the same role in topology that linear isomorphisms play in linear algebra, or that biholomorphic maps play in function theory, or group isomorphisms in group theory, or isometries in Riemannian geometry. In the course of generalizations of the notion of homeomorphism, Maki et al.
    [Show full text]
  • The Fundamental Group
    The Fundamental Group Tyrone Cutler July 9, 2020 Contents 1 Where do Homotopy Groups Come From? 1 2 The Fundamental Group 3 3 Methods of Computation 8 3.1 Covering Spaces . 8 3.2 The Seifert-van Kampen Theorem . 10 1 Where do Homotopy Groups Come From? 0 Working in the based category T op∗, a `point' of a space X is a map S ! X. Unfortunately, 0 the set T op∗(S ;X) of points of X determines no topological information about the space. The same is true in the homotopy category. The set of `points' of X in this case is the set 0 π0X = [S ;X] = [∗;X]0 (1.1) of its path components. As expected, this pointed set is a very coarse invariant of the pointed homotopy type of X. How might we squeeze out some more useful information from it? 0 One approach is to back up a step and return to the set T op∗(S ;X) before quotienting out the homotopy relation. As we saw in the first lecture, there is extra information in this set in the form of track homotopies which is discarded upon passage to [S0;X]. Recall our slogan: it matters not only that a map is null homotopic, but also the manner in which it becomes so. So, taking a cue from algebraic geometry, let us try to understand the automorphism group of the zero map S0 ! ∗ ! X with regards to this extra structure. If we vary the basepoint of X across all its points, maybe it could be possible to detect information not visible on the level of π0.
    [Show full text]
  • Homological Algebra
    Homological Algebra Donu Arapura April 1, 2020 Contents 1 Some module theory3 1.1 Modules................................3 1.6 Projective modules..........................5 1.12 Projective modules versus free modules..............7 1.15 Injective modules...........................8 1.21 Tensor products............................9 2 Homology 13 2.1 Simplicial complexes......................... 13 2.8 Complexes............................... 15 2.15 Homotopy............................... 18 2.23 Mapping cones............................ 19 3 Ext groups 21 3.1 Extensions............................... 21 3.11 Projective resolutions........................ 24 3.16 Higher Ext groups.......................... 26 3.22 Characterization of projectives and injectives........... 28 4 Cohomology of groups 32 4.1 Group cohomology.......................... 32 4.6 Bar resolution............................. 33 4.11 Low degree cohomology....................... 34 4.16 Applications to finite groups..................... 36 4.20 Topological interpretation...................... 38 5 Derived Functors and Tor 39 5.1 Abelian categories.......................... 39 5.13 Derived functors........................... 41 5.23 Tor functors.............................. 44 5.28 Homology of a group......................... 45 1 6 Further techniques 47 6.1 Double complexes........................... 47 6.7 Koszul complexes........................... 49 7 Applications to commutative algebra 52 7.1 Global dimensions.......................... 52 7.9 Global dimension of
    [Show full text]
  • Algebra I Chapter 1. Basic Facts from Set Theory 1.1 Glossary of Abbreviations
    Notes: c F.P. Greenleaf, 2000-2014 v43-s14sets.tex (version 1/1/2014) Algebra I Chapter 1. Basic Facts from Set Theory 1.1 Glossary of abbreviations. Below we list some standard math symbols that will be used as shorthand abbreviations throughout this course. means “for all; for every” • ∀ means “there exists (at least one)” • ∃ ! means “there exists exactly one” • ∃ s.t. means “such that” • = means “implies” • ⇒ means “if and only if” • ⇐⇒ x A means “the point x belongs to a set A;” x / A means “x is not in A” • ∈ ∈ N denotes the set of natural numbers (counting numbers) 1, 2, 3, • · · · Z denotes the set of all integers (positive, negative or zero) • Q denotes the set of rational numbers • R denotes the set of real numbers • C denotes the set of complex numbers • x A : P (x) If A is a set, this denotes the subset of elements x in A such that •statement { ∈ P (x)} is true. As examples of the last notation for specifying subsets: x R : x2 +1 2 = ( , 1] [1, ) { ∈ ≥ } −∞ − ∪ ∞ x R : x2 +1=0 = { ∈ } ∅ z C : z2 +1=0 = +i, i where i = √ 1 { ∈ } { − } − 1.2 Basic facts from set theory. Next we review the basic definitions and notations of set theory, which will be used throughout our discussions of algebra. denotes the empty set, the set with nothing in it • ∅ x A means that the point x belongs to a set A, or that x is an element of A. • ∈ A B means A is a subset of B – i.e.
    [Show full text]
  • Basic Properties of Filter Convergence Spaces
    Basic Properties of Filter Convergence Spaces Barbel¨ M. R. Stadlery, Peter F. Stadlery;z;∗ yInstitut fur¨ Theoretische Chemie, Universit¨at Wien, W¨ahringerstraße 17, A-1090 Wien, Austria zThe Santa Fe Institute, 1399 Hyde Park Road, Santa Fe, NM 87501, USA ∗Address for corresponce Abstract. This technical report summarized facts from the basic theory of filter convergence spaces and gives detailed proofs for them. Many of the results collected here are well known for various types of spaces. We have made no attempt to find the original proofs. 1. Introduction Mathematical notions such as convergence, continuity, and separation are, at textbook level, usually associated with topological spaces. It is possible, however, to introduce them in a much more abstract way, based on axioms for convergence instead of neighborhood. This approach was explored in seminal work by Choquet [4], Hausdorff [12], Katˇetov [14], Kent [16], and others. Here we give a brief introduction to this line of reasoning. While the material is well known to specialists it does not seem to be easily accessible to non-topologists. In some cases we include proofs of elementary facts for two reasons: (i) The most basic facts are quoted without proofs in research papers, and (ii) the proofs may serve as examples to see the rather abstract formalism at work. 2. Sets and Filters Let X be a set, P(X) its power set, and H ⊆ P(X). The we define H∗ = fA ⊆ Xj(X n A) 2= Hg (1) H# = fA ⊆ Xj8Q 2 H : A \ Q =6 ;g The set systems H∗ and H# are called the conjugate and the grill of H, respectively.
    [Show full text]
  • Topological Paths, Cycles and Spanning Trees in Infinite Graphs
    Topological Paths, Cycles and Spanning Trees in Infinite Graphs Reinhard Diestel Daniela K¨uhn Abstract We study topological versions of paths, cycles and spanning trees in in- finite graphs with ends that allow more comprehensive generalizations of finite results than their standard notions. For some graphs it turns out that best results are obtained not for the standard space consist- ing of the graph and all its ends, but for one where only its topological ends are added as new points, while rays from other ends are made to converge to certain vertices. 1Introduction This paper is part of an on-going project in which we seek to explore how standard facts about paths and cycles in finite graphs can best be generalized to infinite graphs. The basic idea is that such generalizations can, and should, involve the ends of an infinite graph on a par with its other points (vertices or inner points of edges), both as endpoints of paths and as inner points of paths or cycles. To implement this idea we define paths and cycles topologically: in the space G consisting of a graph G together with its ends, we consider arcs (homeomorphic images of [0, 1]) instead of paths, and circles (homeomorphic images of the unit circle) instead of cycles. The topological version of a spanning tree, then, will be a path-connected subset of G that contains its vertices and ends but does not contain any circles. Let us look at an example. The double ladder L shown in Figure 1 has two ends, and its two sides (the top double ray R and the bottom double ray Q) will form a circle D with these ends: in the standard topology on L (to be defined later), every left-going ray converges to ω, while every right- going ray converges to ω.
    [Show full text]
  • Basic Topologytaken From
    Notes by Tamal K. Dey, OSU 1 Basic Topology taken from [1] 1 Metric space topology We introduce basic notions from point set topology. These notions are prerequisites for more sophisticated topological ideas—manifolds, homeomorphism, and isotopy—introduced later to study algorithms for topological data analysis. To a layman, the word topology evokes visions of “rubber-sheet topology”: the idea that if you bend and stretch a sheet of rubber, it changes shape but always preserves the underlying structure of how it is connected to itself. Homeomorphisms offer a rigorous way to state that an operation preserves the topology of a domain, and isotopy offers a rigorous way to state that the domain can be deformed into a shape without ever colliding with itself. Topology begins with a set T of points—perhaps the points comprising the d-dimensional Euclidean space Rd, or perhaps the points on the surface of a volume such as a coffee mug. We suppose that there is a metric d(p, q) that specifies the scalar distance between every pair of points p, q ∈ T. In the Euclidean space Rd we choose the Euclidean distance. On the surface of the coffee mug, we could choose the Euclidean distance too; alternatively, we could choose the geodesic distance, namely the length of the shortest path from p to q on the mug’s surface. d Let us briefly review the Euclidean metric. We write points in R as p = (p1, p2,..., pd), d where each pi is a real-valued coordinate. The Euclidean inner product of two points p, q ∈ R is d Rd 1/2 d 2 1/2 hp, qi = Pi=1 piqi.
    [Show full text]
  • Topology and Data
    BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY Volume 46, Number 2, April 2009, Pages 255–308 S 0273-0979(09)01249-X Article electronically published on January 29, 2009 TOPOLOGY AND DATA GUNNAR CARLSSON 1. Introduction An important feature of modern science and engineering is that data of various kinds is being produced at an unprecedented rate. This is so in part because of new experimental methods, and in part because of the increase in the availability of high powered computing technology. It is also clear that the nature of the data we are obtaining is significantly different. For example, it is now often the case that we are given data in the form of very long vectors, where all but a few of the coordinates turn out to be irrelevant to the questions of interest, and further that we don’t necessarily know which coordinates are the interesting ones. A related fact is that the data is often very high-dimensional, which severely restricts our ability to visualize it. The data obtained is also often much noisier than in the past and has more missing information (missing data). This is particularly so in the case of biological data, particularly high throughput data from microarray or other sources. Our ability to analyze this data, both in terms of quantity and the nature of the data, is clearly not keeping pace with the data being produced. In this paper, we will discuss how geometry and topology can be applied to make useful contributions to the analysis of various kinds of data.
    [Show full text]
  • 07. Homeomorphisms and Embeddings
    07. Homeomorphisms and embeddings Homeomorphisms are the isomorphisms in the category of topological spaces and continuous functions. Definition. 1) A function f :(X; τ) ! (Y; σ) is called a homeomorphism between (X; τ) and (Y; σ) if f is bijective, continuous and the inverse function f −1 :(Y; σ) ! (X; τ) is continuous. 2) (X; τ) is called homeomorphic to (Y; σ) if there is a homeomorphism f : X ! Y . Remarks. 1) Being homeomorphic is apparently an equivalence relation on the class of all topological spaces. 2) It is a fundamental task in General Topology to decide whether two spaces are homeomorphic or not. 3) Homeomorphic spaces (X; τ) and (Y; σ) cannot be distinguished with respect to their topological structure because a homeomorphism f : X ! Y not only is a bijection between the elements of X and Y but also yields a bijection between the topologies via O 7! f(O). Hence a property whose definition relies only on set theoretic notions and the concept of an open set holds if and only if it holds in each homeomorphic space. Such properties are called topological properties. For example, ”first countable" is a topological property, whereas the notion of a bounded subset in R is not a topological property. R ! − t Example. The function f : ( 1; 1) with f(t) = 1+jtj is a −1 x homeomorphism, the inverse function is f (x) = 1−|xj . 1 − ! b−a a+b The function g :( 1; 1) (a; b) with g(x) = 2 x + 2 is a homeo- morphism. Therefore all open intervals in R are homeomorphic to each other and homeomorphic to R .
    [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]
  • The Fundamental Group and Seifert-Van Kampen's
    THE FUNDAMENTAL GROUP AND SEIFERT-VAN KAMPEN'S THEOREM KATHERINE GALLAGHER Abstract. The fundamental group is an essential tool for studying a topo- logical space since it provides us with information about the basic shape of the space. In this paper, we will introduce the notion of free products and free groups in order to understand Seifert-van Kampen's Theorem, which will prove to be a useful tool in computing fundamental groups. Contents 1. Introduction 1 2. Background Definitions and Facts 2 3. Free Groups and Free Products 4 4. Seifert-van Kampen Theorem 6 Acknowledgments 12 References 12 1. Introduction One of the fundamental questions in topology is whether two topological spaces are homeomorphic or not. To show that two topological spaces are homeomorphic, one must construct a continuous function from one space to the other having a continuous inverse. To show that two topological spaces are not homeomorphic, one must show there does not exist a continuous function with a continuous inverse. Both of these tasks can be quite difficult as the recently proved Poincar´econjecture suggests. The conjecture is about the existence of a homeomorphism between two spaces, and it took over 100 years to prove. Since the task of showing whether or not two spaces are homeomorphic can be difficult, mathematicians have developed other ways to solve this problem. One way to solve this problem is to find a topological property that holds for one space but not the other, e.g. the first space is metrizable but the second is not. Since many spaces are similar in many ways but not homeomorphic, mathematicians use a weaker notion of equivalence between spaces { that of homotopy equivalence.
    [Show full text]
  • Topologically Rigid (Surgery/Geodesic Flow/Homeomorphism Group/Marked Leaves/Foliated Control) F
    Proc. Natl. Acad. Sci. USA Vol. 86, pp. 3461-3463, May 1989 Mathematics Compact negatively curved manifolds (of dim # 3, 4) are topologically rigid (surgery/geodesic flow/homeomorphism group/marked leaves/foliated control) F. T. FARRELLt AND L. E. JONESt tDepartment of Mathematics, Columbia University, New York, NY 10027; and tDepartment of Mathematics, State University of New York, Stony Brook, NY 11794 Communicated by William Browder, January 27, 1989 (receivedfor review October 20, 1988) ABSTRACT Let M be a complete (connected) Riemannian COROLLARY 1.3. If M and N are both negatively curved, manifold having finite volume and whose sectional curvatures compact (connected) Riemannian manifolds with isomorphic lie in the interval [cl, c2d with -x < cl c2 < 0. Then any fundamental groups, then M and N are homeomorphic proper homotopy equivalence h:N -* M from a topological provided dim M 4 3 and 4. manifold N is properly homotopic to a homeomorphism, Gromov had previously shown (under the hypotheses of provided the dimension of M is >5. In particular, if M and N Corollary 1.3) that the total spaces of the tangent sphere are both compact (connected) negatively curved Riemannian bundles ofM and N are homeomorphic (even when dim M = manifolds with isomorphic fundamental groups, then M and N 3 or 4) by a homeomorphism preserving the orbits of the are homeomorphic provided dim M # 3 and 4. {If both are geodesic flows, and Cheeger had (even earlier) shown that locally symmetric, this is a consequence of Mostow's rigidity the total spaces of the two-frame bundles of M and N are theorem [Mostow, G.
    [Show full text]