Table Of ContentTopology and Geometry
for Physicists
CHARLES NASH
Department of Mathematical Physics, St. Patrick’s College, Maynooth,
Ireland
SIDDHARTHA SEN
School of Mathematics, Trinity College, Dublin, Ireland
ACADEMIC PRESS, INC.
Harcourt Brace Jovanovich, Publishers
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ACADEMIC PRESS INC. (LONDON) LTD
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Copyright © 1983 by
ACADEMIC PRESS INC. (LONDON) LTD.
Third printing 1987
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No part of this book may be reproduced in any form by photostat, microfilm, or any other
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British Library Cataloguing in Publication Data
Nash, C.
Topology and geometry for physicists.
1. Topology
I. Title IT. Sen, S.
514 QA611
ISBN 0-12-514080-0
Printed in Great Britain by Galliard (Printers) Ltd, Great Yarmouth
Preface
One noticeable feature of theoretical physics of the last decade or so
has been rapid growth of the use of topological and geometrical methods.
This book is intended to teach physicists these methods. No previous
knowledge of topology or geometry is assumed.
The prerequisites for this book are those possessed by an advanced
undergraduate or a first-year graduate student. The style and approach of
the book reflect the fact that the authors are physicists—the level of rigour
has, in many cases, been appropriately lowered both to shorten arguments
and, we hope, to improve their clarity. Nevertheless we have tried to
provide the references necessary for those who wish to read a completely
rigorous account.
Applications from condensed matter physics, statistical mechanics and
elementary particle theory appear in the book. An obvious ommission here
is general relativity—we apologize for this. We originally intended to discuss
general relativity. However, both the need to keep the size of the book
within reasonable limits and the fact that accounts of the topology and
geometry of relativity are already available, for example, in The Large
Scale Structure of Space-Time by S. Hawking and G, Ellis, made us
reluctantly decide to omit this topic.
We would like to warmly thank all the colleagues who encouraged us
and criticized the manuscript. In particular we wish to mention David
Simms and Richard Ward. Finally we thank Rose Coyne and Breda O’Neill
for careful typing of the manuscript.
August, 1982 Charles Nash and Siddhartha Sen
To Anita and Edna
Contents
Preface v
CHAPTER 1. Basic Notions of Topology and the Value of Topological
Reasoning
1.1. Introduction ...... Co 1
1.2. Basic topological notions ..........,.06. 8
1.3. Homeomorphisms, homotopy and the idea of ‘topological
invariants 2... ee 20
1.4. Topological invariants of compactness and connectedness. 22
1.5. Invariance of the dimension of R” ...... Le 23
CHAPTER 2. Differential Geometry: Manifolds and Differential Forms
2.1. Manifolds 2... 2... ee 25
2.2. Orientability 2... ee ee ee 33
2.3. Calculus on manifolds ... 2.2.. 2.. ee ee 37
2.4, Infinite dimensional manifolds ...... Foe ee 49
2.5. Differentiable structures... 2... 2... Loe eee 49
CHAPTER 3. The Fundamental Group
. Introduction 2... . ee 31
. Definition of the fundamental group .......... 56
. Simplexes and the calculating theorem .......... 67
. Triangulation of aspace withexamples ......... 70
. Fundamental group ofaproducttXxY ......... 77
CHAPTER 4. The Homology Groups
4.1. Introduction... 2... ee 79
4.2. Oriented simplexes and the definition of thehomology groups . 83
4.3. Abelian groups ... 2... ee ee ee ee ee 90
4.4, Relative homology groups ..............2.- 95
4.5, Exact sequences... ......... Loe ee 99
Vill CONTENTS
4.6. Torsion, Kunneth formula, Euler-Poincaré formula and sin-
gular homology Co ee Loe ee 104
CHAPTER 5. The Higher Homotopy Groups
5.1. Introduction 2...) 109
5.2. Definition of higher homotopy groups .......... 109
5.3. Abelian nature of higher homotopy groups ........ 112
5.4. Relative homotopy groups ...........2.8084 113
5.5. The exact homotopy sequence ...........0.. 116
CHAPTER 6, Cohomology and De Rham Cohomology
6.1. Introduction . 2... 2... ee ee 120
6.2. H"’(M;R) and Poincaré’slemma ............ 123
6.3. Poincaré’slemma ...2.... .ee. 125
6.4. Calculationof H?(M;R) 2.2.... .e e 127
6.5. Generalremarks . 2... 2... 2... ee ee et 136
6.6. Thecup product .. 2... 2... ee 137
6.7. Superiority of cohomology over homology ........ 138
CHAPTER 7. Fibre Bundles and Further Differential Geometry
7.1. Introduction ...............-008- 140
7.2. Fibrebundle .... 2... eee ee 141
7.3. More examplesof bundles ............2.2.. 148
7.4. When isabundletrivial? 2... .....2...02.2.2.. 152
7.5. Sections of bundles and singularities of vector fields .. . 156
7.6. Cutting a bundle down to size: reduction of the group and
contraction of the base space1. 7 .e.e)e 159
7.7. Remarks on almost Hamiltonian and almost complex
Structures . 2. 1 1 ee 165
7.8. G-structures on a compact closed manifold M ...... 171
7.9. Lie derivative 2... eeee eee 171
7.10. Connection and curvature ..........---4 174
7.11. The connection form and the gauge potential . ..... 177
7.12. Parallel transport, covariant derivative and curvature . . . 178
7.13. Covariant exterior derivatives ............-. 181
7.14. The Bianchiidentitiesand*F ............. 182
7.15. Connection inthe tangent bundle ........... 184
7.16. Thetorsiontensor .............-.-04 187
7.17. Geodesics 2... 1. ee 190
CONTENTS ix
7.18. The Levi-Civitaconnection ............0.2. 191
7.19. The Yang-Mills connection .............. 194
7.20. The Maxwell connection .............04. 196
7.21, General remarks ............ toe eee 198
7.22. Characteristic classes . . 2 1. 1. eee ee 200
7.23. Chern, Pontrjaginand Euler classes .......... 204
7.24. Characteristic classes in terms of curvature and invariant
polynomials ...... Co ee rn 206
7.25, Classification of bundles . 2... 2... 211
7.26. The Stiefel-Whitney class . 2... 2.1.. .ee e 212
7.27, Calculation of characteristic classes . . 2. 2... 0.0.4.4. 213
7.28. Generalremarks ............ tee ee 217
7.29, Formulae obeyed by characteristic classes . . ...... 219
7.30. Global invariants and local geometry .......... 221
CHAPTER 8. Morse Theory
8.1. More inequalities ©... .. 2.2.. .ee ee 227
8.2. Morselemma ..............2..20-8. 229
8.3. Symmetry breaking selection rules in crystals . ...... 236
8.4. Estimating equilibrium positions ............ 242
CHAPTER 9. Defects, Textures and Homotopy Theory
9.1. Planar spin in twodimensions ..........4.4.. 244
9.2. Definition of anordered medium ............ 245
9.3. Stability of defects theorem ........ tee ee 246
9.4. Examples . 2... 1 0 wee ee ee 250
9.5. General remarks and crossing of defects, textures and 73(S”) 251
CHAPTER 10. Yang-Mills Theories: Instantons and Monopoles
10.1. Introduction .. 2... 2. ee ee 256
10.2. Instantomns ... 2... ee 259
10.3. Topology and boundary conditions ......4.2.2.. 260
10.4. Instantons and absolute minima ........... 263
10.5. The instanton solution ...........00.0244 265
10.6. The instanton number and the second Chern class 269
10,7. Multi-instantons .............2.2..0 4 272
10.8. Quaternions and SU (2) connections .......2.. 272
10.9. The k =1 instanton in terms of quaternions ...... 276
10,10. Instantons with |k|>1 and quaternions ........ 278
Xx CONTENTS
10.11. Example of instantons with |k|>1 ........2.. 281
10.12. Twistor methods andinstantons ........... 283
10.13. The projective twistor space... 1... ee 283
10.14. Twistor spaceand planesinC* 2... 2... 285
10.15. a-planes and anti-self-dual connections ........ 288
10.16. The equivalence between instantons and holomorphic vec-
torbundles «2... 2... ee en 289
10.17. Construction of an instanton given a holomorphic vector
bundle 2... 2 ee ee 295
10.18. The Minkowski case... 2... 2. ee 297
10.19. Monopoles ... 1... 2 ee ee ee ee ee
10.20. The Bohm-Aharanoveffect ..........0... 301
Further Reading ... 2... . eee ee 305
SubjectIndex .............-, Lo ee ee 306
CHAPTER 1
Basic Notions of Topology and the Value of Topological
Reasoning
1.1. INTRODUCTION
Topology can be thought of as a kind of generalization of Euclidean
geometry, and also as a natural framework for the study of continuity.
Euclidean geometry is generalized by regarding triangles, circles, and
squares as being the same basic object. Continuity enters because in saying
this one has in mind a continuous deformation of a triangle into a square
or a circle, or indeed any arbitrary shape. A disc with a hole in the centre
is topologically different from a circle or a square because one cannot
create or destroy holes by continuous deformations. Thus using topological
methods one does not expect to be able to identify a geometrical figure as
being a triangle or a square. However, one does expect to be able to detect
the presence of gross features such as holes or the fact that the figure is
made up of two disjoint pieces etc. This leads to the important point
that topology produces theorems that are usually qualitative in nature—they
may assert, for example, the existence or non-existence of an object. They
will not in general, provide the means for its construction.
Let us begin by looking at some examples where topology plays a role.
Example 1. Cauchy’s residue theorem
Consider the contour integral for a meromorphic function f(z) along the
path I’, which starts at a and finishes at 6 (c.f. Fig. 1.1). Let us write
r=[ f(z) dz (1.1)
Ty
Now deform the path I’, continuously into the path [2 shown in Fig. 1.1.
Provided we cross no poles of f(z) in deforming I’, into T2, then Cauchy’s
1
2 TOPOLOGY AND GEOMETRY FOR PHYSICISTS
Y
Figure 1.1
theorem for meromorphic functions allows us straightaway to deduce that
| fiz) dz =[ f(z) dz (1.2)
Ty T2
This is just the statement that
[ pera: =0 (1.3)
where C is the closed contour made up by joining I’; to 2 and reversing
the arrow on I, so as to give an anticlockwise direction to the contour C.
The intuitive content of this result is that to integrate f(z) from a to 6 in
the complex plane is independent of the path joining a to 6 (under the
conditions stated). Even if we relax these conditions and allow that the
deformation of IT’, into [, may entail the crossing of some poles of f(z),
then we still have complete knowledge of the relationship of the two
integrals. It is simply
| f(z) dz =| f(z)dz+2aidres _ (1.4)
Ts T2
where the sum is over the residues, if any, of the poles inside C. This simple
example uncovers some topological properties underlying the familiar
Cauchy theorem.
