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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
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Fig. 3. A house with 2 rooms, each having one entrance
