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~~tJbER STUD~ ~l~'6~ C. P. Rourke B. J. Sanderson Introduction to Piecewise-Linear Topology With 58 Figures Springer-Verlag Berlin Heidelberg New York 1982 Colin P. Rourke . Brian J. Sanderson The University of Warwick, England Revised printing of Ergebnisse der Mathematik' und ihrer Grenzgebiete, Vol. 69,1972 AMS Subject Classifications (1970): Primary 57 A XX, 57 C XX, 57 D XX Secondary 50 XX, 52 XX, 53 XX, 54 XX, 55 XX, 57 XX, 58 XX ISBN- 13 : 978-3-540-11102-3 e-ISBN- 13 : 978-3-642-81735-9 001: 10.1007/978-3-642-81735-9 Library of Congress Cataloging in Publication Data Rourke, C. P. (Colin Patrick), 1943- Introduction to piecewise-linear topology. "Springer study edition". Bibliography: p. Includes index. , 1. Piecewise linear topology. 2. Manifolds (Mathematics). 3. Differential topology. I. Sanderson, B. J. (Brian Joseph), 1939-. II. Title. QA613.4.R68 1982 514'.2 81-18311 ISBN-13:978-3-540-11102-3 (U.S.: pbk.) AACR2 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to Verwertungsgesellschaft Wort, Munich. @) by Springer-Verlag Berlin Heidelberg 1972, 1982 214113140-543210 Preface The first five chapters of this book form an introductory course in piece wise-linear topology in which no assumptions are made other than basic topological notions. This course would be suitable as a second course in topology with a geometric flavour, to follow a first course in point-set topology, andi)erhaps to be given as a final year undergraduate course. The whole book gives an account of handle theory in a piecewise linear setting and could be the basis of a first year postgraduate lecture or reading course. Some results from algebraic topology are needed for handle theory and these are collected in an appendix. In a second appen dix are listed the properties of Whitehead torsion which are used in the s-cobordism theorem. These appendices should enable a reader with only basic knowledge to complete the book. The book is also intended to form an introduction to modern geo metric topology as a research subject, a bibliography of research papers being included. We have omitted acknowledgements and references from the main text and have collected these in a set of "historical notes" to be found after the appendices. Note on the Paperback Edition For the paperback edition, the main text has been revised only where necessary in ord.er to correct mistakes in the original edition and we are very grateful to readers of the original edition for pointing out these mistakes. The bibliography has been revised to take account of publi cations, relevant to the topics listed, which have appeared since the ori ginal edition. No attempt however has been made to cover areas which have developed in the meantime (for example the recent work of Thurston et al. on geometric structures). Table of Contents Chapter 1. Polyhedra and P.L. Maps 1 Basic Notation . 1 joins and Cones. . . 1 Polyhedra. . . . . . 2 Piecewise-Linear Maps 5 The Standard Mistake 6 P.L. Embeddings 7 Manifolds . . . . . 7 Balls and Spheres . . 8 The Poincare Conjecture and the h-Cobordism Theorem 8 Chapter 2. Complexes . 11 Simplexes. . . 11 Cells. . . . . 13 Cell Complexes 14 Subdivisions . 15 Si,~plicial Complexes. 16 Simplicial Maps . . . 16 Triangulations. . . . 17 Subdividing Diagrams of Maps 18 Derived Subdivisions. . . . . 20 Abstract Isomorphism of Cell Complexes . 20 Pseudo-Radial Projection. 20 External Joins. . . . . . . . . . . . . 22 Collars. . . . . . . . . . . . . . . . 24 Appendix to Chapter 2. On Convex Cells . 27 Chapter 3. Regular Neighbourhoods 31 Full Subcomplexes. . . . 31 Derived Neighbourhoods . 32 Regular Neighbourhoods . 33 Table of Contents VII Regular Neighbourhoods in Manifolds . .. . . 34 Isotopy Uniqueness of Regular Neighbourhoods . 37 Collapsing . . . . . . . . . . . . 39 Remarks on Simple Homotopy Type . 39 Shelling .... 40 Orientation . . . . . 43 Connected Sums. . . 46 SchOnflies Conjecture. 47 Chapter 4. Pairs of Polyhedra and Isotopies 50 Links and Stars . . . . . 50 Collars .............. . 52 Regular Neighbourhoods ..... . 52 Simplicial Neighbourhood Theorem for Pairs 53 Collapsing and Shelling for Pairs 54 Application to Cellular Moves. 54 Disc Theorem for Pairs. . . . ~ 56 Isotopy Ex~nsion . . . . . . . 56 Chapter 5. General Position and Applications. 60 General Position. . . . . . 60 Embedding and Unknotting ..... . 63 Piping .............. . 67 Whitney Lemma and Unlinking Spheres 68 Non-Simply-Connected Whitney Lemma 72 Chapter 6. Handle Theory . 74 Handles on a: Cobordism . 75 Reordering Handles . . . 76 Handles of Adjacent Index 76 Complementary Handles 78 Adding Handles . . . . . 80 Handle Decompositions. . 81 The CW Complex Associated with a Decomposition 83 The Duality Theorems . . . . . . 84 Simplifying Handle Decompositions 84 Proof of the h-Cobordism Theorem 87 The Relative Case . . . . . . . 87 The Non-Simply-Connected Case 88 Constructing h-Cobordisms ... 90 Chapter 7. Applications . . . . . . 91 Unknotting Balls and Spheres in Codimension ;:;;; 3 . 91 A Criterion for Unknotting in Codimension 2. . . 92 vm Table of Contents Weak 5-Dimensional Theorems 93 Engulfing ........ . 94 Embedding Manifolds . . . 96 Appendix A. Algebraic Results . 97 A. 1 Homology . . . . . 97 A. 2 Geometric Interpretation of Homology 98 A. 3 Homology Groups of Spheres 99 A. 4 Cohomology. . . . 100 A. 5 Coefficients'. . . . 100 A. 6 Homotopy Groups . 100 A. 7 CW Complexes . . 101 A.8 The Universal Cover 102 Ap~ndix B. Torsion . . . . 104 B.l Geometrical Definition of Torsion 104 B. 2 Geometrical Properties of Torsion 104 B.3 Algebraic Definition of Torsion . 106 B. 4 Torsion and Polyhedra . . . . . 106 B.5 Torsion and Homotopy Equivalences. 107 Historical Notes 108 Bibliography . 112 Index .... 119 Chapter 1. Polyhedra and P.L. Maps In this chapter we introduce the main objects of study, polyhedra and p.l. maps. The chapter consists mostly of definitions, examples, and exercises. In a final section we introduce the main results of the book: the Poincare conjecture and the h-cobordism theorem. This section may be omitted until after Chapter 5 if the reader wishes; we have included it here to give a taste of deeper results. Basic Notation A map is a continuous function. cl(X) denotes the closure of X. IR denotes the real numbers and IRn (Euclidean n-space) the space of n-vectors {x=(x1, x2'"'' xn)} ofreal numbers. We will use the product metric on IRn given by d(x, y)=sup IXi-yd. "Linear" always means linear in the affine sense; thus a linear subspace (or just subspace) V c IRn is a translated vector subspace, or equivalently: for each finite set {ail c V and real numbers Ai with LAi=l we have LAiaiEV. A map f: V_IRm is linear if f(L Ai ai) = L Ai f(aJ Joins and Cones Let A, Be IR n be subsets. Define their join AB to be the subset AB = {A a+,u blaEA, pEB} where A, ,uEIR, A, ,u~O and A+,u= 1. Then AB consists of all points on straight-line segments "arcs" with endpoints in each of A and B. If A = Ii> we define AB = B. B Fig.l AB 2 Chapter 1. Polyhedra and P. L. Maps If A={a} is a one-point set then we often abbreviate {a} to a. We say that aB is a cone with vertex a and base B (or simply that aB is a cone) if each point not equal to a is expressed uniquely as Aa + pb with b e B, A, p ~ 0 and A + p = 1. Equivalently a fj; B and the arcs abl and aba, for each pair of distinct points b b e B, meet only at a. l, 2 Example Fig. 2 aB1 is a cone while aB2 is not. The example makes it clear that the property of being a cone depends on the presentation of the set aBo Polyhedra 1.1 A subset PclRn is a polyhedron if each point aeP has a cone neighbourhood N=aLin P, where Lis compact; N is called a star of a in P and L a link and we write N =N,,(P), L=L,,(P). Note that the case L=~ is not excluded so that a point is a polyhedron. Examples of Polyhedra / / / / / / / / Fig. 3. A house with 2 rooms, each having one entrance