Visual Geometry and Topology [Softcover reprint of the original 1st ed. 1994] 3642762379, 9783642762376

Table of contents :
Preface
Table of Contents
1. Polyhedra. Simplicial Complexes. Homologies
1.1 Polyhedra
1.1.1 Introductory Remarks
1.1.2 The Concept of an n-Dimensional Simplex
1.1.3 Polyhedra. Simplicial Subdivisions of Polyhedra. Simplicial Complexes
1.1.4 Examples of Polyhedra
1.1.5 Barycentric Subdivision
1.1.6 Visual Material
1.2 Simplicial Homology Groups of Simplicial Complexes (Polyhedra)
1.2.1 Simplicial Chains
Definition
1.2.2 Chain Boundary
Definition
Definition.
1.2.3 The Simplest Properties of the Boundary Operator. Cycles. Boundaries
Definition. cycle
1.2.4 Examples of Calculations of the Boundary Operator
1.2.5 Simplicial Homology Groups
Definition. H_k (X)
Definition. beta_k
Theorem
1.2.6 Examples of Calculations of Homology Groups. Homologies of Two-dimensional Surfaces
1.2.7 Visual Material
1.3 General Properties of Simplicial Homology Groups. Some Methods for Calculating Homology Groups
1.3.1 Incidence Matrices
1.3.2 The Method of Calculation of Homology Groups Using Incidence Matrices
1.3.3 "Traces" of Cell Homologies Inside Simplicial Ones
1.3.4 Chain Homotopy. Independence of Simplicial Homologies of a Polyhedron of the Choice of Triangulation
Theorem on simplicial approximation.
1.3.5 Visual Material
2. Low-Dimensional Manifolds
2.1 Basic Concepts of Differential Geometry
2.1.1 Coordinates in a Region, Transformations of Curvilinear Coordinates
2.1.2 The Concept of a Manifold. Smooth Manifolds. Submanifolds and Ways of Defining Them. Manifolds with Boundary. Tangent Space and Tangent Bundle
2.1.3 Orientabllity and Non-Orientabllity. The Differential of a Mapping. Regular Values and Regular Points. Embeddings and Immersions of Manifolds. Critical Points of Smooth Functions on Manifolds. Index of Nondegenerate Critical Points and Morse Functions
2.1.4 Vector and Covector Fields. Integral Trajectories. Vector Field Commutators. The Lie Algebra of Vector Fields on a Manifold
2.1.5 Visual Material
2.2 Visual Properties of One-Dimensional Manifolds
2.2.1 Isotopies, Frames
2.2.2 Visual Material
2.3 Visual Properties of Two-Dimensional Manifolds
2.3.1 Two-Dimensional Manifolds with Boundary
2.3.2 Examples of Two-Dimensional Manifolds
2.3.3 Modelling of a Projective Plane in a Three-Dimensional Space
2.3.5 Classification of Closed 2-Manifolds
2.3.6 Inversion of a Two-Dimensional Sphere
2.3.7 Visual Material
2.4 What Distinguishes Two-Dimensional Manifolds? Cohomology Groups and Differential Forms
2.4.1 Differentiall-Forms on a Smooth Manifold
2.4.2 Closed and Exact Forms on a Two-Dimensional Manifold
2.4.3 An Important Property of Cohomology Groups
2.4.4 Direct Calculation of One-Dimensional Cohomology Groups of One-Dimensional Manifolds
2.4.5 Direct Calculation of One-Dimensional Cohomology Groups of a Plane, a Two-Dimensional Sphere and a Torus
2.4.6 Direct Calculation of One-Dimensional Cohomology Groups of Oriented Surfaces, i.e. Spheres with Handles
2.4.7 An Algorithm for Recognition of Two-Dimensional Manifolds. Elements of Two-Dimensional Computer Geometry
2.4.8 Calculation of One-Dimensional Cohomologies of a Surface Using Triangulation
2.4.9 Visual Material
2.5 Visual Properties of Three-Dimensional Manifolds
2.5.1 Heegaard Splittings (or Diagrams)
2.5.2 Examples of Three-Dimensional Manifolds
2.5.3 Equivalence of Heegaard Splittings
2.5.4 Spines
2.S.S Special Spines
2.5.6 Filtration of 3-Manifolds with Respect to Matveev's Complexity
2.5.7 Simplification of Special Spines
2.5.8 The Use of Computers in Three-Dimensional Topology. Enumeration of Manifolds in Increasing Order of Complexity
2.5.9 Matveev's Complexity of 3-Manifolds and Simplex Glueings
2.5.10 Visual Material
3. Visual Symplectic Topology and Visual Hamiltonian Mechanics
3.1 Some Concepts of Hamiltonian Geometry
3.1.1 Hamiltonian Systems on Symplectic Manifolds
3.1.2 Involutive Integrals and Liouville Tori
3.1.3 Momentum Mapping of an Integrable System
3.1.4 Surgery on Liouv1lle Tori at Critical Energy Values
3.1.5 Visual Material
3.2 QUalitative Questions of Geometric Integration of Some Differential Equations. Classification of Typical Surgeries of Liouville Tori of Integrable Systems with Bott Integrals
3.2.1 Nondegenerate (Bott) Integrals
3.2.2 Classification of Surgeries of Bott Position on Liouville Tori
3.2.3 The Topological Structure of Critical Energy Levels at a Fixed Second Integral
3.2.4 Examples from Mechanics. The Equations of Motion of a Rigid Body. The Poisson Sphere. Geometrical Interpretation of Mechanical Systems
3.2.5 An Example of an Investigation of a Mechanical System. The Liouville System on the Plane
3.2.6 The Liouville System on the Sphere
3.2.7 Inertial Motion of a Gyrostat
3.2.8 The Case of ChapJygin-Sretensky
3.2.9 The Case of Kovalevskaya
3.2.10 Visual Material
3.3 Three-Dimensional Manifolds and Visual Geometry of Isoenergy Surfaces of Integrable Systems
3.3.1 A One-Dimensional Graph as a Hamiltonian Diagram
3.3.3 The Simplest Isoenergy Surfaces (with Boundary)
3.3.4 Any Isoenergy Surface of an Integrable Nondegenerate System Falls into the Sum of Five (or Two) Types of Elementary Bricks
3.3.5 New Topological Properties of the Isoenergy Surfaces Class
3.3.6 One Example of it Computer Use in Symplectic Topology
3.3.7 Visual Material
4. Visual Images in Some Other Fields of Geometry and in Its Applications
4.1 Visual Geometry of Soap Films. Minimal Surfaces
4.1.1 Boundaries Between Physical Media. Minimal Surfaces
4.1.2 Some Examples of Minimal Surfaces
4.1.3 Visual Material
4.2 Fractal Geometry and Homeomorphisms
4.2.1 Various Concepts of Dimension
4.2.2 Fractals
4.2.3 Homeomorphisms
4.2.4 Visual Material
4.3 Visual Computer Geometry in the Number Theory
Appendix 1. Visual Geometry of Some Natural and Nonholonomic Systems
1.1 On Projection of Liouville Tori in Systems with Separation of Variables
1.2 What Are Nonholonomic Constraints?
1.3 The Variety of Manifolds in the Suslov Problem
Appendix 2. Visual Hyperbolic Geometry
2.1 Discrete Groups and Their Fundamental Region
2.2 Discrete Groups Generated by Reflections in the Plane
2.3 The Gram Matrix and the Coxeter Scheme
2.4 Reflection-Generated Discrete Groups in Space
2.5 A Model of the Lobachevskian Plane
2.6 Convex Polygons on the Lobachevskian Plane
2.7 Coxeter Polygons on the Lobachevskian Plane
2.8 Coxeter Polyhedra in the Lobachevskian Space
2.9 Discrete Groups of Motions of Lobachevskian Space and Groups of Integer-Valued Automorphisms of Hyperbolic Quadratic Forms
2.10 Reflection-Generated Discrete Groups in High-Dimensional Lobachevskian Spaces
References

Citation preview

Visual Geometry and Topology

Anatolij Fomenko as guest Professor at the Mathematical Institute in Heidelberg

Anatolij Fomenko

Visual Geometry and Topology With 50 Full-page Illustrations and 287 Drawings

Springer-Verlag Berlin Heidelberg New York London Paris Tokyo HongKong Barcelona Budapest

Author: Anatolij T.Fomenko Dept. of Differential Geometry and Applications Faculty of Mathematics and Mechanics Moscow University, Moscow 119899 Russia Translator: Marianna V. Tsaplina Moscow, Russia Title of the Russian edition: Naglyadnaya geometriya i topologia Moscow University Press, 1993 (abridged version)

Mathematics Subject Classification (1991): 49-XX, 51-XX, 53-xx, 55-XX, 57-XX, 58-XX, 70-XX, 81-XX, 83-XX

Library of Congress Cataloging-in-Publication Data Fomenko, A. T. Visual geometry and topology/ Anatolij Fomenko; [translator, Marianna V. Tsaplinaj. p. em. Includes bibliographical references and index. ISBN-13: 978-3-642-76237-6 e-ISBN-13: 978-3-642-76235-2 DOl: 10.1007/978-3-642-76235-2 1. Geometry. 2. Topology. I. Title. QA445.F58 1993 516-dc20 92-39676 CIP

This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprint~, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other way, and storage in data banks. Duplication of this publication or parts thereofis permitted only under the provisions of the German Copyright Law of September 9,1965, in its current version, and permission for use must always be obtained from SpringerVerlag. Violations are liable for prosecution under the German Copyright Law. © Springer-Verlag Berlin Heidelberg 1994 Softcover reprint of the hardcover 1st edition 1994

Typesetting: Springer TEX in-house system 41/3140 - 5 4 3 2 1 0 - Printed on acid-free paper

Preface

Modem geometry and topology take a special place in mathematics because many of the objects they deal with are treated using visual methods. At the same time, these visual methods now successfully undergo formalization and far-reaching abstraction which have contributed much to the remarkable progress in modem geometry and its applications. David Hilbert wrote in 1932: "As far as geometry is concerned, the tendency of abstraction in it had led to grand systematic constructions of algebraic geometry, Riemann geometry and topology where the methods of abstract reasoning, symbolism and analysis are widely used. But non the less, visual perception still plays the leading role in geometry, and not only for being strongly demonstrative in the course of investigation, but also for the understanding and estimation of the results obtained in the course of investigation" [l]. Many geometric concepts arose from concrete problems of mechanics, physics, etc., and we shall point out here some of these links. For important modem mechanisms of the appearance of topological ideas in the framework of classical mechanics and mathematical physics see, for example, the papers by Smale [2], Poston and Stewart [3], Hildebrandt and Tromba [4], Novikov [5], Arnold [7], Nitsche [8]. The mathematical life is now being actively intruded by computer geometry which permits, in particular, visualization of intricate mechanical objects that result from long computational experiments and whose geometric character is hardly perdictable. It is relevant to mention here the remarkable works by Banchoff [9], Mandelbrot [10], Francis [11], Penrose [12], Poston & Stewart [3], and Peitgen & Richter [13]. Some geometrical aspects of the theory of probability and mathematical statistics are elucidated in the book by Kolmogorov [14], B.V. Gnedenko [15], Shiryaev [16] and Chentsov [17]. Some of these ideas are reflected in the sections "Visual Material" of the present book.

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We have selected only the fragments of geometrical knowledge which are most visual and closely related to applications. See, in particular, Chap.3 devoted to some modem visual aspects of symplectic topology and Hamiltonian mechanics. See also the contributions due to Maslov [18], Faddeev & Zakharov [19], Gelfand [20], Matveev [46], [47], Zieschang [65], Bolsinov [70] and Kozlov [21]. The principal aim of the present book is to narrate, in an accessible and fairly visual language, about some classical and modem achievements of geometry in both intrinsic mathematical problems and applications. We do not restrict our consideration to the classics, and also touch upon current problems which are being rapidly developed today. We lay special emphasis upon visual explanation of the statement of problems, the methods of their solution and the results obtained and try to acquaint the reader with geometric ideas as soon as possible, ignoring for the time being the abstract logical aspect of calculations, considerations, etc. After having read this book, the reader will be able himself to comprehend, when reading specialized literature, more formal approaches to the problems pointed out in this book. Many modem fields of mathematics admit visual presentations which do not, of course, claim to be logically rigorous but, on the other hand, offer a prompt introduction into the subject matter. In this connection, in Chap. 1 we give a brief presentation of the classical theory of polyhedra and simplicial homologies because these ideas are now widely used in mathematics, physics, etc., but their logical simplification and perfection is often achieved by a higher level of abstraction. In this respect, the geometric approach to the theory of homologies, which goes back to Poincare, is perhaps more cumbersome (in what concerns theoretical grounds) but appreciably simpler and more visual (and therefore more comprehensive at early stages of acquaintance with the subject). Geometrical intuition plays an essential role in contemporary algebrotopological and geometric studies. Many profound scientific mathematical papers devoted to multi-dimensional geometry use intensively the "visual slang" such as, say, "cut the surface", "glue together the strips", "glue the cylinder", "evert the sphere", etc., typical of the studies of twoand three-dimensional images. Such a terminology is not a caprice of mathematicians, but rather a "practical necessity" since its employment and the mathematical thinking in these terms appear to be quite necessary for

Preface

VII

the proof of technically very sophisticated results. It happens rather frequently that the proof of one or another mathematical fact can at ftrst be "seen", and only after that (and following this visual idea) can we present a logically consistent formulation, which is sometimes a very difficult task requiring serious intellectual efforts. The expediency of these efforts is however psychologically justifted by the visual and beautiful picture already created in the head of the researcher and convincing him that he had taken the right way. Thus, the criterion of beauty of one or another geometric image often serves as a compass for choosing an optimal way of a further formal logical proof. For a more thorough acquaintance with these ideas the reader may be referred to the well-known books by Manin [22], [23], where many interesting details can be found. It was not our aim to give a systematic presentation of individual ftelds of geometry, but we started on a journey' round its rich world and on our way "took photos" of those fragments which we thought of as particularly interesting. The visual and scientiftcally urgent material being exceedingly abundant, a complete or a systematic presentation is out of the question. What we offer is a short "diary", an attempt to narrate to a wide range of mathematicians, mechanics, physicists about the diversity of methods and applications of modem geometry, to help them recognize really exciting geometric and physical objects against the background of sophisticated abstractions. Each chapter of the book is written as autonomously as possible, so that the reader could plunge into the ideas and concepts of each section and choose for himself the order of reading separete chapters. The author is greatful to S.V. Matveev, Ya.V. Tatarinov, A.V. Chernavsky, E.B. Vinberg, V.O. Bugaenko and A.A. Zenkin who kindly presented some materials on visual geometry and topology. Each mathematician has his own system of concepts of the intrinsic geometry of his (speciftc) mathematical world and visual images which he associated with some or other abstract concepts of mathematics (including algebra, number theory, analysis, etc.). It is noteworthy that sometimes one and the same abstraction brings about the same visual picture in different mathematicians, but these pictures born by imagination are in most cases very difficult to represent graphically, so to say, to draw. Part of the graphical material contained in the present book is an attempt to ''photograph from within" the sophisticated, peculiar

VIII

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mathematical world generously endowed with images and concepts that constitute the subject matter of contemporary. These graphical representations are compiled in sections called "Visual Material". Almost each figure of such a section is endowed with a comment in the text. Some of the figures complement the content of the chapters with concepts not reflected in the main text. In this case, such figures give references to the corresponding literature. Our graphical material is either based on concrete geometric constructions, ideas, theorems and depict real mathematical objects and processes or reflects various ways of perception of mathematical concepts, for instance, infinity, homotopy, etc. The first attempt in this direction was the book "Homotopic Topology" by Fomenko, Fuchs and Gutenmacher [24]. This topic was further developed in a new book "The Course in Homotopic Topology" by Fomenko & Fuchs [25]. The reader can also see the author's art book, Mathematical Impressions [26], which contains high-quality reproductions of approximately 80 of the author's works, including some in colour. The album also contains short mathematical and extra-mathematical comments to the pictures. Because they were written with different purposes, these two books complement each other. Different sections of the present book are intended for different levels of mathematical knowledge, but the greater part of the material is intended for the first- and second-year students of mathematics or physics. The book includes some visual aspects of the results in the field of modem computer geometry obtained in the framework of the scientific research "Computer Geometry" headed by AT. Fomenko at the Faculty of Mechanics and Mathematics of Moscow State University. These results have been discussed at the scientific seminar "Computer Geometry" working at the Department of Differential Geometry and Applications, Department of Higher Geometry and Topology and Department of Computational Methods headed by AV. Bolsinov, V.L. Golo, I.Kh. Sabitov, E.G. Sklyarenko, V.V. Trofimov and AT. Fomenko in Moscow State University. The readers who is interested in the details of modem geometrical methods can continue his education using the books by Dubrovin, Fomenko, Novikov [27] and the book by Fomenko [28].

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IX

The book is intended for natural sciences students (beginning from the first year), post-graduates and specialists interested in applications of modem geometry and topology. The book included 50 graphical sheets drawn by the author and presented in sections Visual Material. The majority of them are made in pencil and Indian ink on paper and exhibit half tones and sophisticated light and shade technique. The preparation of precise and high-quality photocopies, which the author submitted to Springer-Verlag for reproduction in the book, was therefore a rather difficult task. It was successfully fulfilled by a professional photographer N.S. Moiseenko (Moscow) to whom the author expresses his gratitude. The author is deeply indebted to Springer Publishers.

Table of Contents

1

Polyhedra. Simplicial Complexes. Homologies

1.1 Polyhedra . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.1 Introductory Remarks . . . . . . . . . . . . . . . . . . . . 1.1.2 The Concept of an n-Dimensional Simplex Barycentric Coordinates . . . . . . . . . . . . . . . . . . 5 1.1.3 Polyhedra. Simplicial Subdivisions of Polyhedra. Simplicial Complexes . . . . . . . . . . . . . . . . . . .. 8 1.1.4 Examples of Polyhedra . . . . . . . . . . . . . . . . . . . 10 1.1.5 Barycentric Subdivision . . . . . . . . . . . . . . . . . . 14 1.1.6 Visual Material ., . . . . . . . . . . . . . . . . . . . . . 16 1.2 1.2.1 1.2.2 1.2.3 1.2.4 1.2.5 1.2.6 1.2.7

Simplicial Homology Groups of Simplicial Complexes (polyhedra) . . . . . . . . . . . . . . . . . . . . . . . .. Simplicial Chains . . . . . . . . . . . . . . . . . . . . .. Chain Boundary . . . . . . . . . . . . . . . . . . . . . . The Simplest Properties of the Boundary Operator Cycles. Boundaries ..... . . . . . . . . . . . Examples of Calculations of the Boundary Operator . . . . Simplicial Homology Groups . . . . . . . . . . . . . . . . Examples of Calculations of Homology Groups. Homologies of Two-dimensional Surfaces . . . . . . . . . Visual Material .,. . . . . . . . . . . . . . . . . . . . .

18 18 23 26 27 29 32 46

1.3 General Properties of Simplicial Homology Groups .. 49 1.3.1 Incidence Matrices . . . . . . . . . . . . . . . . . . . . . 49 1.3.2 The Method of Calculation of Homology Groups Using Incidence Matrices . . . . . . . . . . . . . . . . . . 50

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1.3.3 "Traces" of Cell Homologies Inside Simplicial Ones ... 56 1.3.4 Chain Homotopy. Independence of Simplicial Homologies of a Polyhedron of the Choice of Triangulation ...... 59 1.3.5 Visual Material . . . . . . . . . . . . . . . . . . . . . . . 65

2

Low-Dimensional Manifolds

2.1 Basic Concepts of Differential Geometry . . . . . . . . 2.1.1 Coordinates in a Region. Transformations of Curvilinear Coordinates . . . . . . . . 2.1.2 The Concept of a Manifold. Smooth Manifolds. Submanifolds and Ways of Defining Them. Manifolds with Boundary. Tangent Space and Tangent Bundle . . . . 2.1.3 Orientability and Non-Orientability. The Differential of a Mapping. Regular Values and Regular Points. Embeddings and Immersions of Manifolds. Critical Points of Smooth Functions on Manifolds. Index of Nondegenerate Critical Points and Morse Functions . . . . . . . . . . . . . . . . . . . . 2.1.4 Vector and Covector Fields. Integral Trajectories. Vector Field Commutators. The Lie Algebra of Vector Fields on a Manifold . . . . . . . . . . . . . . . 2.1.5 Visual Material . . . . . . . . . . . . . . . . . . . . . . .

75 75

79

85 91 95

2.2 Visual Properties of One-Dimensional Manifolds . . . . 98 2.2.1 Isotopies, Frames . . . . . . . . . . . . . . . . . . . . . . 98 2.2.2 Visual Material ...... . . . . . . . . . . . . . . . . . 102 2.3 2.3.1 2.3.2 2.3.3

Visual Properties of Two-Dimensional Manifolds . . . . 111 Two-Dimensional Manifolds with Boundary . . . . . . . . 111 Examples of Two-Dimensional Manifolds . . . . . . . . . 113 Modelling of a Projective Plane in a Three-Dimensional Space . . . . . . . . . . . . . . . 115 2.3.4 Two Series of Two-Dimensional Closed Manifolds .... 120

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XIII

2.3.5 Classification of Closed 2-Manifolds . . . . . . . . . . . . 125 2.3.6 Inversion of a Two-Dimensional Sphere . . . . . . . . . . 128 2.3.7 Visual Material ., . . . . . . . . . . . . . . . . . . . . . 130 2.4 2.4.1 2.4.2 2.4.3 2.4.4

2.4.5

2.4.6 2.4.7

2.4.8 2.4.9 2.5 2.5.1 2.5.2 2.5.3 2.5.4 2.5.5 2.5.6

2.5.7 2.5.8

2.5.9 2.5.10

Cohomology Groups and Differential Forms ...... 135 Differential I-Forms on a Smooth Manifold . . . . . . . . 135 Closed and Exact Forms on a Two-Dimensional Manifold 136 An Important Property of Cohomology Groups ...... 138 Direct Calculation of One-Dimensional Cohomology Groups of One-Dimensional Manifolds . . . . . . . . . . 139 Direct Calculation of One-Dimensional Cohomology Groups of a Plane, a Two-Dimensional Sphere and a Torus . . . . . . . . . . . . . . . . . . . . . . . . . 141 Direct Calculation of One-Dimensional Cohomology Groups of Oriented Surfaces, i.e. Spheres with Handles . . 148 An Algorithm for Recognition of Two-Dimensional Manifolds. Elements of Two-Dimensional Computer Geometry . . . . . . . . . . . . . . . . . . . . 152 Calculation of One-Dimensional Cohomologies of a Surface Using Triangulation . . . . . . . . . . . . . . 154 Visual Material . . . . . . . . . . . . . . . . . . . . . . . 154 Visual Properties of Three-Dimensional Manifolds .. . 159 Heegaard Splittings (or Diagrams) . . . . . . . . . . . . . 159 Examples of Three-Dimensional Manifolds . . . . . . . . 162 Equivalence of Heegaard Splittings . . . . . . . . . . . . 164 Spines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 Special Spines . . . . . . . . . . . . . . . . . . . . . . . 168 Filtration of 3-Manifolds with Respect to Matveev's Complexity . . . . . . . . . . . . . . . . . . . . . . . . . 170 Simplification of Special Spines . . . . . . . . . . . . . . 173 The Use of Computers in Three-Dimensional Topology. Enumeration of Manifolds in Increasing Order of Complexity . . . . . . . . . . . . . . . . . . . . . . . 177 Matveev's Complexity of 3-Manifolds and Simplex Glueings . . . . . . . . . . . . . . . . . . . 184 Visual Material . . . . . . . . . . . . . . . . . . . . . . . 188

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3

Visual Symplectic Topology and Visual Hamiltonian Mechanics

3.1 3.1.1 3.1.2 3.1.3 3.1.4 3.1.5

Some Concepts of Hamiltonian Geometry . . . . . . . . 193 Hamiltonian Systems on Symplectic Manifolds ...... 193 Invo1utive Integrals and Liouville Tori . . . . . . . . . . . 196 Momentum Mapping of an Integrable System ..... . . 200 Surgery on Liouville Tori at Critical Energy Values .... 200 Visual Material . . . . . . . . . . . . . . . . . . . . . . . 204

3.2

Qualitative Questions of Geometric Integration of Some Differential Equations. Classification of Typical Surgeries of Liouville Tori of Integrable Systems with Bott Integrals . . . . . . . . 208 Nondegenerate (Bott) Integrals . . . . . . . . . . . . . . . 208 Classification of Surgeries of Bott Position on Liouville Tori . . . . . . . . . . . . . . . . . . . . . . 210 The Topological Structure of Critical Energy Levels at a Fixed Second Integral . . . . . . . . . . . . . . . . . 215 Examples from Mechanics. The Equations of Motion of a Rigid Body. The Poisson Sphere. Geometrical Interpretation of Mechanical Systems . . . . . 216 An Example of an Investigation of a Mechanical System. The Liouville System on the Plane . . . . . . . . . . . . . 219 The Liouville System on the Sphere . . . . . . . . . . . . 220 Inertial Motion of a Gyrostat . . . . . . . . . . . . . . . . 221 The Case of Chaplygin-Sretensky . . . . . . . . . . . . . 223 The Case of Kovalevskaya ...... . . . . . . . . . . . 224 Visual Material . . . . . . . . . . . . . . . . . . . . . . . 225

3.2.1 3.2.2 3.2.3 3.2.4

3.2.5 3.2.6 3.2.7 3.2.8 3.2.9 3.2.10 3.3

Three-Dimensional Manifolds and Visual Geometry of Isoenergy Surfaces of Integrable Systems . . . . . . . 231 3.3.1 A One-Dimensional Graph as a Hamiltonian Diagram ... 231 3.3.2 What Familiar Manifolds Are Encountered Among Isoenergy Surfaces? . . . . . . . . . . . . . . . . 234

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3.3.3 The Simplest Isoenergy Surfaces (with Boundary) ..... 243 3.3.4 Any Isoenergy Surface of an Integrable Nondegenerate System Falls into the Sum of Five (or Two) Types of Elementary Bricks . . . . . . . . . . . . . . . . . . . . 244 3.3.5 New Topological Properties of the Isoenergy Surfaces Class . . . . . . . . . . . . . . . . . . . . . . . 246 3.3.6 One Example of a Computer Use in Symplectic Topology 249 3.3.7 Visual Material . . . . . . . . . . . . . . . . . . . . . . . 252

4

Visual Images in Some Other Fields of Geometry and Its Applications

4.1 4.1.1 4.1.2 4.1.3

Visual Geometry of Soap Films. Minimal Surfaces . . . 255 Boundaries Between Physical Media. Minimal Surfaces .. 255 Some Examples of Minimal Surfaces . . . . . . . . . . . 258 Visual Material . . . . . . . . . . . . . . . . . . . . . . . 260

4.2 4.2.1 4.2.2 4.2.3 4.2.4

Fractal Geometry and Homeomorphisms . . . . . . . . 264 Various Concepts of Dimension . . . . . . . . . . . . . . 264 Fractals . . . . . . . . . . . . . . . . . . . . . . . . . . . 267 Homeomorphisms...................... 268 Visual Material .,. . . . . . . . . . . . . . . . . . . . . 276

4.3

Visual Computer Geometry in the Number Theory

.. 284

Appendix 1 Visual Geometry of Some Natural and Nonholonomic Systems 1.1 1.2 1.3

On Projection of Liouville Tori in Systems with Separation of Variables . . . . . . . . . . . . . . . . 293 What Are Nonholonomic Constraints? ..... . . . . . . 295 The Variety of Manifolds in the Suslov Problem ..... 297

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Appendix 2 Visual Hyperbolic Geometry 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9

2.10

Discrete Groups and Their Fundamental Region . . . . . . 300 Discrete Groups Generated by Reflections in the Plane .. 301 The Gram Matrix and the Coxeter Scheme . . . . . . . . 303 Reflection-Generated Discrete Groups in Space ...... 303 A Model of the Lobachevskian Plane . . . . . . . . . . . 306 Convex Polygons on the Lobachevskian Plane . . . . . . . 308 Coxeter Polygons on the Lobachevskian Plane . . . . . . . 309 Coxeter Polyhedra in the Lobachevskian Space ...... 310 Discrete Groups of Motions of Lobachevskian Space and Groups of Integer-Valued Automorphisms of Hyperbolic Quadratic Forms . . . . . . . . . . . . . . . 314 Reflection-Generated Discrete Groups in High-Dimensional Lobachevskian Spaces . . . . . . . . 315

References

............................ 317

1. Polyhedra. Simplicial Complexes. Homologies

1.1 Polyhedra 1.1.1 Introductory Remarks

Geometry and topology most often deal with geometrical figures, objects realized as a set of points in a Euclidean space (maybe of many dimensions). It is useful to view these objects not as rigid (solid) bodies, but as figures that admit continuous deformation preserving some qualitative properties of the object. Recall that the mapping of one object onto another is called continuous if it can be determined by means of continuous functions in a Cartesian coordinate system in space. The mapping of one figure onto another is called homeomorphism if it is continuous and one-to-one, i.e. establishes a one-to-one correspondence between points of both figures. In this case there exists the inverse mapping with the same properties. For example, a straight line segment and a continuous arc (without self-intersections) on a plane are homeomorphic (Fig. 1.1.1). Homeomorphic are also a square and a circle (Fig. 1.1.1), a cube and a tetrahedron (sometimes called a simplex) (Fig. 1.1.1), a plane and a sphere with one punctured (discarded) point (Fig. 1.1.1). In the latter case the homeomorphism can be realized using the so-called stereographic projection (Fig. 1.1.2). To this end one should put a standard sphere onto a plane, take the point of tangency for the south pole and the very top point for the north pole N. Then one draws a ray from the north pole through an arbitrary point x on the sphere and continue the ray up to its intersection with the horizontal plane. The point obtained will be denoted by f(x). The map x - t f(x) just determines the homeomorphism between the sphere without a point and the Euclidean plane.

'~ ---

I I

I

//--

Fig.1.1.1

Fig.1.1.2

~

__

~~~~-~-

2

Topological and metric properties

Embeddings

Fig. 1.1.3

Mobius strip or Mobius band

Simple arc

1. Polyhedra. Simplicial Complexes. Homologies

The concept of homeomorphism appears to be convenient for establishing those important properties of figures which remain unchanged under such deformations. These properties are sometimes referred to as topological, as distinguished from metrical, which are customarily associated with distances between points, angles between lines, edges of a figure, etc. A cube and a tetrahedron (Fig.1.l.1) are of course different from the metric point of view, but at the same time they are homeomorphic. For many problems, minute metrical properties of a figure are often inessential, and therefore it is of interest to display its more "rough" topological properties which are usually masked. Sometimes, homeomorphism of two figures is not immediately obvious. The simplest example is an ordinary flat circle 8' in a threedimensional Euclidean space R3 and a knotted circle (Fig. 1.1.3). They are of course homeomorphic. To establish this fact, it suffices to cut each of these curves at one point and superpose the two segments obtained. The image of these figures in space seems different only due to the fact that the circle is embedded in R3 differently (in two ways). In a three-dimensional space one cannot continuously and disjointly deform a nontrivial knot into a flat circle. In a four-dimensional space one can already do so. A usual strip (a ring) and a strip twice wound in a threedimensional space (Fig.l.1.3) are homeomorphic in exactly the same manner. We invite the reader to prove it himself. One should not think, however, that any winding of a strip (with any number of loops) leads to a homeomorphic figure. We shall examine, for example, a strip wound only once (Fig. 1.1.3). It can be verified that such a strip is already not homeomorphic to a flat ring (to a usual strip). A strip wound only once is called the Mobius strip (or the Mobius band). The question is: What will happen to a strip if we wind it any even or odd number of times? A set of points homeomorphic to a straight line segment we shall call a simple arc. A simple arc contains two distinguished points, the ends. They are sometimes called arc vertices. Under a homeomorphism of a simple arc onto itself, its vertices are carried into themselves or into one another. The rest of the points of the simple arc (the so-called interior points) are equivalent in the sense that for any pair x and y of interior points of the arc there always exists a homeomorphism of the arc onto itself, such that the point x is mapped into the point y. The visual

1.1 Polyhedra

3

picture of this is given in Fig. 1.1.4 as a graph. Clearly, in one of the two cases depicted the arc ends exchange places, while in the other case they go into themselves, i.e. remain in their own places. A zero-dimensional simplicial complex (a zero-dimensional polyhedron) is a finite set of points (the vertices of the polyhedron). A onedimensional complex or graph (a one-dimensional graph) is a set consisting of a finite number ao of points (''vertices'') and a finite number aj of simple arcs ("edges"). Given this, the following two properties should be satisfied. 1) Any two edges either have no common points or have only one common end-point. Each vertex either does not belong to a single edge (such vertices will be called "isolated") or serves as the end-point of one or several edges. The number of such edges is called the degree or the index of the vertex. 2) Both end-points of each edge are included in the number of vertices of the graph (Fig. 1.1.5). A graph is called connected if it cannot be divided into two subgraphs without common vertices and edges. Otherwise the graph is disconnected and falls into several connected subgraphs called components of this graph (connected components). We have begun with the concept of a graph in order that already at the beginning of Chap. 1 we could visually demonstrate the basic concepts which are further on developed in a more general case of arbitrary polyhedra. The reader who will now "feel" these concepts will easily orientate in the entire material of the book. If we remove one edge from the graph r without removing a single vertex (including the end-points of the edge), we shall have a subgraph r'. This operation may apparently vary the number of connected components in the graph. The number of components of the graph r' will remain the same (as in the graph r) if the removed edge joined the vertices of one and the same component of r ' (Fig. 1.1.5). On the contrary, the number of components of the graph r will increase by one if the removed edge joined vertices of different components of the graph r'. The order of connectedness of a graph is the maximal number of edges which can be removed without changing the number of components of the graph.

~

:f

Fig. 1.1.4

o r_r'

b-l r

-f'

Fig. U.S

Components of the graph

4

Fig. 1.1.6

Simple cycle

Oriented edge

Cycles

Algebraic sum

Chains

1. Polyhedra. Simplicial Complexes. Homologies

A simple cycle is a connected graph all of whose vertices have index 2. It is depicted as a circle. More precisely, such a graph is homeomorphic to a circle (Fig. 1.l.6). It is not true that each graph can be realized as a set of points on a plane. For example, the graph in Fig. 1.1.6 (the set of edges of a tetrahedron) cannot be embedded into a plane. By the way, this graph is not a simple cycle (verify this!). We shall soon get acquainted with the general concept of homologies. Now we shall only discuss the simplest version of this concept. Let us consider an arbitrary graph r. We choose on each of its edges an arbitrary direction and fix it. An edge endowed with a direction (shown by an arrow) will be called an oriented edge and will be denoted by ..1J, the subscript i running through the values from 1 to O!] (see above, where O!] is the number of edges in a graph). The same edge, but oppositely aligned, will be denoted by - ..11. Each simple cycle on the graph with the chosen direction of detour we shall write formally as an algebraic sum of the constituent edges, taking each edge either with the "+" or " -" sign depending on whether its direction coincides with that on the cycle or is opposite to it. If a certain edge of the graph has not entered in the cycle, we assume that it enters in the algebraic sum with zero coefficient. Consequently, each simple cycle z will be written as the sum z = C]..1j + ... + ck..1l, where the numerical coefficients Ci are equal to +1, -lor zero. We shall consider the various linear combinations of such linear forms, i.e. of simple cycles written in the algebraic language. Such general forms will not, generally speaking, correspond to simple cycles. We shall call them cycles (without the adjective "simple"). After "grouping of like terms" such a general cycle becomes a linear form a]..1j + ... + ak..1l, where the coefficients ai are integers already generally distinct from +I, -1, O. At the same time it is clear that the obtained numerical coefficients aj are not quite arbitrary: they satisfy some linear relations. We might consider arbitrary linear combinations of the form A]..11 + ... + Ak..1l, where the coefficients Ai are already arbitrary integers. Such linear combinations ar called chains. Clearly, a simple cycle and a cycle are particular cases of chains. Not any chain is however a cycle. A cycle is a linear combination of simple cycles.

1.1 Polyhedra

5

Now we can give an algebraic definition of the order of connectedness of a graph: The order of connectedness of a graph is equal to the algebraic number of linearly independent cycles in the graph. The reader can verify that the new definition of the order of connectedness of the graph coincides with the geometrical definition given above. From this example we can see that certain geometric concepts can be transformed into algebraic concepts through an introduction of rather simple and natural objects of algebro-geometric character (cycles, chains, etc.). This technique proves to be very useful first of all because it allows involving very powerful algebraic methods for studying geometric properties of figures. Employing this technique one can, in particular, calculate qualitative topological characteristics of figures and compare them with one another. We now proceed to the simplest concepts of the combinatorial geometry of polyhedra.

Simplex

Barycentric coordinates

1.1.2 The Concept of an n-Dimensional Simplex. Barycentric Coordinates

Centre of gravity

Suppose in a Euclidean space ]Rn+l we are given n + 1 linearly independent points Ao, ... , An. We assume the points to be independent, so there are n + 1 vectors going from the origin into these points (Fig. 1.1.7). We place at each point Ai a non-negative m~ss mi (i.e. assume Ai to be a material point) and require that the resultant mass of all the points be equal to unity. This means that we have tI,.e equality mo + ... + mn = 1. The centre of gravity of these masses (points) is the point A, the tip of the vector OA which has the form of the linear combination OA = ~Al +.. ·+mnOAn of the vectors OAi joining the origin o with the points Ai. The numbers mo, ... , mn are called barycentric coordinates of the point A. They are related as mi ~ 0, mo+" ·+mn = 1. By changing the masses mo, ... , ffin we force the point A to change its position. As a result, it runs through a certain set ,1n, which is called an n-dimensional simplex. In other words, the simplex,1n is a convex linear hull of the points Ao, ... ,An+l • Therefore it is occasionally referred to as a rectilinear (or Euclidean) simplex. Figure 1.1.7 gives the simplest examples. A zero-dimensional simplex is represented by a point, one-

n-dimensional simplex

m,

6

1. Polyhedra. Simplicial Complexes. Homologies

dimensional by a segment on a plane, two-dimensional by a triangle in a three-dimensional space. A three-dimensional simplex can be realized as a filled tetrahedron in a three-dimensional space. The points Ao, ... , An are called the vertices of the simplex. Clearly, a rectilinear simplex is completely defined provided that all of its vertices are given. A simplex is therefore denoted simply by the set of its vertices. We shall now examine the set of points of an n-dimensional simplex whose ith barycentric coordinate is equal to zero, i.e. mi = O. From the definition of simplex it is immediate that this set is in turn an (n - 1)dimensional simplex. Besides, this simplex (which we denote by Ll~-l) is embedded into the initial simplex Lln and is opposite to its vertex Ai -+-:--......... ... 'f---_I (Fig.U.8). The simplex Ll~-l is called the ith (n-l)-dimensionalface o AI. Ao AI. of the simplex Lln. Thus, an n-dimensional simplex has n + 1 faces of dimension n - 1. Fig. 1.1.8 Next, a k-dimensional face of the simplex Lln is the set of such points of the n-dimensional simplex Lln for which some n-k barycentric Simplex edges coordinates are equal to zero, while the rest k +1 barycentric coordinates change so that the corresponding masses (coordinates) are non-negative and their sum is equal to unity. Clearly, the k-dimensional face of the Simplex vertices simplex Lln is itself a k-dimensional simplex. One-dimensional faces of a simplex are usually called the simplex edges. How many k-dimensional faces has an n-dimensional simplex? Each face is uniquely defined by its k + 1 vertices which at the same time constitute part of vertices of the initial simplex Lln. Hence, the total Face number of k-dimensional faces in the simplex Lln is equal to C~!l, i.e. to the number of permutations of n + 1 taken k + 1 at a time. By the definition, an n-dimensional rectilinear simplex is a convex set, i.e. a segment joining any two of its points lies entirely inside the simplex. The geometric boundary of an n-dimensional simplex is the union Boundary of all of its (n - I)-dimensional faces. Each simplex has a uniquely determined centre, i.e. a point whose barycentric coordinates are equal to each other. This means that mo = ... = mn = I/(n + 1). Figure 1.1.8 shows the centres of one-, two- and three-dimensional simplexes. In what follows it will be convenient for us to work with oriented Centre of the simplex simplexes. We shall say that a simplex is oriented if a defmite order of its vertices is given. We assume that two orders of vertices of a simplex

...A-

1.1 Polyhedra

7

differing by an even permutation determine one and the same orientation of the simplex. If the orders of vertices differ by an odd permutation, they determine opposite orientations of the simplex. For example, if two neighbouring vertices of a simplex exchange places (within a given ordering), the orientation changes (Fig. 1.1.8). We shall sometimes denote an oriented simplex by +L1n and a simplex with opposite orientation by

- L1n. Here we can trace out the connection with linear algebra. The above definition of orientation is connected with the fact that a linear mapping of an n-dimensional simplex onto itself determined by some permutation of its vertices has either a positive or a negative determinant depending of whether this permutation is even or odd. So, we shall in some cases speak of a positive or a negative simplex orientation. It should be recalled here that any permutation of vertices of a simplex can be obtained as a composition of elementary permutations understood as a permutation of neighbouring vertices. At the same time, a linear mapping of a simplex onto itself, determined by permutation of two neighbouring vertices, has a negative determinant. In what follows we shall consider not only rectilinear (Euclidean) simplexes themselves, but also their various homeomorphic images, i.e. images of a rectilinear simplex under homeomorphisms. Such homeomorphic images of a simplex are called topological or curvilinear simplexes. Let us, for example, examine a standard tetrahedron inscribed in a sphere, i.e. positioned inside a sphere so that all of its vertices lie on the sphere. Then the centre of the tetrahedron coincides with the centre of the sphere. Projecting the edges of the tetrahedron from the centre of the sphere onto the sphere, we obtain curvilinear triangles which are the images of the equilateral faces of the tetrahedron (Fig. 1.1.9). Consequently, the sphere falls into the union of four curvilinear triangles - homeomorphic images of equilateral rectilinear triangles. Figure 1.1.9 shows the process of transformation of a rectilinear triangle into a curvilinear one. Curvilinear one-, two- and three-dimensional simplexes are also depicted. Thus, on each curvilinear simplex there exist vertices, curvilinear faces and edges. One can compose many more objects from curvilinear simplexes than from rectilinear convex simplexes. The concept of convexity is generally not so meaningful for the curvilinear simplex as it is for the Euclidean rectilinear one.

Curvilinear simp/ex

Positive and negative orientation

Fig. 1.1.9

Convex simp/ex

8 Rectilinear simplex

Linear map

1. Polyhedra. Simplicial Complexes. Homologies

At the same time, each curvilinear simplex "remembers" its origination from a rectilinear simplex. To this end one could fix a concrete homeomorphism of a rectilinear simplex onto a curvilinear one. From this it is immediately clear how the linear map of two topological curvilinear simplexes should be naturally defined. This is a topological map under which the pre-images of points of curvilinear simplexes (the pre-images belonging to rectilinear simples) are carried into one another through a linear map of the corresponding rectilinear simplexes.

1.1.3 Polyhedra. Simplicial Subdivisions of Polyhedra. Simplicial Complexes

Topological space

Simplicial space

Using simplexes one can compose more complex objects and figures. Let a set X of points of a Euclidean space (or, more generally, the so-called "topological space") be represented as a union of a finite or countable number of curvilinear simplexes of dimensions from zero to a certain n. In other words, the set X is "glued" of curvilinear simplexes. It will be useflli if we henceforth think of a topological (curvilinear) simplex as a pair consisting of a rectilinear simplex and its topological map into a certain set of points of a Euclidean space. The set of simplexes covering X must include all the faces of these simplexes. For simplicity we assume from now on the number of simplexes covering the set X to be finite. We now impose simple and natural restrictions upon the indicated subdivision. We shall say that curvilinear simplexes form a finite simplicial subdivision of the set X if the following two conditions are met. 1) There is a finite number of simplexes and each point of the set X gets into a certain simplex (is covered by a certain simplex). 2) Either two simplexes do not intersect at all (do not have common points), or one of them is a face of the other, or they have a cOmnlon face which is the intersection of these simplexes.

Fig. 1.1.10

Figure 1.1.1 0 represents all the versions of mutual disposition of twodimensional simplexes. Figure 1.1.11 shows some "forbidden" situations. As has already been mentioned above, with each curvilinear simplex

1.1 Polyhedra

9

there is naturally associated a corresponding topological map onto it of a rectilinear simplex. In this case one should be aware of the condition that two simplexes have, for instance, a common face. It is required that under glueing together of two simplexes along their common face, into one and the same point of the set X there go such points of faces of the rectilinear simplexes which are associated with one another through a linear map of one face into another. To put it differently, to the glueing of topological simplexes (faces) there must correspond the glueing of corresponding rectilinear simplexes (faces) through their linear map. If a set of points of a Euclidean space (more generally, a topological space) is divided into simplexes so that there hold the indicated conditions 1 and 2, this set is called a polyhedron. There exist different ways to subdivide one and the same set of points into the union of simplexes satisfying conditions 1 and 2. If one such subdivision if fixed, we shall say that we are given a simplicial subdivision of a given polyhedron or we are given a simplicial complex. In other words, a simplicial complex is a set of all simplexes (and their faces) of a given subdivision of a polyhedron, where the particular manner in which these simplexes are glued together is indicated. One and the same polyhedron may have many distinct simplicial subdivisions, that is, there exist many different ways to represent one and the same polyhedron in the form of a simplicial complex. In the sequel, to simplifY calculations we shall sometimes consider divisions of a space into simplexes which are allowed to have several common vertices (see examples for two-dimensinal polyhedra below). Furthermore, to calculate simplicial homology groups one can employ cells (for their exact definition see below). This means that it suffices to represent the space as a union of closed sets the interior of each of which is homeomorphic to an open disc (of some dimension). In particular, in covering the space with simplexes one can reject condition 2. This does not affect the calculation of the homology groups. It is not at all difficult to define polyhedra glued of a countable number of simplexes (infinite polyhedra). To do so one should require that each point of a given set X be covered with only a finite number of simplexes and that the neighbourhood of each point in the set X be obtained as a union of neighbourhoods of this point in each of the

Fig. 1.1.11

Polyhedron

SimpliCial complex

Infinite polyhedra

Neighbourhood of the point

10

Incident

Simplicial complex

Euclidean polyhedron

1. Polyhedra. Simplicial Complexes. Homologies

simplexes it belongs to. But further on we shall be basically interested in finite polyhedra. Two simplexes of a simplicial complex X are called incident if one of thttm is a face of the other. A simplicial complex (and the corresponding polyhedron) is called n-dimensional if it contains at least one n-dimensional simplex and does not contain higher-dimensional simplexes. We now sum up what has been said above. 1) A polyhedron is a set of points which can be subdivided in a regular way into simplexes. 2) A simplicial complex is a set of simplexes and their faces which constitute this subdivision. 3) The polyhedra defined above should naturally be called curvilinear or topological. 4) The set of points of a Euclidean space which is the sum of a finite or a countable number of rectilinear simplexes satisfying the conditions listed above is occasionally called a Euclidean polyhedron.

From what has been said it is clear that the topological polyhedron can be defined in an abstract manner regardless of its embedding into . a Euclidean space. We are not, however, interested in the extension of generality since one can actually prove (we shall not do this here) that Glueing any finite topological polyhedron is homeomorphic to a certain Euclidean polyhedron i.e. can be embedded into a Euclidean space (maybe, of a Common face large enough dimension).

1.1.4 Examples of Polyhedra

Fig.1.1.12

From the preceding subsection one can see that a polyhedron can be represented as a result of glueing of a certain (finite or infinite) number of simplexes along some of their common faces. Glueing is understood here as follows. We take two simplexes, in each of them take one face and identify these faces through a linear map (in case the simplexes are rectilinear) or through the corresponding topological map (in case the simplexes are curvilinear) (Fig. 1.1.12). This procedure gives a more complex object composed of two simplexes glued along one common

1.1 Polyhedra

11

face. This provides great possibilities for visual construction of concrete polyhedra since the process of glueing of different simplexes can be infmitely continued.

Example 1. A Euclidean space is an infmite polyhedron. For example, a plane can be covered (titled) with equilateral triangles. Such and similar tilings of a plane can be seen in many gmphical canvases of a well-known painter M.C. Escher. A three-dimensional space can be tiled with tetrahedra (Fig. 1.1.13). In this figure one can see one of the many ways to subdivide a standard three-dimensional cube into a union of simplexes (tetrahedra). These tetrahedra are not equilateml. Covering the entire space with cubes (this is easy to do) and subdividing each of them by the indicated recipe we just obtain the subdivision of space into simplexes, as required. A flat ring and a cylinder are homeomorphic and are polyhedra (Fig. 1.1.13). Taking a rectangle and identifying its opposite sides with orientation reversal, we obtain a Mobius strip (Fig. 1.1.13) which is a polyhedron as well. It is natural that an individual simplex is (the simplest) polyhedron too. Example 2. A circle and a two-dimensional sphere are finite polyhedra. For a circle this fact is obvious, while for a two-dimensional sphere 8 2 we have already constructed a simplicial subdivision (see Fig. 1.1.9). It can be readily proved, by way of a simple extension, that an n-dimensional sphere sn is also a finite polyhedron (prove this!). Recall that the sphere 8n is given as a set of points in a Euclidean space ]Rn+l, such that there holds the equation (xli+···+(xn+l)2 = r2, where xl, ... ,xn+l are Cartesian coordinates of the point x. The number r is called the mdius of the sphere. Clearly, a sphere is homeomorphic to the boundary of a threedimensional cube. To make sure of this, it suffices to inscribe a sphere into a cube in such a fashion that its centre coincide with the cube centre (Fig.1.1.14) and then project the sphere from its centre onto the cube boundary. One can readily make sure that this projection determines the homeomorphism. Consequently, in some cases it is convenient to regard a sphere as a two-dimensional polyhedron which is the boundary of a standard cube. Since a square can be represented as glueing together two

Fig. 1.1.13

Fig. 1.1.14

12

1. Polyhedra. Simplicial Complexes. Homologies

triangles (Le. two-dimensional simplexes) (Fig. 1.1.14), it follows that by subdiving all the cube faces into pairs of triangles, we obtain another representation of a two-dimensional sphere in the form of a polyhedron. Here the number of two-dimensional simplexes constituting the sphere is obviously equal to 12. It should be recalled that a sphere is the boundary of a standard ball. We shall drill several holes in a ball, as shown in Fig. 1.1.15. As a result, we obtain a three-dimensional set of points with a two-dimensional boundary. This two-dimensional set appears also to admit the polyhedron Fig.1.1.15 structure.

Fig. 1.1.16

r------ -----------

fB r

I

" I

~

Example 3. Two-dimensional spheres with several "handles". We shall begin with a particularly simple case. We shall represent a sphere in the form of a polyhedron, as shown in Fig. 1.1.15, i.e. in the form of the plate boundary. Not to overload the figure, we shall not depict subdivision of side faces of the plate into triangles. We remember that any square can be triangulated, i.e. subdivided into triangles. Then we drill a square hole from the plate, as shown in Fig. 1.1.16. Clearly, we have obtained a new polyhedron. Indeed, the two-dimensional boundary of a "plate with a hole" consists of several rectangles each of which can be divided into two triangles (by drawing a diagonal in the rectangle). Considering the obtained two-dimensional polyhedron up to a homeomorphism, one can easily see that it is equivalent to a "roll" (Fig. 1.1.16). This polyhedron is called a torus (a two-dimensional torus). Applying one more homeomorphism, one can easily transform this torus into a sphete to which one "handle" is glued (Fig.1.1.16). So, a torus can be thought of as a sphere with one handle. A torus can also be glued from a square, i.e. by identifying its opposite sides as indicated in Fig. 1.1.16. The first step is to glue two sides b to obtain a cylinder. The second step is to glue the bases of the cylinder to obtain a torus. Now let us examine a plate with two holes drilled (Fig. 1.1.17). Clearly, we again obtain a two-dimensional polyhedron composed of two rectangles each of which can be subdivided into simplexes. Applying an appropriate homeomorphism, one can transform a ''plate with two holes" into the object depicted in Fig. 1.1.17 and called in the mathematical

rr

____ ..,r

I

Fig.1.1.17

1.1 Polyhedra

13

slang a ''pretzel''. If we go on transforming this figure, we can represent it in the form of a sphere with two handles, as shown in Fig. 1.1.17. The described process of drilling holes in a plate can be continued. It is clear that in a plate one can drill an arbitrary finite number of square holes (Fig. 1.1.18). Applying appropriate homeomorphisms, we obtain spheres with an arbitrary number of handles. Hence, we have discovered an infinite series of two-dimensional polyhedra - a sphere with 9 handles. The number 9 is called the genus of a surface. As has already been mentioned above, we shall allow some simplexes to have two common vertices. This will not affect our calculations of homology groups and will simplify significantly (in some cases) the representation of topological spaces as unions of simplexes. Example 4. A projective plane. Recall that polyhedra can be obtained by means of glueing simplexes along their common faces. Figure 1.1.19 Fig. 1.1.18 gives an example of a polyhedron obtained by glueing eight triangles which compose the subdivision of the square. Those sides of the triangles which should be glued together are labelled with identical letters, the directions of the arrows showing the orientation for glueing correspondJ. ing sides (segments). Ifwe digress from the system of triangles and view the whole square, it becomes obvious that the polyhedron in question will c/ be obtained if on the sides of the square we identify the pair a and a . a (with allowance made for the arrows) and the pair (3 and (3 (also with d., J allowance made for the arrows) (Fig. 1.1.19). The polyhedron obtained "" ::.....::... :-: ::. c is called a projective plane or a two-dimensional projective space. In the sequel we shall repeatedly encounter this polyhedron and get acquainted Fig.1.1.19 in more detail with its various interesting properties.

J..~.::.:..

.1

~cJ

:; ••::: ••• :.:.

Example 5. A Klein bottle. We again take a square divided into eight triangles, but change (as compared to Example 4) the rule of glueing the sides of the square according to Fig. 1.1.20. The polyhedron thus obtained is called a Klein bottle. It is convenient to represent it in a threedimensional space as follows. First we glue two sides (3 and (3 to obtain a cylinder (Fig. 1.1.20) on whose boundary circles we see the arrows a and a. To identify these circles (so that the directions of the arrows coincide), one should, unfortunately, "puncture" the surface, which leads to self-intersection. Fig. 1.1.20

I

a

14

Topological manifolds

Bing house

+M Fig. 1.1.21

, ,,

, ,,

,,

I

I

I

,

1. Polyhedra. Simplicial Complexes. Homologies

Among the variety of all polyhedra there exists an important class of polyhedra called manifolds (topological manifolds). A polyhedron of dimension n is called an n-dimensional manifold (or an n-manifold) if each of its points has a neighbourhood (inside the polyhedron) which can be homeomorphically mapped onto the interior of a standard ndimensional ball Dn. Recall that a standard open ball is the set of points x in a Euclidean space ]Rn, satisfying the inequality (xl )2+ .. .+(xn)2 < 1. A closed ball (or a ball with boundary) is the set of points x satisfying the inequality (xl)2 + ... (xni :::; 1. Clearly, a closed ball is obtained from an open one by adding a boundary sphere. The examples of polyhedra listed above (Euclidean space, sphere, sphere with handles, projective plane, Klein bottle) are in fact manifolds. Figure 1.1.21 demonstrates two polyhedra (one-dimensional and twodimensional) which are not manifolds. In both cases the point 0 does not have a neighbourhood (in the polyhedron) homeomorphic, respectively, to a one-dimensional disc (i.e. to an interval) and a two-dimensional disc.

Example 6. "Bing house". This two-dimensional polyhedron is needed for dealing with the problems of three-dimensional topology discussed in chapters to follow. To construct this polyhedron, we take a cube and divide it into two parts by the equatorial horizontal plane (Fig. 1.1.22). On the upper face of the cube we start "eating out" a vertical hole downwards. We reach the equatorial plane, "eat it away" to emerge in the lower chamber and "eat out" the whole of it except for the figure B which has been formed by a similar construction starting with the lower face of the cube. Analogously, we "cut our way" into the cube from below, "eat out" a hole inside the B, puncture the equatorial plane, run Fig. into the upper chamber and "eat out" its interior, except for the walls of 1.1.22 the construction A.

1.1.5 Barycentric Subdivision Open ball Closed ball

As has already been mentioned, there exist many ways to represent each polyhedron in the form of a simplicial complex, i.e. to divide it into simplexes. Such ways are infinitely many. To have a deeper insight into

1.1 Polyhedra

15

polyhedra, it turns out useful to distinguish a special class of simplicial divisions called barycentric subdivisions (of a given simplicial division). Generally speaking, the operation of subdivision (reduction) of a simplicial complex consists in the following. We divide each simplex into smaller simplexes in such a way that we again come to a simplicial complex. Clearly, the polyhedron remains unchanged under this operation. Affected will be the simplicial complex whose support (''body'') is a given polyhedron. In the sequel it is good practice to view a polyhedron as a set of points and a simplicial complex as a certain "scheme" of this set which results from representing the set as a union of simplexes. And again we can reduce the simplex to fragments in many ways. We are particularly interested in one special way of this reduction. We now proceed to its exact description. We consider all the vertices of a given simplicial complex. We shall not change them (a point cannot be reduced to smaller fragments). Then examine all one-dimensional edges, i.e. one-dimensional simplexes. We introduce (add) new vertices taking the centres of the one-dimensional simplexes. Each edge splits into two one-dimensional simplexes. Then we take all the two-dimensional simplexes and the new vertices will be the centres of these simplexes. We assume the concept of the centre of a curvilinear simplex be well defined since all our simplexes are always topological images of rectilinear simplexes whose centre (the centre of gravity) is uniquely defmed (Fig. 1.1.23). From the centre of the two-dimensional simplex we project (using straight lines on the rectilinear simplex) already subdivided one-dimensional boundaries (i.e. edges). In other words, we draw medians in the triangle. They intersect in the centre of gravity of the triangle (Fig. 1.1.23). Hence, each one-dimensional simplex has fallen into two one-dimensional simplexes. Each two-dimensional simplex has fallen into six two-dimensional simplexes. Now take three-dimensional simplexes. In each of them we establish the centre and project from it already subdivided two-dimensional faces of this simplex. As a result, we divide the three-dimensional simplex into 6 . 4 = 24 smaller three-dimensional simplexes. Clearly, this process can be extended to higher dimensions. As a result, we are led to a new simplicial complex which is called a barycentric subdivision of the initial one.

Barycentric subdivision

Centre of a curvilinear simp/ex

Fig. 1.1.23

Centre of gravity

16

Process of barycentric subdivision

Fig. 1.1.24

1. Polyhedra. Simplicial Complexes. Homologies

The process of barycentric subdivision can be repeated, i.e. one can realize a two-, three- and m-stage barycentric subdivisions. It is intuitively clear that by increasing m we shall be able to divide the initial simplicial complex into simplexes of arbitrarily small size. So, reducing the initial simplicial division into smaller fractions, we cover the initial polyhedron with increasingly small simplexes whose sizes are simultaneously and uniformly decrease. Visually, this process leads to the following. If a polyhedron is divided into a small number of simplexes, then it is a sufficiently rigid body provided that each simplex is regarded as a rigid body and that simplexes are allowed to rotate about their common faces. Increasing the number of simplexes, we introduce a larger number of edges about which the neighbouring simplexes can rotate. As a result, the polyhedron becomes "softer", pliable and deformable by a larger number of ways. Suppose 0 is a vertex of a simplicial complex. Consider all the simplexes incident to this vertex. Their union is called the simplicial star of the given vertex. Clearly, it is the neighbourhood of the given vertex in the polyhedron (Fig.1.1.24).

1.1.6 Visual Material Simplicial star of a vertex

Cubic homologies Cubic partitions

Cubic polyhedron

In this section we provide a visual demonstration of polyhedra as an example of geometric constructions. Figure 1.1.25 gives an example of a rather complicated polyhedron, partitioned into a union of cubes and parallelepipeds. As is known, all these objects can in turn be trianglated, which will just give us triangulation of polyhedra. But it turns out that one can consider cubic partitions of polyhedra, i.e. take for the elementary object a cube instead of a simplex. This theory has both advantages and disadvantages. On the basis of this theory one can determine the so-called cubic homologies (compare with the material of Sect. 1.2). A similar idea (but in a more rectilinear version) is expressed in Fig. 1.1.26. For the sake of visualization, it is generally convenient to think of polyhedra (even if they are smooth manifolds) as of constructions composed (glued) of cubes, parallelepipeds. This image is apparently psychologically more convenient than the one composed of tetrahedra. Figure 1.1.26 illustrates a "cubic polyhedron" with a cut. This image

Page 17, Fig. 1.1.25. Simplicial polyhedra. Cubic partitions and cubic homologies. To M. Bulgakov's "The Master and Margaret": night talk between Pontius Pilate and Afranii (story of Judas)

18

Common boundary Boundary of the chain

M.A. Bulgakov

1. Polyhedra. Simplicial Complexes. Homologies

will soon be of use in the discussion of the concept of chains and of the important algebro-geometric fact that summation of chains leads in some cases to mutual annihilation of common pieces of the boundary. "Glueing" the crack in the polyhedron in Fig. 1.1.26, we thus obviously decrease its boundary: the crack boundaries are mutually annihilated. Here the "crack" separates the simplicial chain (polyhedron) into two parts with "large boundaries", but when glueing the crack, we decrease the common boundary of the chain, i.e. 8(A+B) =8A+8B-(8An8B). Figure 1.1.27 represents prisms into the union of which there splits the direct product of the polyhedron by a segment. Some of the mathematical figures included in the book have in addition to their primary mathematical meaning also another, "non-formal", extra-mathematical, associative meaning. For example, the drawings by the author, listed above, were once intended not only as mathematical illustrations, but also as illustrations to the remarkable novel by M.A. Bulgakov "Master and Margaret". About 40 illustrations were created by the authors many years ago for the first Russian edition of Bulgakov's novel. But this book was published without any pictures. Now the reader of the present mathematical book can see not only the geometrical content of these drawings but also can try to connect them with remarkable story about Jesus Christ and Pontius Pilate (from Bulgakov's novel): Christ is delivered to Pilate, story of Judas, cruciftxion of Christ, story of Matthew. These are Figs. 3.3.16; 3.1.12; 1.3.13; 1.3.14; 2.5.39; 1.3.10; 4.1.6; 2.5.37.

1.2 Simplicial Homology Groups of Simplicial Complexes (Polyhedra) 1.2.1 Simplicial Chains Simplicial chains Simplicial homology groups

The language of polyhedra and simplicial homology groups (see below) is rather obvious and convenient for the first acquaintance with the important geometric concepts. This language is primary enough, based on

Page 19, Fig. 1.1.26. Cubic polyhedron and the action of boundary oper-

ator. To M. Bulgakov's "The Master and Margaret"; crucifixion of Jesus Christ and hurakan over Jerusalem Page 20, Fig. 1.1.27. Simplicial prisms and cubic homology groups. To M. Bulgakov's "The Master and Margaret"; crucifixion of Jesus Christ; story of St. Matthew - the Jesus' pupil; Gospel according to St. Matthew

1.2 Simplicial Homology Groups of Simplicial Complexes (polyhedra)

21

a small number of concepts, although in concrete calculations it is sometimes cumbersome. In modem geometry, the calculation of homology groups is usually carried out in other, more versatile terms, for instance in terms of the so-called cell homologies. But the relative simplicity of calculation of cell homologies rests on a fairly sophisticated procedures of the very definition of cell homology groups. The advantages and disadvantages of the different definitions of homology groups can be tabulated as follows. Simplicial homologies Cell homologies

The definition is The proof of the invariance simple is complicated The definition is The proof of rather complica- the invariance ted of simplicial is simple homologies

Cell homologies Cell homology groups

Concrete calculations are rather complicates Concrete calculations are much simpler than those

This shows the advantage of cell homologies since in concrete investigations it is preferable to deal with easily computable (even though complicatedly defmed) objects. We have however preferred to acquaint the reader with simplicial homologies (and, accordingly, with polyhedra) since our prime concern is visual and elementary character of the material. The acquaintance with simplicial homologies makes it possible to comprehend very quickly the very idea of homologies which plays an important role in modem geometry, mechanics and physics. The same general idea underlies the concept of cell homologies, and therefore we think it fruitful to apply to the language of polyhedra and simplicial complexes at early stages of studying the fundamentals of topology. After this it is easy to go over to cell homologies. We shall consider a polyhedron X and fix its arbitrary simplicial division. For simplicity, we shall further on denote the corresponding simplicial complex by the same letter X. We shall examine the set of all k-dimensional simplexes of the polyhedron X, number them in some (arbitrary) order and associate with each simplex (also in an arbitrary way) some orientation. We denote these simplexes by Llf, where i is the ordinal number and k is the dimension of the simplex. The number i can change from unity to infmity. Simplex numbering and orientation are assumed to be fixed. Let us consider some Abelian group G, for example, a group of integer number Z, a group of real numbers I. or

Idea of homologies

Abelian group

22

Integer-valued chains Integer-valued function

Elementary chains

Group of simplicial integer-valued chains

1. Polyhedra. Simplicial Complexes. Homologies

a finite group Zp of residues modulo p. The indicated groups are most frequently used in topology. For definiteness, we shall for the present be concerned with the case G = Z. An extension of all further constructions to the case of an arbitrary Abelian group is made automatically and does not add any new essential points. We shall examine linear combinations of the form c = 2:i aiLl7, where ai are integers (either positive or negative) and ..17 are kdimensional simplexes, and assume the sum to contain only a finite number of nonzero ai. Such linear combinations are called integer valued k-dimensional chains (or simplicial integer-valued k-dimensional chains). Such a chain can be interpreted as an integer-valued function defined on the set of all k-dimensional simplexes, i.e. c: ..17 --t ai for all i. This function is assumed to be nonzero only on a finite number of simplexes and to be odd, i.e. c ( - ..17) = -ai = -c (..17), where by -..17 we denote the simplex ..17 with a reverse orientation. Working with chains we shall henceforth mainly use the language of linear forms for it is convenient for calculations. The simplest examples of chains are those of the form 1 . ..17 and ( -1)· ..17 which are sometimes called elementary. Chains can be summed up as ordinary linear forms. Namely, the sum Cl + C2 of two chains Cl = 2:i aiLl7 and C2 = 2:i biLl7 is the chain Cl + C2 = 2:/ai + bi ) ..17· Consequently, the set of all k-dimensional integer-valued chains forms an Abelian group which we denote by Ck(X). The generators of this group are elementary chains of the form 1 . ..17. Definition. The group Ck(X) is called the group ofk-dimensional simplicial integer-valued chains ofthe polyhedron X (ofthe simplicial complex X.)

This shows how one can define chains with coefficients from an arbitrary Abelian group. To this end one should consider linear forms 2:i gi Ll where the coefficients gi belong to the abelian group G. The arising Abelian group of chains with coefficients in the group G will be denoted by Ck(X, G). In this notation, the above-defined group Ck(X) of integer-valued chains will be written as Ck(X, Z). This is afree Abelian group because it is represented in the form of the direct sum of a certain number of infinite Abelian groups Z.

7,

Free Abelian group

1.2 Simplicial Homology Groups of Simplicial Complexes (Polyhedra)

23

If the polyhedron X is finite, the group of its k-dimensional integervalued chains has ajinite number ofgenerators (i.e. is a finitely generated Abelian group). Thus, we have associated with each polyhedron X (each simplicial complex X) a set of Abelian groups Co(X), CI (X), ... , Cn(X). Clearly, for k < 0 and k > n the groups of chains are undefined since there are no simplexes with dimension k less than zero or greater than n = dim X. However, it is sometimes convenient to assume (by definition) all the groups Ck(X), where k > n, to be zero.

1.2.2 Chain Boundary We shall consider an integer-valued elementary chain 1 . Llk (which we denote for simplicity by Llk) and defme its algebraic boundary as a certain (k - 1)-dimensional integer-valued chain which we shall now represent by an explicit formula. As a preliminary step we shall discuss the concept of orientation induced on the ith face Ll~-I of the simplex Llk. Recall that the orientation of the simplex Llk is assumed to be given and fixed. The simplex Llk is given by the set of its vertices Ao, AI'"'' Ak (see Sect. 1.1). Clearly, the (k - 1)-dimensional faces Ll~-I of the simplex Ll k are obtained by a successive elimination of its vertices, i.e. Ll~-I = (Ao, ... ,Ai-I,Ai+l, ... ,Ak). We assume that the orientation of the face Ll~-I (with the number i) induced on it by the orientation of the simplex Llk is determined by the sign (_1)i. We now give several examples. A one-dimensional simplex (a segment) Lli is given by its two vertices Ao and Al (Fig.1.2.l). Then its faces are represented by a pair of points Ao and Al taken one (Ao) with a "-" and the other (AI) with the "+" sign. Indeed, Lli = (Ao, AI), Ll~ = (-1)0 Al = AI; Ll~ = (_1)1 Ao = -Ao. Geometrically, the difference in the sign is explained by the fact that at the point Ao (Fig.1.2.l) the arrow indicating the simplex orientation is turned inside the simplex, while at the point Al outside the simplex. So, when moving along the straight line on which the segment lies, we first enter the segment (the "-" sign) and then leave it (the "+" sign). Therefore, the concept offace orientation introduced above corresponds to the intuitive idea. We shall consider the following example.

Finite number of generators

Chain boundary

Algebraic boundary

Induced orientation

One-dimensional simplex

.

At .... . ...

-zEi

A.

,11.

A1 +

+: f).:.~

.......... Ao - A:z. Fig. 1.2.1

~

6 -+

24 Two-dimensional simplex

1. Polyhedra. Simplicial Complexes. Homologies

A two-dimensional simplex is given by three vertices Llz = (Ao, AI, Az) (Fig. 1.2.1). The simplex has three one-dimensional faces (sides of the triangle), namely, an edge Ll~ = (-I)o(AI,A z) = (AI, Az), an edge .11 = (-1)1 (Ao,Az) = -(Ao,Az) = (A2,Ao), and an edge = (-Ii (Ao,AI) = (Ao,A I ). In Fig. 1.2.1 the arrows are placed on one-dimensional faces of the simplex to show the orientation induced on these faces. It should be emphasized that the edges (AI, A2) and (A o, AI) are positively oriented and the edge (Ao, A2) is negatively oriented. Therefore, we have replaced the edge -(Ao, A2) by an equivalent edge (A2, Ao). From Fig. 1.2.1 it is seen that the formal definition of induced orientation, given above, is in close agreement with geometrical intuition. Indeed, from the calculations we have revealed that the boundary edges endowed with induced orientation give one and the same direction ofrotation on the triangle. Going round its boundary clockwise, it would be natural to expect that the passed-by edges are endowed with arrows indicating the same direction of motion. This is precisely what we have obtained on the basis of the algebraic definition of induced orientation. Now we can define an important concept of algebraic boundary of a simplex.

.11

Boundary edges

Algebraic boundary of a simplex

Induced orientation

Definition. The boundary 8Llk of an oriented simplex Llk is the sum of all its (k - I)-dimensional faces taken with induced orientation.

Boundary operator

We shall write the simplex boundary in the algebraic language. We obtain: k

aLlk = L(_1)i Ll~-I = Ll~-I -

Ll~-I

+ Ll~-I - ... Ll~-I

i=O

In terms of vertices, this formula becomes 8(Ao, ... , Ak) =

k

L( _I)i (Ao, ... , Ai-J, ... , Ai-I, ... , Ak) i=O

In other words, we eliminate successively the simplex vertices, obtain the corresponding faces (of dimension smaller by unity) and sum up (in the form of a formal linear combination) taking them with the sign (_I)i, where i is the number of the eliminated vertex. It should be clarified

1.2 Simplicial Homology Groups of Simplicial Complexes (Polyhedra)

25

that the (k - I)-dimensional simplex ..1~-1 obtained after the vertex Ai has been eliminated is depicted geometrically as the face opposite (in the simplex ..1k) to the vertex Ai (Fig. 1.2.2). It is now obvious that this algebraic definition of the simplex boundary (the algebraic character manifests itself in the allowance made for the sign, i.e. for orientation) is quite natural. Indeed, the boundary of a one-dimensional simplex is the sum of its vertices of the form:

Opposite face

The boundary of a two-dimensional simplex is the sum of its edges:

All these three edges determine one and the same clockwise rotation, that is, precisely that orientation which is induced on the boundary of a triangle if inside it one sets a clockwise rotation. Weare now in a position to derme the boundary of an arbitrary chain. Suppose e = al..1f + ... + aq..1~ is a simplicial integer-valued chain.

Boundary of a chain

Definition. The boundary of a k-dimensional chain e is a (k - 1)dimensional integer-valued simplicial chain 8e given by the following explicit formula:

Given this, we assume that the meaning of the expression 8..1~ has been defined above. In other words, on setting the operator 8 on elementary chains 1 . ..1j, we then extend it "by linearity" to arbitrary linear combinations of elementary chains thus obtaining a well dermed operator on the entire Abelian group of chains Ck(X). This operator is called the boundary operator, or the operator of the boundary, or the operator of taking the boundary.

Boundary operator

26

1. Polyhedra. Simplicial Complexes. Homologies

1.2.3 The Simplest Properties of the Boundary Operator. Cycles. Boundaries

a

Abelian group of k-dimensional chains

Square of the boundary operator

1) The operator is linear. This means that for any k-dimensional chains CI, C2 and for any integer coefficients a, b there holds the identity o(aci + bC2) = aoC) + bOC2. To say it differently, the operator a determines a homeomorphism of the Abelian group of k-dimensional chains Ck(X) into the Abelian (k - I)-dimensional chain group Ck-I (X). We shall sometimes denote this operator by Ok indicating explicitly the dimensions of the chains on which the operator is defined. Consequently, in the preceding subsection we have in fact defined the whole family of operators 80, 01, ... , Ok, ... , an, where n = dim X (Le. the dimension of the complex X). Thus, Ok: Ck(X) -+ Ck-I (X). 2) The square of the boundary operator ais identically zero.

a

Proof. Since the operator is linear, it suffices to prove the statement only for elementary chains, Le. for a single k-dimensional simplex ..1k . It is sometimes convenient to write the boundary operator in the form O..1 k = Ef=o(-1 )i(Ao, ... , Ai, ... ,Ak),where the sign" (the hat) implies that the symbol under it is omitted, eliminated. We must prove that 02..1 k = O. Applying the operator for the second time, we obtain a long sum which necessarily involves the following two summands: A

{j2 == 0

"

a

(-I)i(-IY (Ao, ... ,Aj,'" ,Ai'''' ,Ak) Formula for boundary operator

and

.

. 1

A

A

(-1)1(-1)'- (A o, ... , Aj , ... , Ai, ... , Ak )

We can assume without loss of generality that j < i. In the first summand the vertex Ai was discarded with the first application of the operator Le. before the vertex Aj was discarded (already with the second application of the operator a). In the second summand the picture is different. Here the vertex Aj was first to be discarded (with the first application of a), and only after that it was the vertex Ai with the second application of a). Consequently, to know the sign when we discard the vertex Ai, we should count the number of vertices leftward of Ai. It is clear that leftward of it there are i-I vertices, for the vertex Aj was discarded in the preceding step. Since the signs (-I)i-J and (-1 )i+j-l are opposite, the

a,

1.2 Simplicial Homology Groups of Simplicial Complexes (Polyhedra)

27

two summands that we have distinguished enter in the overall sum with opposite sign and therefore are mutually annihilated, which completes the proof. Its geometrical meaning is absolutely clear: the boundary of a simplex has no boundary. 3) Ijo(ae) = 0 and at-O, then oe= O. This assertion is obvious, as we are examining integer-valued chains and from the equality o(ae) = 0, where at-O, it follows that aoe = 0, i.e. a(oe) = 0 and oe = 0 (the coefficient at- 0 can be cancelled out).

Boundary of a simplex has no boundary

Definition. A chain z is called a cycle if its boundary is equal to zero, i.e. OZ = O. A chain b is called a boundary if it is representable as b = oh, i.e. in the form of the boundary of a certain chain h whose dimension is greater by unity.

Cycle

Clearly, all these objects (boundary operator, cycles, boundaries) can also be defined for an arbitrary Abelian group of coefficients. We shall not repeat here the corresponding definitions and invite the reader to do it himself. 4) The set of cycles forms an Abelian subgroup in a chain group. This subgroup is denoted by Zk(X). The set of boundaries forms an Abelian subgroup in a chain group. This subgroup is denoted'by Bk(X). Each boundary is a cycle. This means that the subgroup Bk(X) is always contained in the subgroup Zk(X). These statements follow from linearity of the operator 0 and from the equality & == 0 proved above. A cycle should not necessarily be a boundary. This means that in the general case the group Zk(X) can be larger than the subgroup Bk(X). So, Ok: Ck(X) --+ Bk_I(X); Ok: Zk(X) --+ 0, Ok-I: Bk_I(X) --+ O.

1.2.4 Examples of Calculations of the Boundary Operator We shall assume (by definition) that the boundary ofany zero-dimensional chain is equal to zero (since there exist no nonzero chains of negative dimension). So long as any zero-dimensional chains a linear combination of points (vertices of a polyhedron) with integer coefficients, our

Boundary

Set of boundaries Each boundary is a cycle

28

1. Polyhedra. Simplicial Complexes. Homologies

agreement is demonstrative: a point has no boundary (i.e. the boundary of a point is equal to zero). Accordingly, any zero-dimensional chain is a cycle. We shall consider a two-dimensional simplex ..12 as an elementary chain. Then its boundary 8..12 has the form Cl + C2 + C3 (Fig. 1.2.3). Therefore, the one-dimensional chain c =Cl +C2 +C3 is the boundary. At the same time, it is obviously a one-dimensional cycle. We shall consider a square divided into four two-dimensional simplexes (Fig. 1.2.3) on each of which we determine a clockwise rotation. We shall take the chain which is the sum of these four simplexes with coeffi(;ients 1 and calculate its boundary. To do so, we should calculate the boundaries of all the four simplexes and then sum up these boundaries. Given this, the following will happen. Each one-dimensional edge lying inside a square will enter this sum twice, each time with a different sign. The point is that orientations induced on an edge by two adjoining triangles are reverse. Consequently, all the edges p, q, T, 5 will cancel out and will not enter the final sum. At the same time, each of the onedimensional edge a, b, c, d will enter the sum only once, and therefore the boundary of the square regarded as a two-dimensional chain will coincide with its geometric boundary - the sum of the four oriented edges a+b+c+d. Let us consider a square with a hole (Fig. 1.2.3) triangulated into eight triangles. We endow each of them with orientation (clockwise rotation) and take a two-dimensional chain c which is the sum of all these triangles with coefficients 1. Acting as in the preceding case we see that Fig. 1.2.3 all the interior one-dimensional edges are mutually annihilated. As a result, the boundary of the chain will consist of two sumes: a + b + c + d (with a "+" sign) and m+n+h+l (with a "- sign). Thus, the algebraic Clockwise rotation boundary of a square with a hole coincides with its geometric boundary, the exterior part of the boundary being taken with a "+" and the interior with a "-" sign. Consequently, 8c =a+b+c+d - m - n - h -t. Since a square with a hole is homeomorphic to a ring, the algebraic boundary of a ring consists of an exterior circle (with a "+" sign) and an interior circle (with a "-" sign.) From the demonstrative point of view, a cycle may therefore be thought of as a "closed surface", i.e. a surface without boundary. The Closed surface

Any zero-dimensional chain is a cycle

1.2 Simplicial Homology Groups of Simplicial Complexes (polyhedra)

29

boundary ofa chain is its geometric boundary if all the chain-constituting simplexes are taken with coefficient 1 (Fig. 1.2.4). In the preceding subsection we have mentioned that a cycle is not necessarily a boundary. For example, if we triangulate a two-dimensional sphere and take the chain which is the sum of all the triangles obtained with coefficients 1, we obviously come to a two-dimensional cycle. At the same time, it cannot be represented as the boundary of a certain three-dimensional chain for the simple reason that our complex does not have a single nonzero three-dimensional chain (the sphere dimension is equal to two). We could also give more delicate examples (we shall soon present them) when cycles are not boundaries.

Fig. 1.2.4

1.2.5 Simplicial Homology Groups Let Z be a k-dimensional cycle, i.e. 8z = O. We shall say that it is homologic to zero if it is the boundary, i.e. z = 8h for a certain (k + 1)dimensional chain h. By the way, in this case the chain h is generally speaking not uniquely defined. If z = 8h for some chain h, then there also holds the equality z = 8(h+I), where l is an arbitrary (k+ I)-dimensional cycle, i.e. 8l = O. Two k-dimensional cycles z\ and Z2 will be called homologic, if their difference z\ - Z2 is homologic to zero, i.e. z\ - Z2 = 8h for a (k + I)-dimensional chain h. The fact that the cycle z is homologic to zero is sometimes written as z rv O. Then the homology of the two cycles z\ and Z2 is written as Zl - Z2 rv 0 or Zl rv Z2. The relation z\ rv Z2 is occasionally called homology. If the cycle Zl is homologous to the cycle Z2, they differ by a boundary, i.e. there holds the equality Z\ = Z2 + 8h. The homology relation is determined not only for cycles, but for chains as well. The chains Cl and C2 are called homologous (ct rv C2) if they differ by a certain boundary, i.e. C\ = C2 + 8h or ct - C2 rv O. We now return to cycles. Suppose X is an arbitrary polyhedron, Zk(X) is its k-dimensional cycle group, Bk(X) the group of its kdimensional boundaries. Since the group Bk(X) is always contained in the group Zk(X) and both the groups are Abelian, it follows that the factor quotient group Hk(X) = Zk(X)/ Bk(X) is well defined.

Geometric boundary

Homologic cycles

Homology

Homologous chains

Group of boundaries

30

1. Polyhedra. Simplicial Complexes. Homologies

Definition. The group Hk(X) is called the k-dimensional integer-valued simplicial homology group of a polyhedron X. It is an Abelian group.

Integral-valued simplicial homology group

Homology groups with coefficients in the group G

Group of coefficients

Finite-order elements

Homologies are indivisible

If we consider chains with coefficients in an arbitrary Abelian group G, then, repeating almost literally all the previous constructions, we would arrive at the definition of the homology groups of the polyhedron X with coefficients in the group G. These groups are denoted by Hk(X, G), i.e. the group of coefficients is indicated In this notation, the integer-valued homology groups Hk(X) will be written as Hk(X, Z). We shall use both depending on the context. We shall discuss the concept of homology group in terms of homologous cycles. Let us consider the group of cycles Zk(X). Any linear combination of cycles homologous to zero is again a cycle homologous to zero. This just means that Bk(X) is a subgroup in Zk(X). Therefore, for each cycle z one can consider its coset with respect to the subgroup Bk(X), i.e. a set of cycles of the form z + oh, where h runs through all (k+ I)-dimensional chains. The coset obtained can be denoted by {z}. Then it is clear that the two classes {zI} and {Z2} either coincide or do not intersect. Clearly, the linear combination of the classes {Zl} and {Z2} is well defined, namely: a{zl}+b{Z2} = {azl +bZ2}. Thus, the group Hk(X) is interpreted as a group of classes of the form {z}. However, when speaking of elements o~ the group Hk(X) we shall often bear in mind the cycles themselves, i.e. some representatives of the class {z}. We shall point out an essential property of the homology group Hk(X). From the fact that the cycle mz is homologous to zero (where m is an integer) it does not at all follow that in the general case the cycle z itself is homologous to zero. In other words, "homologies are indivisible". From the point of view of the group Hk(X) this means that there may exist nonzero elements {z} from the group Hk(X), such that their integer multiple m {z} is equal to zero (in the group Hk ), i.e. mz '" 0, although z rf O. Such elements are therefore finite-order elements in the group Hk(X). This implies that the homology groups Hk are not generally free Abelian groups. They may contain subgroups consisting of finite-order elements. Thus, each group Hk can be represented as the direct sum of its two subgroups Ak and Bk, where Ak is a free Abelian group (the direct sum of a certain number of copies of the group Z) and Bk is a finite Abelian group. The group Ak is uniquely characterized by the number

1.2 Simplicial Homology Groups of Simplicial Complexes (Polyhedra)

31

(3k of constituent copies of the group Z. In the algebraic language, the number (3k is the rank of the group Ak, i.e. the minimal number of genemtors of this group. At the same time it is clear that the number (3k is also the mnk of the entire group Hk, i.e. (in this case) the minimal number of free generators (infinite-order genemtors).

Definition. The number (3k is called a k-dimensional Betti number of a polyhedron X. To summarize we shall say that to each n-dimensional polyhedron X we have assigned a set of Abelian groups Ho(X), ... , Hn(X) (homology groups). These groups are Abelian. The homology groups Hk(X) are equal to zero in all the dimensions k exceeding n (the dimension of the polyhedron). These groups have been calculated for a certain simplicial polyhedron partition. If the simplicial partition is fixed then associated with it is the following sequence of free Abelian groups Ck(X) (chain groups) and homomorphisms Ok (boundary operators): 0 ~ Co(X) ~ C1(X) ~ ... ~ Ck(X) ~ ... ~ Cn(X) ~ O. This sequence is called a chain complex (of a given simplicial complex). As we know, however, one can represent one and the same polyhedron in different ways in the form of a simplicial complex, i.e. triangulate it differently. But to different simplicial complexes there genemlly correspond different chain complexes. The simplicial homology groups Hk(X) might therefore be expected to depend on the choice of polyhedron triangulation. But this is not the case. The simplicial homology groups appear to be determined only by the polyhedron itself but not by its triangulation. Theorem. The simplicial homology groups of the polyhedron do not depend on the way the polyhedron is represented as a simplicial complex. .This fact increases sharply the importance of homology groups transforming them into algebmic objects that characterize the polyhedron itself and not the way of its determination (representation). In what follows we shall mainly deal with finite polyhedra. In this case all the homology groups Hk(X) are finitely generated. Each group Hk(X) is represented

Rank of the group

k-dimensional Betti number

Chain complex

In variance of the simplicial homology groups

Different triangulation of the polyhedron

32

1. Polyhedra. Simplicial Complexes. Homologies

in the form of the direct sum Hk(X) = Z EEl .•. EEl Z EElZPI EEl ••• EEl Zp, '---..-'

,

(3k

Betti number

Torsion coefficients

where 13k is the rank (the Betti number), ZPi are fmite cyclic Abelian groups of orders Pi, and each number Pi may be assumed to be a divisor of a preceding one. The numbers PI , ... , Ps are sometimes referred to as torsion coefficients of the simplicial complex X in dimension k. 1.2.6 Examples of Calculations of Homology Groups. Homologies of Two-dimensional Surfaces

Zero-dimensional homology groups Linearly connected polyhedron

Any two vertices of a connected polyhedron are homologous

Fig. 1.2.5

1) Zero-dimensional homology groups. To begin with, we shall calculate a zero-dimensional homology group Ho(X) of a linearly (arcwise) connected polyhedron X. A polyhedron (a simplicial complex) is called linearly connected if any pair of its points can be joined by a continuous arc lying wholly in the polyhedron. We state that the zerodimensional homology group of a linearly connected simplicial complex is isomorphic to a group Z. If the simplicial complex is disconnected, its zero-dimensional homology group is isomorphic to the direct sum Z EEl ••. EEl Z = zq, where the number of summands q equal to the number of components of linear connectedness of the complex X. As we already know, the zero-dimensional cycle group coincides with the zero-dimensional chain group, i.e. Co(X) = Zo(X). A zerodimensional cycle is a linear combination of vertices of a simplicial complex with arbitrary integer coefficients. We fix an arbitrary vertex a of the complex X and consider it as an elementary zero-dimensional chain (cycle). We shall prove that any other zero-dimensional chain is homologous to a chain rna, where m is an integer. We shall prove that elementary zero-dimensional chains corresponding to any two vertices a and 13 of the complex X are homologous. Since the complex X is linearly connected, one can always construct a continuous path "f such that consists of one-dimensional edges of simplexes, starts at the vertex a and ends at the vertex 13 (Fig. 1.2.5). This path can be regarded as a one-dimensional chain. To this end it suffices to prescribe orientation to each of its links (we indicate it with arrows going from the vertex a

1.2 Simplicial Homology Groups of Simplicial Complexes (Polyhedra)

33

towards the vertex (3) and then to assign to each link, i.e. to each onedimensional simplex, an integer equal to unity. As a result, we obtain a chain, = ..11 +... +..11, where N is the number of edges composing the path. Let us find the boundary of this chain. Consider an interior vertex ai of the path, i.e. a vertex which is the end of a certain edge ..11 and the beginning of the subsequent edge ..11+1 (Fig. 1.2.5). Then we have: 8..11 = ai - ai-\, 8 ..11+1 = ai+1 - ai. Therefore,

Consequently, each interior vertex will enter the sum twice, once with the "+" and then with the "-" sign, which entails mutual cancellation of these two summands. This does not happen only to the two boundary vertices a = ao and (3 = aN, i.e. to the initial and terminal points of the path ,. Hence, 8, = aN - ao =(3 - a. This means that the zero-dimensional chain (3 - a is homologous to zero (it is represented as the boundary of some one-dimensional chain ,). This implies our statement. Indeed, since any zero-dimensional vertex (with coefficient 1) is homologous to the chosen vertex a, it follows that any zero-dimensional chain I:f=1 ai..1~ = c is homologous to the zero-dimensional chain rna = (I:f=1 ai) a. Thus, to calculate the zero-dimensional group Ho(X) it suffices to consider only one vertex a. From this we obtain that the group Ho(X) of the' connected complex X is the same as that of the complex consisting of a single point a. But the latter group is isomorphic to Z all the cycles here have the form rna, where m is an arbitrary integer (there are no nonzero boundaries). From the above arguments one can immediately deduce the proof of the fact that the group Ho(X) of the disconnected complex X is isomorphic to 7!}, where q is the number of arcwise connected components. We shall not do it here and leave the proof to the reader. 2) Homologies of a circle. Let us consider a simplicial partition of a circle 8 1. Clearly, it is given by the set of vertices ao, ... ,aN and the one-dimensional simplexes confined between neighbouring vertices. We prescribe a clockwise orientation to all these one-dimensional edges (Fig. 1.2.6). The zero-dimensional homology group Ho(8 1) is already known to us - it is isomorphic to Z. The groups Hi(8 1), where i > 1, are equal to zero since the complex 8 1 is one-dimensional.

Interior vertex

Boundary vertices

Homologies of a circle

,i

Fig. 1.2.6

34 One-dimensional and two-dimensional homology groups of 2-manifolds

1. Polyhedra. Simplicial Complexes. Homologies It remains to calculate the group HI(SI). We state that HI(SI) = z.

An arbitrary one-dimensional chain has the form e = ao'Yo + ... +an'YN,

where aj are integers. In what case is such a chain a cycle? Calculating the chain boundary, we obtain oe = ao(al - ao) +al (a2 - al)+'" +aN(ao - aN)

We remove the brackets amd group the like terms. As a result, we are led to the following zero-dimensional chain: oe=(-ao+aN)ao+(ao-aJ)al + ... +(aN-I-aN)aN . 2-dimensional topological manifold is a polyhedron

Thus, the one-dimensional chain e is a cycle if and only if oe = 0, Le. if the coefficients ai satisfy the following system of equations: -aD + aN

Sphere, torus, sphere with handles

= 0,

ao - al

= 0, ... , aN-I -

aN

=0 .

From this we obtain ao = al = ... = aN-I = aN. Thus, the chain e is a cycle if and only if all the coefficients are equal to one another. In other words, we habe described all one-dimensional cycles. They have the form z = abo +... +'YN), where a is an arbitrary integer. The group of one-dimensional boundaries BI(SI) is equal to zero since there are no chains of dimension two. Therefore, the group of one-dimensional homologies HI (Sl) coincides with the group of one-dimensional cycles ZI (Sl). Since all one-dimensional cycles have the form a (E 'Yi), where a E Z, it follows that the one-dimensional homology group is isomorphic to Z. Thus, for a circle we have: Ho = Z, HI = Z, Hi = 0 for i 0, 1.

r

..::::..:...

::,. .. ~ :: :::':

:'.

'"

.: .. ...... .

'

~CJ32

Fig. 1.2.7

3) Two-dimensional homology groups of two-dimensional polyhedra, which are manifolds without boundary. We shall say that a twodimensional topological manifold has no boundary ifeach of its points has a neighbourhood homeomorphic to an open two-dimensional disc (the centre of this disc is a point). Any two-dimensional topological manifold is a polyhedron. We shall for simplicity examine only those manifolds which are finite polyhedra. For example, a sphere, a torus, a sphere with handles are manifolds without boundary. To demonstrate this defmition, Fig. 1.2.7 gives examples of manifolds with boundary. Points lying on the boundry (boundary points) differ from interior points in that they possess a neighbourhood homeomorphic to half a disc, the point itself lying on the diameter of this half disc. A manifold without boundary has no such points.

1.2 Simplicial Homology Groups of Simplicial Complexes (polyhedra)

35

So, suppose we are given a finite polyhedron which is a manifold without boundary. Then its triangulation possesses the following important property: each one-dimensional edge (a one-dimensional simplex) is incident to two and only two adjacent triangles. In other words, each edge enters the composition of exactly two triangles (glued along it) (Fig. 1.2.8). Clearly, we essentially employ here the absence of boundary in the manifold. If the manifold had boundary, there would exist an edge lying on the boundary and incident only to one triangle (approaching it from within the manifold) (Fig. 1.2.8). We call a manifold oriented if each two-dimensional simplex of its triangulation can be so oriented that any adjacent triangles incident along the edge (having a common edge) induce on it opposite orientations (Fig. 1.2.8). Otherwise a manifold is called non-oriented. It is characterized by the fact that for any way of orientation of triangles there exists at least one edge on which adjacent triangles induce one and the same orientation.

Theorem. The two-dimensional homology group H2(M) of a finite connected polyhedron which is an orientated manifold without boundary is isomorphic to a group of integers Z. If the manifold is non-oriented, the group H2(M) is equal to zero. Proof. Suppose the manifold is at first orientable. We consider its triangulation and orient all the triangles L1l in such a way that on each edge adjacent triangles induce opposite orientations. An arbitrary twodimensional chain has the form e = E~1 aiL1f, Our goal is to describe all two-dimensional cycles. Calculation the boundary of the chain e, we obtain ae =E aia L1f. As a result, we have obtained a one-dimensional chain. The chain e will be a cycle if and only if the one-dimensional chain ae is equal to zero on each one-dimensional simplex (edge of triangulation). We take an arbitrary edge ,11. It is incident only to two triangles L1l and and is their common edge. By the condition, both the triangles induce opposite orientations on the edge ,11 (Fig. 1.2.9). Consequently, the edge ,11 will enter the sum ae (after grouping like terms) with the coefficient ai - aj, where ai and aj are the values of the two-dimensional chain e on the triangles ,1; and L1J, respectively.

Adjacent triangles

Fig. 1.2.8

Orientable and non-orientable manifolds

.1;

Opposite orientation

36

Fig.1.2.9a

Fig.1.2.9b

1. Polyhedra. Simplicial Complexes. Homologies

Since Be = 0, it follows that ai = aj. Thus, we have proved that a twodimensional chain e is a cycle if and only if it acquires equal values on any two adjacent triangles. But the manifold being connected, any two triangles t and T can be joined by a chain of adjacent triangles (Fig. 1.2.9). The chain e will, accordingly, assume one and the same value on all the triangles of triangulation. Hence, e = a . (~i ,1;). Thus, we have completely described all cycles in two dimensions. It turns out that the group Z2(M) is isomorphic to the group Z seing that the coefficient a can take on arbitrary integer values. So long as the manifold is two-dimensional, it has no nonzero three-dimensional chains, and therefore the group of boundaries B2(M) is equal to zero. From this it is immediate that the homology group H2(M) is equal to Z. This proves the theorem in the first case. Suppose now the manifold is non-orientable. We fix orientations of all triangles of triangulation. We again write a two-dimensional cycle e in the form ~~l aiL1f. Repeating the previous arguments we can see that if two adjacent triangles L1r and ,1] induce distinct orientations on their common side, then ai = aj, whereas if they induce one and the same orientation, then ai = -aj. Consequently, by virtue of connectedness of the manifold all the coefficients aj are equal to one another in the absolute value, i.e. lail = lajl for any i and j. We take two adjacent triangles ,1;, ,1; and orient them so that they induce distinct orientations on their common edge. Then ai = aj. We take successively triangles adjacent to the two preceding ones and also orient them in a compatible way (as long as possible). Then on all these triangles the cycle e will acquire the value equal to the initial value ai. But since the manifold is non-orientable, the moment will necessarily come when two adjacent triangles will meet that induce one and the same orientation on their common edge (Fig. 1.2.9). Given this, the cycle z assumes one and the same value ai on both triangles. On the other hand, the equality ai + ai = 0 must hold. From this we obtain that ai = O. Therefore, any two-dimensional cycle on a non-orientable manifold is equal to zero, which completes the proof of the theorem. We shall give examples of orientable and non-orientable manifolds. A plane, a square, a sphere, a torus, a ring, a cylinder, spheres with

1.2 Simplicial Homology Groups of Simplicial Complexes (Polyhedra)

37

handles are orientable manifolds. A Mobius strip, a projective plane, a Klein bottle are non-orientable manifolds. The reader may verify it. Thus, if a polyhedron is a two-dimensional closed manifold, we have calculated its zero- and two-dimensional homology groups. It remais to calculate the one-dimensional homology group. The argumentation will be more complicated, although they are essentially elementary as well. We shall define the concept of cycle support which will be useful for our further purposes. Let z be some k-dimensional cycle. The union of all k-dimensional simplexes on which the cycle z takes nonzero values will be called the support of the cycle. 4) Auxiliary lemma on deformation of one-dimension cycles.

Lemma. 1) Let in a polyhedron these exist a one-dimensional cycle z assuming values a, b, c respectively on oriented edges t, T, (J (Fig. 1.2.10) which are sides of a two-dimensional simplex (triangle) ,12. Then the cycle z is homologous to another cycle z' which on the edges t, T, (J assumes respectively the values 0, b + a, c + a. Given this, the values of the cycles z and z' coincide on all the other edges of triangulation (Fig. 1.2.10). Figuratively speaking, the cycle z can be taken out of the edge t. 2) Let in a polyhedron there exist a one-dimensional cycle z assuming values a, b, c respectively on oriented edges t, T, (J (Fig. 1.2.10) which are sides of a triangle ,12. Let the cycle z be equal to zero on all the other edges of triangulation incident to the vertex A of the triangle ,12 and different from the edges T and g. Then necessarily b = c and the cycle z is homologous to another cycle z' which on the edges t, T, (J assumes respectively the values a + b, 0, O. Given this, the values of the cycles z and z' coincide on all the other edges of triangulation. In other words, the cycle z can be taken from the sum of the edges T and (J, and its value b can be directed along the edge t. 3) Let in a polyhedron there exist a one-dimensional cycle z assuming a certain value a on an edge t of the polyhedron one of whose vertices is indicent only to such edges of the polyhedron on which the cycle z is equal to zero (Fig. 1.2.10). Then a = 0 and the cycle z is homologous to another cycle z', in the composition of whose support the edge t does not enter. Given this, the values of the cycles z and z' coin-

Cycle support

Lemma on deformation of the cycles

38

1. Polyhedra. Simplicial Complexes. Homologies

cide on all the other edges or triangulation. In other words, the cycle z can be taken from the edge t. Proof of the lemma

Edges of triangulation \

/ \

l-l'

q

,.

() \v"~/ -

-q

i

~

q

0

d(a.t/-) =!

Fig. 1.2.11

Elimination of the edge

Proof. Let the edges be oriented as shown in Fig. 1.2.10. We shall calculate the difference z - z' of the cycles z and z'. From Fig. 1.2.11 it seen that as a result we obtain a cycle assuming the values a on t, -a on 7, and -a on fl. We shall consider a two-dimensional chain e acquiring the value a on a triangle Ll and equal to zero on all the other triangles of triangulation. Let us calculate its boundary 8e. Clearly, we obtain a one-dimensional chain equal to zero on all the edges of triangulation different from t, 7, fl and equal respectively to a, -a and -a on these latter edges. Here we make use of the fact that the oriented boundary of the triangle Ll consists of edges t, - fl, and -7 (Fig. 1.2.11), and therefore, (8e)(t) = +a, (8e)( -fl)) = a, (8e)( -7) = -a. Thus, the cycles z and z' are homologous since z - z' = 8e. The proof of the second and third parts of the lemma is carried out in a similar way (see Figs. 1.2.10 and 1.2.11). We shall not dwell on this proof and only point out that the equality b = e in item 2 the equality a = 0 in item 3 follow from the condition 8z = O. This statement is based on the fact that the vertex A is incident only to two edges of triangulation on which the cycle acquires a nonzero value, namely, to the edges 7 and fl. The other edges, incident to A, make a zero contributiop. to the zero-dimensional chain 8z. This completes the proof of the lemma. This lemma can be imparted a clear geometrical meaning. It turns out that we can deform the initial cycle z by displacing its support along the surface. We replace its edge which enters into some triangle by the sum of two other edges of this triangle, and the value a which was earlier ascribed to the edge t ''transits'' onto these edges and is added to the former values of the cycle on these edges. As a result, the cycle z is replaced to the one homologous to it, and the edge t can be eliminated from its composition since on this edge the cycle z' acquires zero value. Similarly, if a certain triangle has a "free" vertex, i.e. such a vertex to which no other nonzero edges of the cycle are incident (except for two sides of the triangle), then the cycle can be taken from the triangle through deformation of the sum of two sides onto the third side (Fig. 1.2.12). In other words, we can continuously deform one-dimensional cycles about the

1.2 Simplicial Homology Groups of Simplicial Complexes (polyhedra)

39

polyhedron sliding along two-dimensional simplexes (without changing the homology class of the cycle). Figure 1.2.12 demonstrates the process of cycle extrusion along the triangle and its replacement by the sum of two other sides of the triangle. We shall apply this simple but useful lemma to calculate one-dimensional homologies of two-dimensional manifolds. It is useful to think of a one-dimensional cycle as a closed rubber thread located on the manifold. We deform the cycle by expanding or contracting this thread without breaks (glueings are admitted). We can thus essentially simplify the cycle by disposing it on the surface in an optimal way. All these operations can be visually interpreted in terms of the support of the one-dimensional cycle z. According to the lemma, we can subject the support to the following operations (surgery). 1) Any edge of the support entering in the composition of a twodimensional simplex can be replaced by the sum of two other edges of this simplex (Fig. 1.2.12) with a corresponding change in the numerical values of the cycle. 2) Some pairs of edges converging at a free vertex of the triangle can be replaced by one edge (Fig. 1.2.12). 3) An edge of the support protruding sideways and having a free end can be removed (Fig. 1.2.12). These operations can be called combinatorial disassembly (simplification) of the cycle. Clearly, each of the operations 1-3 decreases the number of edges in the cycle support and therefore we actually simplify the support. If we digress from triangulation and consider the support as a onedimensional set consisting of segments of polygonal lines on a manifold, then the operation of support simplification introduced above can be interpreted in another manner. If several polygonal lines in some region are rather close and parallel to one another (but, perhaps, differently oriented), this system can be replaced by another one (Fig. 1.2.12) by glueing together all the previous polygonal lines. In doing so, one should sum up (with allowance made for signs) all the numbers which stood on the previous polygonal lines and ascribe the result obtained to the new segment. One can simplify the cycle by cutting from it (or pulling onto it) some "branches", i.e. polygonal lines with free ends which come from a

Fig. 1.2.12

Combinatorial disassembly of the cycles

40 Polygonal lines with free ends

A#\~

V@]~O}tJ 11; .; ,.,w

......

Fig. 1.2.13

Homologies of a 2-sphere

Case of n-dimensional sphere

Deformation of multidimensional cycles

1. Polyhedra. Simplicial Complexes. Homologies

certain point of the cycle and do not return onto this cycle. Thus, a system of rubber threads representing a cycle can be continuously deformed about the manifold, by glueing the threads and simplifying the cycle support. For example, a closed curve drawn on a torus and going round it twice along the parallel (and evidently representing a one-dimensional cycle) can be continuously deformed into a curve which is also a parallel but taken with coefficient 2 (Le. into a parallel passed twice). 5) Homologies of a sphere. We shall consider an arbitrary fmite triangulation of a two-dimensional sphere (Fig. 1.2.13). Let z be an arbitrary one-dimensional cycle. We state that it is homologous to a zero cycle, i.e. to a cycle which is identically zero on all edges of triangulation. We apply the preceding lemma. It states that one can always remove any edge ("erase" it) from the composition of the cycle z by adding the numerical value given initially on this edge to two adjacent edges of the triangle which contained this edge. As a result, one obtains a new cycle homologous to the initial one. We start this operation with an arbitrary edge. We erase it, redistribute the number ascribed to it between the adjacent edges, take the next edge, erase it, etc. Repeating the process a fimte number of times and combining it with operations 2 and 3 (see the lemma), we run over all the edges of triangulation (they are finite). In the penultimate step we obtain a cycle equal to zero everywhere except on the edges of one triangle, on which the cycle assumes vales a, b, c. But since we are dealing with a cycle, a = b = c. Consequently, in the last step, extruding one of the edges onto the sum of two other edges. we liquidate all the three numbers and as a result obtain a cycle homologous to the initial one but equal to zero on all the edges. We have proved that the one-dimensional homology groups of a sphere is trivial (equal to zero). Thus, HO(S2) = 'l, HI (S2) = 0, H2(S2) = 'l. We invite the reader to prove that the homology groups of an n-dimensional sphere have the form: Ho(sn) = 'l, Hi(sn) = 0 for 1 ~ i ~ n - 1, Hn(sn) = 'l. At the same time the reader will have to prove a multidimensional analogue of the lemma on cycle deformation. From the proof of our lemma one can see that one can similarly deform supports of multidimensional cycles by extruding them inside a multidimensional cycle and replacing one face of the simplex by the sum of the rest of its faces (with altered numerical values).

1.2 Simplicial Homology Groups of Simplicial Complexes (Polyhedra)

41

6) Homology of a torus. We represent a torus in the form of a square with opposite sides identified (Fig. 1.2.14). Consider triangulation of a torus depicted in Fig. 1.2.14. We have deliberately indicated orientations of all the simplexes to show that a torus is an orientable manifold. One can see that all the orientations are coordinated, i.e. adjacent triangles induce different orientations on their common sides. Suppose we are given a one-dimensional cycle z on a torus. We shall prove that it is homologous to a cycle of the form ma + n(3, where a and (3 are the two cycles shown in Fig. 1.2.14 and m and n are some integers. More precisely, the cycle a is equal to a+b, where a and b are oriented edges Ai e (Fig. 1.2.14), and the cycle (3 is equal to c + d. We shall apply the three Fig. 1.2.14 above-described operations of cycle simplification (deformation). As a result (Fig. 1.2.14) we can take the support of the cycle z from all the interior edges of triangulation which come from the centre of the square. We obtain a cycle z' whose support consists only of boundary edges of the square. One cai:L see that a = a +b and (3 = c +d. It remains to prove that the cycle z' = m(a +b) +n(c +d) is homologous to zero if and only if the coefficients m and n are equal to zero. Indeed, suppose that at least one of the numbers m and n is nonzero and that there exists a two-dimensional chain q such that 8q =z'. Since the cycle z' is equal to zero on all interior edges of triangulations (incident to the centre of the square), it follows that the chain q takes necessarily equal values on all adjacent (by a common edge) triangles. Consequently, the chain q takes one and the same value A on all the triangles of triangulation. The point is that any pair of triangles is joined by a chain of triangles adjacent by a common edge. But in such a case the boundary aq of the chain q takes zero on all the edges a, b, c, d. Indeed, e.g. two copies of the side a (on opposide sides of the square) evidently induce reverse orientations of the square, and therefore when taken with coefficients equal in absolute value but different in sign, they annihilate each other in the boundary of the chain q. These arguments are also applicable for the edges b, c, d. As a result we obtain 8q = O. But since 8q = z', we have z' = O. Therefore, m(a + b) +n(c +d) =O. It remains to make sure that the cycles a =a +b and (3 = c +d are linearly independent. But this is obvious because the cycle (3 is equal to zero on edges c, d, and the cycle (3 is equal to zero on edges a, b. From this we obtain that m = n = O. One should bear in

Homology of a torus

Values of the chain on the edges of triangulation

42

1. Polyhedra. Simplicial Complexes. Homologies

mind that all the four vertices of the square, AI,A2,A3,~, represent actually one and the same polyhedron vertex. This fact is dictated by the rule of identification of the sides of a square in the case of a torus. This proves the statement. Thus, each one-dimensional cycle is homologous to a cycle of the form ma+n{3, the cycles a and {3 being linearly independent. Therefore, the one-dimensional homology group HI (T2) of the torus is isomorphic to the direct sum Z EB Z two copies of the group Z. Returning to the model of a torus in the form of a ''roll'' (Fig. 1.2.14) we can see that the generator a corresponds to the torus parallel and the generator (3 to the torus meridian. As has been proved above, any other cycles on the torus are a linear combinations ofthe parallel and meridian taken with arbitrary integer coefficients.

Fig. 1.2.15

7) Homologies of the sphere with 9 handles. It is now clear how one should calculate the one-dimensional homology group of a sphere with an arbitrary number of handles. Recall that a torus is a sphere with a single handle. A sphere with 9 handles is convenient to represent as shown in Fig. 1.2.15. One can see that the consideration is largely reduced to a multiple repetition of the construction which we have realized on a torus. To this end we construct a convenient triangulation of a sphere with 9 handles. We shall cut the surface curvilinearly, as shown in Fig. 1.2.15. As a result, we obtain a flat polygon with 4g sides. On a handle with number i we denote the parallel by ai and the meridian by {3i. Then the sides of the polygon obtained are numbered as follows (we write down the letters in the order of their appearance in a clockwise detour on the boundary: 0'.1 {31 all {31 1... a g{3g a; I{3; I. We triangulate the polygon (and thus the surface) as shown in Fig. 1.2.16. Clearly, we can now repeat the arguments used above for the torus. First we prove that each cycle z is homologous to a cycle z' which has the form z'

mlal +nl{31 + .. ·+mgag+ng{3g ml(al +bl)+nl(ci +dl)+ .. ·+mg(ag+bg)+ng(cg+dg)

Fig. 1.2.16

Here mj, ni are integers. Next, that all the cycles aj = aj + bi g, are linearly independent. We the basis in the group H1(Mg),

we prove, as in the case of the torus, and (3j = Cj + dj, where 1 ~ i, j ~ therefore point out in an explicit form where Mg is a sphere with g handles.

1.2 Simplicial Homology Groups of Simplicial Complexes (polyhedra)

43

Namely, the basis consists of the cycles ai, ... , a g and (31, ... , (3g. Thus, HI(Mg) = Z E!1'" E!1 Z (29 times), i.e. HI (Mg) = Z29. Summarizing the results obtained above, we now can formulate the following theorem.

Homologies of a sphere with handles

Theorem. Integer-valued homology groups of a sphere with 9 handles, where 9 is a non-negative integer, have the form: Ho = Z, HI = Z2g ,H2 = Z. 8) Homologies of a projective plane Rp2. Let us consider the triangulation of a projective plane shown in Fig. 1.2.17. It is distinct from the triangulation of a torus. Although in both cases we triangulate a square, glueings on its sides are organized differently. Given this, we have prescribed specially the orientation on two-dimension~simplexes, as on the triangluation of a torus, in order that non-orientability of the projective plane might be clearly seen. Indeed, on the square boundary, on the sides a, b, c, d, identical orientations are induced. This fact is indicated in Fig. 1.2.17 by the "+" signs. Proceeding by analogy with the torus we obtain that anyone-dimensional cycle z is homologous to the cycle z', whose support consists of boundary edges of the square. In other words, z rv ma + n(3 = m(a + b) + n(c + li) = z', where m and n are integers. Here a and (3 (separately) are not cycles so long as oa = A2 - Al f 0, 0(3 = A3 - A2 f O. In the case of torus a and (3 are on the contrary cycles in so far as on the torus we have Al = A2 = A3 = AJ. In the case of projective plane there is, however, an additional circumstance which makes our further calculations different from those in the case of torus. Indeed, let us calculate the boundary of the cycle z'. On the one hand, this is a zero-dimensional chain of the form oz' = o(ma+n(3) = o(ma+mb+nc+nli) = m(A2 - AI)+n(A3 - A2). But by the rule of identifications of sides of a square only two of the vertices A I,A2,A3,AJ (Fig. 1.2.17) are distinct since Al = A3 and A2 = A4• Consequently, 0 = oz' = (m - n)A2 + (n - m)AI. Considering that Al f A2, we have m = n. Thus, we have proved that on a projective plane anyone-dimensional cycle is homologous to a cycle z' of the form (m + n)(a + (3) = q(a + (3) = n, where "1 = a + (3 and q is an arbitrary integer. Next, we state that the cycle "1 taken twice is homologous to zero, i.e. 2"1 rv O. Indeed, let us consider a two-dimensional chain T which takes the value 1 on each oriented triangle of our triangulation.

Fig. 1.2.17

Homologies of a projective plane

Non-orientability of projective plane

44

Projective plane and Klein bottle

1. Polyhedra. Simplicial Complexes. Homologies

Calculating its boundary, we obviously obtain from Fig. 1.2.17 the relation 8T = 2a + 2b + 2c + 2d = 2"1, and the statement follows. We have proved therefore that the group HI (RP2) is isomorphic to the group Z2 whose generator is the cycle "I depicted in Fig. 1.2.17, i.e. "I = a+b+c+d.

Theorem. Integer-valued homology groups of a projective plane have the form Ho = Z, HI = Z2, H2 = O.

Fig. 1.2.18

9) Homologies of a Klein bottle. We consider triangulation of a Klein bottle K illustrated in Fig. 1.2.18. One can see that this manifold is non-orientable. As in the preceding cases, we immediately obtain that anyone-dimensional cycle z is homologous to a cycle z' of the form z' = rna: + n(3, where rn and n are some integers and a: and (3 are cycles, and a: = a + b, (3 = c + d. We have: 8a: = 8(3 = 0 since all the four vertices of the square represent actually one vertex on the Klein bottle, i.e. Al = A2 = A3 = A4 (as on a torus). We shall examine a two-dimensional chain T which takes the value 1 on all the oriented triangles (fig. 1.2.18). Calculating its boundary, we obtain 8T =a: +(3 a + (3 = 2(3. Accordingly, a double cycle is homologous to zero. Thus, the one-dimensional homology group is isomorphic to the direct sum of two Abelian groups Z E9 Z2, where the subgroup Z is generated by the generator a (an infinite-order cycle), while the subgroup Z2 is generated by the generator (3 (a second-order cycle, i.e. 2(3 rv 0).

Theorem. Integer-valued homology groups of the Klein bottle have the form Ho = Z, HI = Z E9 Z2, H2 = O. 10) Homologies of non-orientable two-dimensional closed surfaces.

Fig. 1.2.19

We shall consider the polygon depicted in Fig. 1.2.19. It is a natural generalization of the polygon which determines a projective plane (Fig. 1.2.17). The reader may verify that identifying edges on the polygon boundary in accordance with the indications (letters and arrows), we shall obtain a two-dimensional closed compact manifold. Besides, this manifold will be non-orientable. Omitting the calculational details which we have repeatedly demonstrated already, we obtain that anyone-dimensional cycle z is homologous to a cycle z' composed of

1.2 Simplicial Homology Groups of Simplicial Complexes (polyhedra)

edges - cycles z'

0:1,"" O:k

45

taken with arbitrary integer coefficients, i.e.

= ml 0:1 +... +mkO:k. Given this, the cycle 2(0:1 +... +O:k) is homo-

logous to zero since it is the boundary of a two-dimensional chain which takes the value 1 on each oriented triangle. Consequently, for the generators in a one-dimensional homology group one can take the following cycles: 0:1, ... , O:k-I, h, where h = 0:1 + ... + O:k-I +O:k. Then 2h 0, and the cycles 0:1, ... , O:k-I are free. As a result, the homology group HI is isomorphic to the direct sum Zk-I Ell Z2.

Homologies of non-orientable 2-surfaces

Theorem. Integer-valued homology groups of a non-orientable twodimensional closed surface have the form: Ho = Z, HI = Zk-I Ell Z2, H2=0.

Homologies with coefficients in the group 2:2

tv

We have hitherto calculated integer-valued homologies of polyhedra. As a useful (and simple) exercise we advise that the reader calculate homologies of all the above-mentioned polyhedra with coefficients in the group Z2 or more generally in the group Zp for an arbitrary prime p. We shall only present the result withOl~t going into detail Of the proof (although we have in fact presented all the necessary material). Homology groups of a two-dimensional manifold M with coefficients in the group Z2 have the following form. 1) In the case of a sphere with 9 handles we have: Ho = Z2, HI = Z2 Ell ... Ell Z2(2g times) = z~q, H2 = Z2. 2) In the case of a non-orientable manifold we have: Ho = Z2, HI = Z2 Ell··· Ell Z2(k times) = Z~, H2 = Z2. We should emphasize the following: while the two-dimensional integer-valued homology group of a non-orientable manifold is equal to zero, the two-dimensional homology group of the same manifold, but with coefficients in the group Z2, is non-zero and equal to Z2.

Sphere with handles

Sphere with Mobius strips

46

1. Polyhedra. Simplicial Complexes. Homologies

1.2.7 Visual Material

Algebraic chain

Figure 1.2.20 represents an infinite polyhedron composed of threedimensional simplexes (tetrahedra). A similar picture is obtained by unfolding a ball triangulated so that the simplexes have bases on the ball boundary (on the sphere) and one vertex at the ball centre. In Fig. 1.2.21 one can see an algebraic chain composed of discs = cells endowed with numerical coefficients (numbers are conditionally represented by various figures). The condition that the chain is a cycle implies that all these numbers are equal, i.e. all the figures must be identical. In case the figure~ are distinct (as in the Fig. 1.2.21), the chain is not a cycle.

Page 47, Fig. 1.2.20. Simplifical subdivision of a topological space; spines of 3-manifolds. Valley of ancient kings; egyptian myth on the forgotten tombs of gods and pharaohs Page 48, Fig. 1.2.21. Random walk and random sequences. Homology chains with coefficients in Abelian group. Medieval legend on the Flying Hollander; Scandinavian myths

1.3 General Properties of Simplicial Homology Groups

49

1.3 General Properties of Simplicial Homology Groups. Some Methods for Calculating Homology Groups 1.3.1 Incidence Matrices

Incidence matrices

We shall consider an arbitrary polyhedron X and its arbitrary triangulation. We shall number all of its oriented simplexes by symbols Llf, where k indicates the simplex dimension and i shows its ordinal number (in a particular dimension), i.e. 1 ::; i ::; G:k. Here G:k is the number of k-dimensional simplexes in the polyhedron. Besides, 1 ::; k ::; n, n = dimX, i.e. n is the polyhedron dimension. We assume for simplicity that the polyhedron is finite, then G:k < 00. From what follows it will be clear how we extend most of the constructions to the case of infinite polyhedra. Let us consider the boundary operator 8. For each simplex LlJ+l we obtain the relation 8LlJ+l = L~kl atLlf, where afj are arbitrary integers. Given this, we have 1 ::; j ::; G:k+l, 1 ::; i ::; G:k. The number afj is equal either to +I or to -1 depending on the orientation with which the simplex Llf enters in the boundary of the simplex LlJ+l. In other words, afj is equal to +1 if the orientation given on LlJ coincides with that induced on it by the simplex LlJ+l, and otherwise is equal to -1. The number is called the incidence coefficient of the simplex Llf and simples Llj+l. In case the simplexes Llf and LlJ+l are not incident (i.e. if the simplex Llf is not a face of the simplex LlJ+ 1), then by definition afj = O. Clearly, the numbers afj are organized in a natural way into a rectangular matrix which we denote by Ek. It is convenient to write this matrix as

at

at

Incidence coefficients Boundary operator Rectangular incidence matrix

50 k-dimensional incidence matrix

1. Polyhedra. Simplicial Complexes. Homologies

This matrix has Cik rows and Cik+I columns. It is called a k-dimensional incidence matrix. Thus, with each triangulation of the polyhedron there is associated a set of incidence matrices EO, ... , En-I, where n = dim X. Let us consider the simplest examples. In Fig. 1.3.1 one can see two polyhedra: a segment and a triangle. In the case of a segment we have a single incidence matrix EO of the form .1 1 EO=.10

-1

.1~

+1

1

For a triangle we have two incidence matrixes: EO and EI, namely .1 11 .1i

Ef
e.&t'c:O ... ..

~;

codes

of manifolds

Fig. 2.4.13

Cartography process

Algorithm for enumerating all 2-surfaces

154 Algorithm for recognition and incidence matrices

1-dimensional cohomology groups and triangulation

Fig. 2.4.14

1. ~ "fJ...... •.

..

:::.L13 '

/1;"

2. Low-Dimensional Manifolds

Such an algorithm for recognition does exist, although in practical work it may require much time to give an answer. This necessary time can however be efficiently estimated from above using a calculable constant which depends only on the size of the initial table T(M). This procedure is described below. 2.4.8 Calculation of One-Dimensional Cohomologies of a Surface Using Triangulation We should now return to the material of Chap. 1. Recall that the real cohomology groups of smooth manifolds coincide with real simplicial homology groups of these manifolds calculated using incidence matrices. The problem therefore reduces to the analysis of the incidence matrices which are in fact already known to the geodesist since he has already compiled the table T(M) corresponding to a given triangulation. The geodesist should calculate all the boundary operators (Fig.2.4.14) and find all the one-dimensional cycles and boundaries (Fig.2.4.15). Then, using personal computer, he can quickly reduce the incidence matrices to the normal form and calculate the real homology groups. Moreover, he can determine whether the surface is orientable or not. Using the familiar to him theoretical information on the structure of homology groups of 2-manifolds, the geodesist answers the question.

Fig. 2.4.15

2.4.9 Visual Material Spectral sequence method

In Sect. 2.4 we have described only the simplest methods of calculation of homologies and cohomologies. In addition, the spectral sequence method is frequently employed. See e.g. Refs. [27], [24]. Figure 2.4.16 represents the initial stage of triangulation of a two-dimensional surface, necessary for algorithmic clarification of the topological structure of the object on which the geodesist works (see Sect. 2.4.7). Using a "gas burner" one can cut the surface into triangles and then glue of them a flat funda-

Page 155, Fig. 2.4.16. Triangulation and coding of 2-surfaces. Medieval German and Scandinavian myths about Sigfried and Brunhilda (death of Sigfriedj

156

Triangulation of a polyhedron

Memory of an individual block Complex system

2. Low-Dimensional Manifolds

mental polygon - the code of the surface. Figure 2.4.17 is an infmite sequence of spheres and in the background there is a sphere subjected to homeomorphism. Triangulation of a polyhedron and the process of its cutting into elementary pieces, blocks are depited in Fig. 2.4.18. The polyhedron is transformed here into a chaotic cluster of simplexes (or parallelepipeds) each of which remembers, however, its neighbours since the incidence matrices restore completely the polyhedron. Therefore, all the blocks may, if necessary, again unite to form a polyhedron. The necessary memory of an individual block (simplex) is very small - it must remember only its immediate neighbours to which it is incident. Thus, a complex system can be effectively constructed of a large number of individual elements with a comparatively small memory.

~

Page 157, Fig. 2.4.17. Differential geometry and topological properties of a smooth 2-surfaces; the main step in the proof of Gauss-Bonnet formula. Expulsion of ancient Greek gods from Olymp Page 158, Fig. 2.4.18. Incidence matrices determine the topological structure of a complex objects; memory af an individual block reconstructs its neighbourhood

2.5 Visual Properties of Three-Dimensional Manifolds

159

2.5 Visual Properties of Three-Dimensional Manifolds 2.5.1 Heegaard Splittings (or Diagrams) The problem of classification of 3-manifolds is much more complicated than that of2-manifolds and it has not yet been completely solved even in the case of closed 3-manifolds. The solution of the classification problem consists in the construction of two algorithms: the algorithm for enumeration of manifolds (repetitions are admitted) and the algorithm for identification of manifolds. The latter establishes in each pair of manifolds (codes) whether they are identical or not. There exist several good algorithms for enumerating closed 3-manifolds. Each of them i~ based on a special way of specifying 3-manifolds. A complete algorithm for identifying (recognizing) 3-manifolds has not been constructed up to now. There does not known even an effective simple algorithm for recognizing a standard sphere 83, alth9ugh partial algorithms, for example, in the class of so-called 3-manifolds of genus 2 do exist. Birman and Hilden [41] have proved algorithmic recognizability of Heegaard diagrams of the sphere 83 in a certain class of Heegaard diagrams of 3-manifolds, this class containing, in particular, all Heegaard diagrams of genus 2. But they have reduced the problem to the famous Haken algorithm (recognition of a trivial knot) which cannot be practically realized using computer. So, the problem of finding an effective algorithm admitting computer realization remained urgent. This algorithm (for genus 2) was proposed and experimentally worked out by I.A. Volodin and A.T. Fomenko in 1974 and realized using computer together with V.E. Kuznetsov (see Ref. [42]) (so-called cut-vertex algorithm). The efficiency of this algorithm was finally grounded by Homma, Ochiai and Takahashi [43] in 1980. A more detailed and demonstrative presentation of the algorithm is given in the books [28] and [29]. Here this material is omitted. One of the most frequently used classical ways of specifying 3manifolds is Heegaard splittings and their modifications (Heegaard diagrams, nets, special splittings). For brevity, we shall henceforth call a sphere with 9 handles a pretzel (of genus g). Embedding such a pretzel into R3, we may consider a three-dimensional manifold (body) which is restricted by this pretzel and is called afull (or solid) pretzel of genus g.

Classification of 3-manifolds

Algorithm for enumeration Algorithm for identification

Heegaard diagrams Partial algorithms Haken algorithm

Cut-vertex algorithm for recognition of a standard sphere

Solid pretzel of genus g

160

Glueing the pretzels

Heegaard splitting

Genus of the Heegaard splitting

2. Low-Dimensional Manifolds

Suppose there exist two copies Kg, K~ of a solid pretzel of genus 9 and a certain homeomorphism h: 8Kg --t 8K~ of their boundaries. Glueing the pretzels via this homeomorphism we obtain a closed 3manifold M. Note that the manifold M is represented as a union of the two solid pretzels (the images of the pretzels K~ and Kg under their glueing) lying in it. Definition 1. The Heegaard splitting of a closed orientable 3-manifold M is the representation of this manifold as a union of two pretzels with common boundary, lying it it. Since the pretzel boundaries coincide, the pretzels are of the same genus. It is called the genus of the Heegaard splitting. Theorem 1. Any closed orientable 3-manifold admits Heegaard splitting of some genus. The proof is carried out using the same technique as in the proof of the two-dimensional surface classification theorem. We preliminarily introduce a very important concept of a handle. Let M be an n-manifold and let h : 8 A- 1 X Dn-A --t 8M be a certain embedding. Then we say that the manifold MI = M Uh(D A X Dn-A)

n-dimensional handle of index'\ Base of the handle

Fig. 2.5.1 Splitting into union of handles Duality between the handles

is obtained from the manifold M by glueing an n-dimensional handle

of index A. The image of the manifold DA X Dn-A in the manifold MI under glueing is called a handle and the manifold h(8 A- 1 X Dn - A) is called the base of it. The disc, the strip and the patch from the proof of Theorem 2, Sect. 2.4.3 are two-dimensional handles of indices 0, 1 and 2 respectively. The glueing of a handle of index 0 to an n-manifold consists in the addition of an n-ball taken separately; the glueing of a handle of index n consists in glueing up the spherical component of the boundary by an n-ball. Examples of three-dimensional handles of indices 1 and 2 are given in Fig. 2.5.1. An n-manifold is said to be splitted into handles if it is obtained

from a ball by a successive glueing of handles of different indices. There exists the following duality: any handle of index A glued to the union of the preceding handles can be regarded as a handle of index n - Aglued to the union of the remaining handles.

2.5 Visual Properties of Three-Dimensional Manifolds

161

The proof ofTheorem 1 is carried out by the following scheme. First we prove that an arbitrary 3-manifolds can be trianglulated, i.e. partitioned into curvilinear tetrahedra in such a way that any two tetrahedra either do not intersect at all or intersect at a vertex, along an edge or a face. To db so, one should be able to make triangulations of Euclidean neighbourhoods of points compatible. As was proved by Moise, this can always be done in three dimensions. It should be noted that one cannot, generally, achieve such a compatibility in higher dimensions. Then the vertices, edges and triangles are replaced by handles of indices 0, 1 and 2 respectively. The remaining "middle" parts of tetrahedra are handles of index 3 (Fig. 2.5.2). If two handles of index 0 are joined by a handle of index 1, their union is assumed to be a new handle of index 0, which will decrease the total number of handles. In a dual manner, two handles of index 3, which a handle of index 2 joins with its free sides, can be united with this latter one to form a new handle of index 3. Repeating this argument, we arrive at the case of one handle of index 0 and one handle of index 3. Then the union of a handle of index 0 and handles of index 1 is a ball with handles, i.e. a solid pretzel. Its complement is a union of handles of index 2 and a handle of index 3, i.e. from the dual point of view the union of handles of index I with a handle of index 0, which is a solid pretzel too. As a result we obtain a Heegaard splitting. Obviously, the only 3-manifold of genus 0 (i.e. the one obtained by glueing two balls) is the sphere. The sphere also posseses the Heegaard splitting of genus 1 (and the Heegaard splitting of any higher order). This can be most easily seen from the "differentiation" formula: 8 3 = a(D2 x D2) = aD 2 x D2 U D2 x aD 2, where aD 2 x D2 and D2 x aD2 are pretzels of genus 1 (solid tori). Identification of points of the sphere a(D2 x D2) symmetric with respect to zero yields a three-dimensional projective space lRp3. Identification of points of each of the solid tori also yields solid tori. Thus we have constructed an example of the Heegaard splitting of genus 1 of a manifold ]Rp3. The Heegaard splittings of a sphere and a projective space can be seen in Fig. 2.5.3 and 4. The sphere 8 3 is represented in Fig. 2.5.3 as a space lR3 compactified by an "infinitely remote" point. One of the tori of this splitting is obtained by rotation of the ring A around the axis l which, together with the infinitely remote point, composes a circle. The complement of this torus is splitted into

Triangulation of 3-manifolds

Fig. 2.5.2

Ball with handles

Fig. 2.5.3

Heegaard splittings of a sphere and a projective space 3-sphere

162

2. Low-Dimensional Manifolds

discs each of which intersects the i-axis at exactly one point and is obtained by rotation of a corresponding "solenoidal" arc. Therefore, this complement is also a solid torus. Note that meridians and parallels of solid tori exchange places under glueing: a meridian becomes a parallel and vice versa. In Fig. 2.5.4 the space ID'3 is represented as a space JR.3 with improper points added by one for each direction (the class of parallel straight lines). The one-sheeted hyperboloid splits JR.p3 into two solid tori. The meridional disc of one torus coincides with the disc spanned by the neck cross-section of the hyperboloid, the meridional disc of the other one is composed of parts (31, f32 of the plane passing through the hyperboloid axis. As the parallel of both the tori one can take the rectilinear generator of the hyperboloid.

2.5.2 Examples of Three-Dimensional Manifolds Fig. 2.5.4

Projective space lRp3 Direct product 8 1 x 82 3-torus Lens spaces

We are already acquainted with the simplest examples of three-dimensional manifolds. This is the standard sphere 8 3 given by the equation (xli + (x 2)2 + (x 3i + (x4)2 = I in a Euclidean space JR.4; the threedimensional projective space JR.p3 whose points are, for example, straight lines in JR.4 passing through the origin; the direct product of a circle by a two-dimensional sphere, i.e. 8' x 8 2 and, finany, the three-dimensional torus T3 which is a product of three circles, i.e. 8 1 x 8 1 X 8 1. Besides, in concrete applications we often encounter so-called lens spaces (or simply "lenses") Lp,q defmed like this. We consider a sphere 8 3 embedded standardly in JR.4 which is identified with the complex space C2(z, w). The sphere is given here by the equation Izl2 + Iwl 2= 1. Let us consider the action of the Abelian group Zc on the sphere 8 3, which is given by the formula (z, w) - t (e27riP/C. z, e27riq/c • w). The linear transformation defined by this formula obviously maps the sphere into itself Considering the iterations of this transformation, we determine the action of the other elements of the group Zc. Factorizing the sphere by the action of this group (Le. identifying the points obtained from one another via group transformations) we obtain a new space which is just called the lens space. In a particular case, considering a lens Ll,1 for c = 2, we obtain a projective space. In this case the action of the group Z2 is given by

2.5 Visual Properties of Three-Dimensional Manifolds

0;

163

the formula (z, w) - (-z, -w) which describes an involutive self-map of the sphere. Each point of the sphere is mapped into a diametrically opposite one. New three-dimensional manifolds can be alternatively obtained like this. Suppose we are given two n-manifolds Ml and M2. In each of them we single out an n-dimensional ball D and discard it. We obtain two manifolds with boundary homeomorphic to a sphere sn-l . Consider a cylinder (a tube) Dl X sn-l, where Dl is a segment. The cylinder boundary consists of two spheres. We glue these spheres to the boundary spheres of the manifolds Ml \D and M2 \D (Fig. 2.5.5). We obtain a new manifold which we denote by Ml UM2 and call it the connected sum of the manifolds Ml and M2. Note that the angles arising in the course of glueing the cylinder (the tube) to the holes can be smoothed, as shown in Fig. 2.5.5, so that we again have a smooth manifold. The operation of taking a connected sum is especially obvious on an example of2-manifolds (Fig. 2.5.6). If we take a connected sum ofa sphere with any manifold, we shall have nothing new: the result will be homeomorphic to the initial manifold (Fig.2.5.6), i.e. Mn Usn ~ Mn. In particular, sn U..• usn = sn. A connected sum of two tori gives a pretzel. Generally, a connected sum of a 2-manifold with a torus is equivalent to glueing one handle. Thus, any closed orientable connected 2-manifold, i.e. a sphere with handles results from taking a connected sum of a corresponding number of tori. In this sense S2 and the torus T2 are "independent" (in the class of orientable manifolds). The other orientable manifolds are decomposed into connected sums o~ these manifolds. On this basis we give the natural defmition. A manifold M n is called irreducible (in the topological sense) if it cannot be represented as a connected sum of two manifolds Ml and M2 distinct from the standard sphere, i.e. M f Ml UM2. The other manifolds are called reducible. Therefore, the manifold M is reducible provided that M = Ml ~ M2, where Mi f sn, i = 1,2. From this point of view, the standard sphere is assumed to be an irreducible manifold (although it falls into a connected sum of an arbitrary number of spheres).

~

,gOl 1

._~ .., ......

~I I..... G I,

_

1

.....-

Fig. 2.5.5

0"'" .~

Fig. 2.5.6

'\ ,

164

2. Low-Dimensional Manifolds

2.5.3 Equivalence of Heegaard Splittings Equivalent Heegaard splittings Heegaard 2-surface

Two Heegaard splittings of a 3-manifold M are called equivalent if there exists a homeomorphism of the manifold M onto itself which maps the Heegaard surface (i.e. the common boundary of cracknels) of one splitting into the Heegaard surface of the other splitting. In this connection we may present the following information. 1) Any two Heegaard splittings of a sphere, of equal genus, are equivalent (the result of Waldhausen). 2) Any two splittings of genus 1 of a manifold of genus 1 are equivalent. 3) There exist non-equivalent splittings of genus 2 of a connected sum of two lens spaces. 4) Any two Heegaard splittings of one and the same manifold are stable-equivalent, i.e. become equivalent after each of them undergoes several stabilization operations, namely, local addition ofa trivial-handle splitting to a pretzel (Fig. 2.5.7).

Fig. 2.5.7 Homeomorphism group of the pretzel

Homeomorphism group of a solid pretzel

Note that none of these facts is trivial. The first of them admits a beautiful reformulation in the language of homeomorphisms of a pretzel surface. We denote a homeomorphism of the surface of a standard pretzel, which maps its meridians into parallel~, by cpo Then the homeomorphism h determines a sphere if and only if h has the form acpb, where the homeomorphisms a and b are continued onto the interior of a solid pretzel. The structure of the homeomorphism group of the pretzel surface is well-known. Lickorish has found the generators and Weinreeb the relations (see e.g. Referativnyi Zhumal (Review Journal) 1984, 4A597). The structure of a homeomorphism group of a solid pretzel is also well known. For the review of these results see Ref. [29]. The problem of algorithmic recognition of a sphere is reduced to the problem of recognition of bilateral cosets of a known group by a known subgroup. From the second class one can readily obtain the topological classification of lens spaces (without the use of Rademeister torsion). Using the fourth fact one can reuce the problem of classification of all 3-manifolds to the problem of classification of bilateral cosets of a stabilized homeomorphism group of a pretzel surface by a stabilized homeomorphism group of a solid pretzel.

2.5 Visual Properties of Three-Dimensional Manifolds

165

Let M3 be represented as a union of two solid pretzels Kg and K~ with a common boundary Hg = 8Kg = 8K;. The meridians of the pretzels Kg and K; form a set of curves on the 2-manifold Hg which may be thought of as standard. This set of curves is called the Heegaard diagram of the manifold M3. In order that the curves CJ, C2,.··, cg , ~,ci, ... ,g~ on Hg form the Heegaard diagram of a certain manifold M3 it is necessary and sufficient that there hold the following conditions: 1) the curves Ci do not intersect, and after one makes a cut along them one obtains a connected surface, 2) the same for the curves Referred to as the Heegaard diagram is sometimes a set of curves {s} on the boundary Kg (or a set of curves {Ci} on the boundary K;). The corresponding 3-manifold is obtained by glueing 9 handles of index 2 (thick discs) to Kg along the curves {cn and one handle of index 3 (a ball).

Heegaard diagram of a 3-manifold

c:.

2.5.4 Spines

The classical concept of a spine (not to be confused with "spline") is a convenient means for defining 3-manifolds; it is based on the idea of reducing the study of 3-manifolds to the study of two-dimensional polyhedra. See, for example [44]. Coding 3-manifolds by spines is favourable from the viewpoint of a succesive enumeration of manifolds with their increasing "complexity" (for details see below). The new theory of almost special spines and its applications to the problems of three-dimensional topology (including computer geometry) has been thoroughly developed by S.V. Matveev. Some of his results admitting a particularly obvious interpretation are presented in this subsection.

Spine of a 3-manifold

Coding 3-manifo/ds by spines

Definition 2. A polyhedron P c M is called the spine of an n-manifold M with boundary if M -P ~ 8Mx(O, 1). The polyhedron P is called the spine ofa closed n-manifold if P is the spine of the manifold M - Int Dn , where IntDn is an open n-ball in M. An example of the spine of a manifold is depicted in Fig. 2.5.8. In Fig. 2.5.9 one can see that any tree (a graph without cycles) in a ball or in a sphere of any dimension is the spine of the ball or the sphere. A circle

Fig. 2.5.8

166

Fig. 2.5.9

g. . . . . .-

j

Fig. 2.5.10

~

."... --~~

...........

Fig.2.5.11

Fig. 2.5.12

2. Low-Dimensional Manifolds

is a natural spine of a ring 8 1 x I and of a Mobius strip. For any M, the manifold M - (8M), where O(8M) is a "collar" of the boundary, is the spine of M. It is hardly relevant to consider here all possible spines ofa manifold. It is convenient to restrict the consideration to the so-called noncollapsible (or incompressible, closed, non-decreasing, etc.) spines. We shall recall the defmition of a collapse. Let K be a simplicial complex (e.g. a triangulated manifold); (In its basic open simplex, i.e. a simplex which is not a boundary of any simplex from K, and let on-I be its free boundary, i.e. such that is not the boundary of any other simplex from K. Then the operation of discarding simplexes (J and 8 from the complex K is called an elementary collapse (contraction). Figure 2.5.10 presents collapses of dimensions 1,2 and 3. Collapse is a sequence of elementary collapses. The collapse of a polyhedron is the collapse of some of its triangulations. In this case, elementary collapses can be enlarged by including in their number the discarding of n-cells together with their free (n - I)-boundaries (Fig. 2.5.11). The concept of a collapse appears to be very closely related to the concept of a spine.

Theorem 2. If P is the spine of a manifold M and if P collapses on Q, then Q is the spine of the manifold M. The proof of this theorem consists in the fact that the structure of the direct product M - P ~ 8M x (0,1) can be changed locally (in the neighbourhood of some given simplexes) so as to obtain the structure of the direct product by M - Q. It is sufficient to verify the possibility of such a change for a standard simplex in a Euclidean space (Fig. 2.5.12) because all the simplexes in a manifold are positioned standardly up to homeomorphism. A spine is called non-collapsible if it cannot be collapsed into a smaller spine. From among the above examples of spines it is only the circle (the spine of a ring or of a Mobius strip) and point (the spine of a ball or of a sphere) that are non-collapsible for the reason of "the absence of the beginning", i.e. free boundaries from which the collapse could start. It can be readily proved that a point is the only non-collapsible spine of a two-dimensional sphere (a two-dimensional disc). Indeed, a

2.5 Visual Properties of Three-Dimensional Manifolds

167

non-collapsible spine of a 2-disc cannot contain 2-simplexes since in that case there would exist a 2-simplex with a free boundary. Therefore it must be a polyhedron of dimension not greater than unity and, moreover, a homotopically contracible polyhedron, i.e. a tree. The only tree without free vertices is a point. The first part of this consideration is valid for any dimension.

Theorem 3. Any spine of an n-manifold collapses to become a polyhedron of dimension not greater than n - 1. In particular, any n-manifold has a spine of dimension :::; n - 1. The second part of the consideration showing that all one-dimensional spines of a disc are collapsible into a point is invalid already in three dimensions: there exist examples of non-collapsible two-dimensinsional spines of the sphere 8 3• One of such examples is the so-called "Bing house" with two rooms depicted in Fig.2.5.13 and familar to us from Chap. 1. The collapse of a b~l onto this manifold is like this. First, along the tube we penetrate through the upper disc inside the lower room and exhaust it up to the membrane between the tube and the wall. Then, along the tube we penetrate through the lower disc inside the upper room and exhaust it up to the membrane between the upper tube and the wall. Another example of a non-collapsible tow-dimensional spine of a ball is a "fool's cap" drawn in Fig.2.5.14. It is obtained from a triangle through identification of its sides. Neither the "Bing house" nor the "fool's cap" is collapsible for the reason ofthe "absence ofthe beginning" of the collapse. These examples show that one and the same manifold may have several different spines. Another ambiguity of this problem is that distinct manifolds may have homeomorphic (equal) spines. In other words, the polyhedron may get thickened up to the manifold in several ways if at least one thickening does exist. Figure 2.5.15 gives examples of two thickenings of a circle with diameter up to a manifold of dimension two. This is a "genuine buckle", i.e. a disc with two holes and a "piratic buckle", i.e. a torus with a hole. Figure 2.5.16 gives similar examples of dimension 3: a full torus with two discarded balls and a manifold 8 2 x 8 1 with two discarded balls have one and the same twodimensional spine, namely, the surface of a torus with two discs glued along meridianal curves.

-Sexiis

Fig.2.5.16

Fig.2.5.14

:'0. . .-:-.0 0

• • •:

I I I I I I

t

I I I I .: I

:

~. . . . . . .I • • • • •"

Fig. 2.5.15

I I I I I I

J

168

2. Low-Dimensional Manifolds

2.S.S Special Spines

We introduce the concept of a special spine to eliminate the second ambiguity - a possible existence of several different thicknesses. Definition 3. A two-dimensional polyhedron P is called special if the following conditions are satisfied: 1) Each of its points has a closed neighbourhood homeomorphic to a cone over a circle with zero,two or three radii (Fig. 2.5.17). 2) Each connected component of nonsingular points (i.e. points with a neighbourhood of the first type) is a 2-disc. Fig. 2.5.17

Fig. 2.5.18

The set of singular points of a special polyhedron is a regular graph of degree 4, i.e. a graph from each of whose vertices there go exactly 4 edges. Note that singularities of the above-mentioned type are observed in soap films (the Plateau principles). For details see the book by Fomenko [29]. The spine of a 3-manifold is called special if it is a special polyhedron. Clearly, each special spine is non-collapsible. An example of the special spine of a 3-sphere is the "Bing house". It has exactly two vertices (points of the third type), three 2-components (connected components of nonsingular points), and the set of its singular points is of the form shown in Fig. 2.5.18. Theorem 4. Any 3-manifolds has a special spine.

Fig. 2.5.19

The idea of the proof is as follows. We divide a manifold Minto handles - one handle of index 0 for each vertex, one handle of index 1 for each edge and one handle of index 2 for each 2-simplex of an arbitrary triangulation. Handles of index 3 correspond to 3-simplexes (Fig. 2.5.19). The union Po of the boundaries of these handles is a special polyhedron (this is directly seen). We denote the manifold obtained by discarding an open ball from the interior of each handle by Mo. Then, clearly, Po is a special spine of the manifold Mo. From this one we can obtain the spine P of the manifold M by puncturing its 2-components and thus uniting the balls removed into one ball. If the boundary of the manifold

2.5 Visual Properties of Three-Dimensional Manifolds

169

M is non-empty, it must be punctured in one plane. The spine P will have free edges appeared at the boundaries of the pierced holes, and therefore it admits collapse and does not, generally, lead to a special spine. But if puncturing is made carefully lest free edges arise, then we obtain a special spine. Such a careful puncturing using so-called arches is presented in Fig. 2.5.20. The same consideration shows that any 3-manifold has an infmite number of various special spines. The advantage of special spines is the possibility of an unambiguous restoration of a 3-manifold from its special spine. This means that a special spine carries the whole information about the topological structure of the manifold. Theorem 5. (see Ref. [44]). If two 3-manifolds have homeomorphic special spines, they are homeomorphic. The idea of the proof is the following. Since a circle with three radii is embedded into a two-dimensional sphere in a unique (up to a homeomorphism) way, the neighbourhoods of the vertices of the special polyhedron get thickened up to balls in a unique way. This thickening is uniquely continued up the thickening of the edges of the special polyhedron. It remains to prove that the thickening of the neighbourhood of the boundary of each 2-component, which is observed in this case, i.e. the direct product of this neighbourhood by a segment, is continued up to the thickening (i.e. the direct product by a segment) of the 2-component itself in a unique way. This easily follows from the fact that all 2-components are cells - this is why the condition was imposed. Similar consideration show that the neighbourhood of the set of singular points of any special polyhedron (not necessarily the spine of a 3-manifold) thickens up to the 3-manifold. The continuation of this thickening up to the thickening of 2-components may be hampered by a nontrivail character of the fibre bundle of segments appearing near the boundary. Such an obstacle is really encountered: let a special polyhedron be obtained by glueing a disc to a projective plane along a projective straight line. Then it is not embedded into any 3-manifold just for the reason indicated above. Thus, not all special polyhedra are spines of 3-manifolds, i.e. get thickened. But if a special polyhedron does get thickened, it does it in a unique way!

Fig. 2.5.20

3-manifolds with homeomorphic special spines are homeomorphic

Special polyhedron

Filtration in the set of all 3-manifolds

170 Matveev's complexity

2. Low-Dimensional Manifolds

2.5.6 Filtration of 3-Manifolds with Respect to Matveev's Complexity

Theorem 5 implies that the problem of enumerating 3-manifolds reduces to the problem of enumerating special thickened polyhedra. An important fact is that for each number k there exists only a finite number of various 3-manifolds having special spines with k vertices. All Special thickened polyhedra special polyhedra with k vertices can be enumerated as follows. First we should construct all regular graphs of degree 4 with k vertices. This is not difficult to do. Then for each such graph r we should run through all special polyhedra whose set of singular points is homeomorphic to the graph r. For this purpose we should do this. We embed the graph r into "Detail" ]R3 and replace each of its vertices by a so-called "detail" homeomorphic to the cone above the circle with three radii (Fig. 2.5.21). Problem of enumerating of all 3-manifolds

IAL

X~ Triod

2-

(42.3) '23 ( i 2. 3) ~ 13! II

(13 2. It3) ~

3 and L3,1 have zero complexity. Their minimal almost special spines are respectively a point, a bouquet of a sphere and a circle, a projective plane and a Mobius cup. The connected sums of these manifolds also appear to be of complexity O. It turns out that there exist no other closed orientable manifolds of complexity O. Theorem 6. (S.V. Matveev). Let the complexity of a closed orientable 3-manifold M be equal to k. Then M = Ui Mi, where each Mi is homeomorphic to 82 x 8 1, m>3 or has a special spine Pi the number of whose vertices coincides with the complexity ki of the manifold Mi. Given this, there holds the equality k =Ei ki.

2.5 Visual Properties of Three-Dimensional Manifolds

173

The idea of the proof is the following. If the minimal almost special spine P of the manifold M contains points with one-dimensional neighbourhoods, it contains the principal edge, i.e. an edge not adjoint by the two-dimensional part of the polyhedron P. When this edge is cut, the spine P either falls into two parts, which corresponds to the manifold decomposition into a connected sum, or does not fall into parts at all, which corresponds to singling out the summand 8 2 x 8 1. If some of the 2-components of the polyhedron P contains a simple closed curve not restricting a disc, then the polyhedron P should be cut along this curve, and one of the circles resulting from the cut should be glued up by a disc. The manifold will then fall into a connected sum of two manifolds with simpler spines. If the spine of P contains no principal edges and if all simple closed curves in its 2-components restrict discs, then this polyhedron is either special or homeomorphic to ]R2, which corresponds to the case of projective space. The proof of the inequality 2:i ki ~ k is obvious since if we join the minimal almost spines of the manifolds Mi with arcs, we obtain the almost special spine of the manifold M. The inverse inequality 2:i ki ::; k is proved using the Haken theory of normal surfaces [45], and the proof is rather cumbersome. This theorem implies the above-mentioned result concerning orientable manifolds of complexity 0 since the only special spine with 0 vertices is a Mobius cap. From the theorem it also follows that up to connected sums with manifolds 8 2 x 8 1, ]Rp3, L 3,1 there exists only a finite number of manifolds of complexity k. In other words, if we restrict ourselves to closed orientable manifolds containing no non-splitting spheres, projective planes and Mobius caps, then there exists only a finite number of such manifolds of complexity k, and the minimal almost special spine of such a manifold is special.

Idea of the proof

Haken theory of normal surfaces 3-manifolds of complexity 0

Only a finite number of irreducible manifolds with a fixed complexity k

2.5.7 Simplification of Special Spines

As mentioned above, a 3-manifold may have an infinite number of various special spines. How are different special spines of one and the same 3-manifold connected? How shall we describe the set of all special spines of this manifold and how is the minimal one to be distinguished among them? We shall describe the technique, called re-laying of2-component,

Set of all special spines Re-Iaying of 2-components

174 Simplification of special spines

2. Low-Dimensional Manifolds

which enables one special spine to be transformed into other special spines. For simplicity we shall consider only closed 3-manifolds. Let P be a special spine of a closed 3-manifold M and let D c M be a disc in M having no common interior points with P, such that the curve aD lies in P and is in general position to its singular graph r. This means that the curve aD does not pass through the vertices of the graph rand crosses its edges transversally. The disc D splits an open ball M - P into two balls Vi and Y2. An important and easily verified (proceeding from the structure of singular points of the polyhedron P) fact is that the intersection K of closures of the balls Vi and Y2, i.e. their common boundary is a closed 2-manifold composed entirely of 2-components of the polyhedron P. The polyhedron MUD is a speGial spine of the manifold M with two removed balls. If we puncture the disc D, we go back to the spine P of the manifold M with one removed ball, i.e. simply to a spine of the manifold M. But if we puncture one of the 2-components on the surface K, then after collapse we obtain another special (or almost special) spine PI of the manifold M. Figure 2.5.27 presents two particularly simple re-Iayings TI and T2 made in the neighbourhood of a vertex or an edge of the polyhedron P. Inverse transformations TI- I and T2- 1 are also re-Iayings. It turns out that even these very simple re-Iayings suffice to transform one special spine of a 3-manifold into another.

Theorem 7. (S.V. Matveev). Two special polyhedra are special spines of one and the same 3-manifold if and only if they can be joined by the chain of transformations T~I , T2±1.

Fig. 2.5.27

The proof of this theorem is based on the fact that any two triangulations of a 3-manifold have a common subdivion. Note that the complexity (the number of vertices) of a special polyhedron increases by 2 under transformation TI and by 1 under transformation T2. Let us analyze the change in the complexity of a special polyhedron under an arbitrary re-Iaying. Let a curve aD intersect the graph r at k points and let the boundary of the punctured 2-component a of the surface K contain m vertices. After the disc D is glued to the polyhedron P, the number of its vertices increases by k, and after the 2-component is punctured and the polyhedron collapses, the number of its vertices decreases not less than by m. Therefore, the complexity p,(PI) of the polyhedron

2.5 Visual Properties of Three-Dimensional Manifolds

175

obtained as a result of re-Iaying of the special polyhedron PJ does not exceed J.L(P) + k - m, where J.L(P) is the complexity of the polyhedron P. We shall call the re-Iaying simplifYing if J.L(PJ) < J.L(P) and balanced if J.L(PJ) = J.L(P). The number k will be called the degree of re-Iaying. Some experience is dealing with spines shows that re-Iaying of degree ~ 6 are of particular importance.

Simplifying re-Iaying Degree of re-Iaying

Hypothesis. Any special spine of a 3-manifold is reduced to any minimal (in the sense of the number of vertices) one using simplifYing and balanced re-layings of degree ~ 6. Note that a positive solution of this hypothesis would imply the possibility of algorithmic recognition of homeomorphic 3-manifolds. This follows from the fact that the number of special polyhedra of fixed complexity is finite as well as from the fact that re-Iayings of fixed degree can be applied to a given special polyhedron also in a finite number of ways. Let PJ and P2 be special spines of manifolds MJ and M2. We shall apply to them simplifying and balanced re-Iayings as long as possible. If the two finite sets of special spines which we obtained intersect, then MJ and M2 are homeomorphic and if not, then MJ and M2 are distinct. The validity of the hypothesis can be grounded as follows.

Balanced re-Iaying Hypothesis about algorithmic recognition of homeomorphic 3-manifolds

1) The hypothesis is valid for all manifolds of complexity ~ 6. Practically, re-Iayings of degree ~ 4 are sufficient. 2) The hypothesis is valid for the series of special spines of a sphere which are constructed in a canonical way using the Heegaard diagram of a sphere simplifyied by the Volodin-Fomenko cut-vertex and "wave" method [29]. 3) The hypothesis is valid for special spines of a sphere obtained using the construction of "combinatorially disassembled" (i.e. remaining after removal of 3-simplexes collapsing to a point) triangulations described in the proof of Theorem 6.

Cut vertex and wave method

We shall point out the simplest and therefore most easily applicable in practice particular cases of simplifying re-Iayings, i.e. simplification by the counterun and by the short boundary curve. Recall that any special polyhedron is obtained from a certain regular graph of degree 4 by glueing several 2-discs via some maps of their boundary circles. The curves in the graph corresponding to these maps will be called boundary

Boundary curves of 2-components

Validity of this conjecture for special types of a 3-manifolds

176 Counterrun Short boundary curve

2. Low-Dimensional Manifolds

curves of 2-components; We say that a boundary curve has a counterrun if it passes on one of the edges of the graph twice in opposite directions. We say that a boundary curve is short if it passes through not more than three vertices of the graph, once through each.

Theorem 8. If a special spine of a closed 3-manifold has a boundary curve with counterrun or a short boundary curve, then it is not minimal.

p Fig. 2.5.28

Re-/aying procedure

hole Fig. 2.5.29

Bing house

Proof. Suppose in a special spine P of a closed 3-manifold M there exist a boundary curve with counterrun. Then in M there exists a disc D which has no common interior points with the polyhedron P and whose boundary contains exactly one singular point and lies in a 2-component with counterrun on the boundary (Fig. 2.5.28). The re-Iaying procedure consisting in adding the disc D to the spine P and puncturing one of the side walls (as shown in the figure) is simplifying. Indeed, in puncturing and collapse there vanish at least the vertex added and the singular edges, starting at this vertex, together with other vertices (the endpoints of these edges). Suppose now that in the special spine P of the closed 3-manifold M there exists a 2-component a: with a short boundary curve passing through k ::; 3 different vertices. If inside the manifold we glue to P a disc parallel to the component a: and puncture the side wall of the cylinder obtained, then after a collapse we obtain an almost special polyhedron with a smaller number of vertices. Indeed, k vertices are added and at least 2 vanish for k = 1 and 4 for k = 2,3 (Figs, 2.5,29, 30). Note that one can easily realize the indicated simplification technique and construct, in fact, an almost special spine with a smaller number of vertices. The Bing house has both a short boundary curve and a counterrun. Using any of these procedures, one can simplify the Bing house to a point. It is noteworthy that in the case of a boundary curve of length 4, such a re-Iaying is balanced.

2.5 Visual Properties of Three-Dimensional Manifolds

177

2.5.8 The Use of Computers in Three-Dimensional Topology. Enumeration of Manifolds in Increasing Order of Complexity The results of subsections 2.5.6 and 2.5.7 allow enumeration of all closed orientable (containing no non-splitting spheres, projective planes and Mobius caps) manifolds using computer. Such an enumeration of manifolds of complexity :::; 5 was made in 1972 by S.V. Matveev and V.V. Savvateev and then repeated in an extended version (including some manifolds of complexity 6) by S.V. Matveev in 1984 [46]. In 1986, S.V. Matveev enumerated all manifolds of complexity 6 [47]. Recently, in 1987, Matveev and Fomenko [48] used computer for analysis one of the problems of three-dimensional topology, namely, for estimation the least volume of a compact hyperbolic manifold. Restriction to complexity 6 is caused both by the lack of machine time (calculation of each graph with 6 vertices takes up 5 to 8 hours depending on its type) and by difficulties of manual identification of manifolds yielded by computer. Regular graphs of degree 4 with a given number of vertices are to be enumerated manually (for the first time this was done in 1972) or using computer (1984). Then each graph is set into computer which enumerates all possible glueings to toroides (see Sect. 2.5.6) and subjects the boundary curves of the special polyhedron obtained to the check: 1) there exists a boundary curve oflength :::; 3; 2) there exists a counterrun; 3) the glueing of thickened details gives Heegaard diagrams of the orientable manifold. If the answer to one of the first two questions is affirmative and to the third question is negative, the given choice of toroide glueing is rejected and a new version is set for verification. Otherwise the computer calculates the one-dimensional homoloy group of the manifold and conveys the result, along with the triode glueing system, on a printer. The results of computer calculations are listed in Tables 1-9. The tables include all irreducible manifolds of complexity :::; 5 and those manifolds of complexity:::; 6 which were modelled on the one-dimenisonal octa-

1 0, then the moment is sure to come when our trajectory will pass near the chosen point at a distance not exceeding the number c:. In other words, in some time such a trajectory will pass arbitrarily close to any aforegiven point of the torus. True, it may be a long time before this event happens. If we now close such an irrational trajectory, we obtain the whole torus. In this case the Liouville torus is a set of points arbitrarily closely approximated by points of the given integral trajectory. It is known from

Special coordinates in the neigbourhood of a Liouville torus Angle-action variables

Rectilinear winding = almost periodic motion

a Fig.3.1 .3

Periodic = rational solutions Almost periodic = irrational solutions

Irrational trajectories are everywhere dense

198 Liouville torus as the closure of its irrational integral trajectory

Phase space as a symplectic manifold in mechanics

Non-resonant integrable systems

Evolution and bifurcations of a Liouville tori show the "topology of a given system"

3. Visual Symplectic Topology and Visual Hamiltonian Mechanics

mechanics, that for analytic systems the case described above is the case of general position, i.e. typical in the sense that if we choose randomly the initial point of the integral trajectory, send a trajectory from this point, continue it to infinity and close, we obtain a complete Liouville torus. The cases when an integral trajectory returns to the initial point (i.e. is homeomorphic to a circle) are "exceedingly rare" and correspond to socalled resonances. In systems of general form, such resonance points fill up a set of measure zero, i.e. a very "thin" set. Thus, the behaviour of integral trajectories of Liouville-integrable systems is described rather thoroughly, at any rate from the topological point of view. The points of a symplectic manifold (in mechanics it is called the phase space of a mechanical system) on which a completely integrable system is given appear to split into three groups (types). Points of type 1 are characterized by the fact that the integral trajectory sent from them never returns to the same point but starts winding round a Liouville torus which is its closure (Fig. 3.1.4). Points of type 2 are characterized by the fact that the integral trajectory sent from them moves along the Liouville torus and in some time goes back, i.e. the trajectory is closed. In this case, the trajectory continues moving along itself winding on itself infinitely many times. All the rest ofthe points are of type 3. We shall not specify here the behaviour of the integral trajectory for points of type 3 since it may prove to be very complicated. We shall mention them later. It turns out that for analytic non-resonant systems points of type 1 are typical, of general position and constitute the "majority" of all points. Points of types 2 and 3 occupy a thin subset of measure zero. Consider a point x of type 1 in a phase space M2n and denote by Tn(x) the Liouville torus which is the closure of the integral trajectory sent from the point x. When the point x changes, the torus m(x) also undergoes some changes. Moving along the phase space, it can at some instant degenerate into a circle. This happens, for example, if when moving the point x gets into the set of points of type 2. Thus, Liouville tori may deform within the phase space transforming, splitting into a union of other Liouville tori, merging with tori, degenerating, etc. What is happening with Liouville tori is important for understanding the geometry of a given mechanical system since they show what is happening with the solutions of a Hamiltonian system of equations when the initial data are changed.

3.1 Some Concepts of Hamiltonian Geometry

199

In other words, the topological deformations and surgery on Liouville tori demonstrate the qualitative character of the dependence of the solutions of a Hamiltonian system on the initial data. To the reader acquainted with mechanics we shall explain in addition that a point of a phase space is given by two sets of n number: n coordinates and n momenta. Thus we have come to the natural problem of classification of topological surgery on Liouville tori arising within a phase space. In the subsection to follow we shall specify the formulation of this problem. The mechanism forcing a Liouville torus drift about a phase space should be described in more detail. Above we have restricted our consideration to an arbitrary deformation of an initial point x. But in physics and mechanics of particular interest are Liouville torus deformations caused by variations of the energy value. Let v be an integrable non-resonant Hamiltonian system on a symplectic manifold M2n. Let Hbe its Hamiltonian and 12, ... ,in additional independent smooth integrals in involution (including the Hamiltonian). Consider an arbitrary Liouville torus. We state that it is specified only by the Hamiltonian H = i1 and by the very fact of existence of other additional integrals h,.'.. , in, i.e. a Liouville torus in general position does not depend on a concrete choice (concrete form) of the integrals h, ... ,in' To put it differently, if instead of the initially given integrals h, ... , in one takes some other integrals 12,"" In (which as before are independent of H and are in involution with it), then the Liouville tori remain unaltered. This simple but important circumstance is explained by the fact that a Liouville torus in general position coincides with closure of the integral trajectory sent from an arbitrary point and forming an irrational winding on the torus. The integral trajectory itself, i.e. the solution of the system is completely specified by the initial point and by the Hamiltonian H and is absolutely insensitive to the choice of a concrete form of the other integrals h, ... , in (they must only exist). The very fact of existence of additional integrals implies that the trajectory is winding round a torus (in non-resonant case). It should be clarified here that additional integrals of the system are not, generally, uniquely determined. The functions 12, ... ,in can be replaced by some other ones, for instance, by those depending functionally on them and on H. Instead of the set of integrals h, ... , in one can take, for example, 12 = 12 + 13'/3 = 12 - h h = Ii, 4 ~ i ~ n, etc.

Classification of a transformations of a Liouville tori

Evolution of

a Liouville tori caused by variations of the energy value

Liouville torus in general position does not depend on a choice of the additional integrals

Additional integrals are not uniquely determined

200 Momentum mapping

Fig. 3.1.5

Critical points of a momentum mapping Critical values of a momentum mapping Bifurcation diagram

3. Visual Symplectic Topology and Visual Hamiltonian Mechanics

3.1.3 Momentum Mapping of an Integrable System Let v be an integrable system with a set of independent involutive integrals H = Ii, Jz, ... , In. Then with this system there is naturally associated an essential mapping, called the momentum mapping. It is defmed as follows. Suppose F(x) = (fI(x), ... , In(x», i.e. each point of the symplectic (phase) manifold is assigned a sequence of numbers - the values of the integrals at this point. As a result, there arises a smooth map of the manifold into a Euclidean space ~n (Fig. 3.1.5). Setting a certain point a E ~n, w~ define the common level surface Ta of all the integrals b, h ... , In. It is given by the equations b(x) = ai, ... ,In(x) = an, where a = (ai, ... ,an), i.e. the numbers ai are components of the vector (point) a in the space ~n. From the implicit function theorem we know that the level surface is a manifold if all the integrals on it are independent. How can the independence of the integrals be verified? To do this, one should first calculate their gradients at points of the surface under consideration. Then the functions b, h ... , In are independent in a neighbourhood of the point x if at this point their gradients grad b, ... , grad In are linearly independent. We recall that grad} = (a I / axi , ... , aI / ax2n ). At each point x of the manifold M2n , the differential dFx of the map F is defined. Recall that the point x is called critical for the map F if at this point the rank of the differential dF is less than n, i.e. less than the dimension of the space ~n into which the manifold is mapped. The image F(x) of the critical point will be called the critical value of the momentum map. The set of all critical values will be denoted by E. It is sometimes called the bifurcation set (the bifurcation diagram). Thus, if N is the set of all critical points of the map F (in the manifold M 2n ), then E = F(N). The set E is known to be the set of measure zero in the space ~n (the complement of E is open and everywhere dense in ~n).

3.1.4 Surgery on Liouv1lle Tori at Critical Energy Values Regular values of a momentum mapping

Points a from ~n which are not critical values will be called regular values (for a given momentum map). Let F-i(q) be a complete preimage of the point q under the map F. If a is a regular value and the

3.1 Some Concepts of Hamiltonian Geometry

201

pre-image p-l(a) is compact then, according to the Liouville theorem, it consists of Liouville tori (Fig. 3.1.5). If a moves along a smooth curve 'Y in the space I.n and remains a regular value for the momentum map then its pre-images deform smoothly in the manifold. In other words, each of the Liouville tori Tn deforms smoothly inside the manifold M 2n Deformation of and remains a Liouville torus. But the situation changes radically when a Liouville tori a approaches the diagram E and at some instant of time punctures it. inside 2n-manifold Let c be the point where the path 'Y meets the set E. Since c is the critical value of the momentum map, its pre-image may be not a tours (or a union of tori). To say it differently, it is precisely at this moment that there occur surgeries (bifurcations) of the tori. Figure 3.1.5 illustrates Surgery of schematically the surgery of tori at the moment when the path 'Y punctures a Liouville tori the bifurcation set E. We distinguish between two types of the sets E. Sets oftheftrst type Two types of are characterised by the fact that their dimension is equal to n - 1. Sets of a bifurcation diagrams the second type are such that their dimension is strictly less than n - 1. "large" and "small" Note that since P is a smooth map, there always holds the inequality bifurcation diagrams dimE:=;n-1. Suppose for a given integrable system there holds the inequality dim E < n - 1. In this case the pre-images p-l(a) and P-l(b) of any two regular values a and b from I.n are "the same" in the sense that they consist of one and the same number of Liouville tori, the manifold p-l (a) being diffeomorphic to the manifold p-l (b). Indeed, to prove this assertion it suffices to join the values a and bby a smooth path 'Y in I.n , which does not intersect E. This can be done since by the assumption we have dim E :::; n - 2 (Fig. 3.1.6). Therefore, according to the implicit function theorem, when the point 'Y(t) moves along the path 'Y from the Fig.3.1.6 value a to b, its complete pre-image is deformed in the manifold M via a diffeomorphism. As a result we obtain that each of the Liouville tori sweeps a cylinder Dl x rn and that cylinders corresponding to different tori do not intersect. This implies our assertion. In this respect, the Hamiltonian systems in which dim E :=; n - 2 are organized rather simply (form the topological point of view): any two sets of Liouville tori of the form p-l(a) and P-l(b) can be deformed smoothly into each other in the class of Liouville tori. Indeed, consider corresponding points a and b and join them first by an arbitrary continuous path 'Y. It may generally speaking meet the bifurcation set E.

202

Chambers of momentum mapping

The case of "large" bifurcation diagrams is vel}' important in physics

-.' ----------- .--c::> ~.~

A

r i:: c:::::)

-- ---- --- -- _.c::> ':.

::. 0 0

'"

o o

""

Fig.3.l.8

Physical example

3. Visual Symplectic Topology and Visual Hamiltonian Mechanics

But since dim E ::; n - 2 and the path "{ is one-dimensional, one can remove the path "{ from the set E using a small deformation. As a result, we obtain the path joining the points a and b and not intersecting E. Then the above arguments are valid. In this sense, when the "majority" of Liouville torus deformations occur in the manifold M2n , the Liouville tori undergo no surgery (and if some surgery does accidentally occur, it can be removed by means of small "stirring"). We see quite a different picture when dim E = n - 1. Here the set E, generally speaking, splits the space ]Rn into individual isolated "chambers" (Fig. 3.1.7). The point is that here E consists of pieces of smooth hypersurfaces (perhaps with singularities). Therefore, E separates ]Rn. Consequently, choosing the points a and b in different chambers we can see that any smooth path joining these points will necessarily puncture the set E at some point c (or in several such points). Hence, when moving along this path, a Liouville tours (in the manifold M) will necessarily come across the set N of critical points and may undergo a serious topological surgery. Moreover, any way we choose to send a torus from F-l(a) into a torus from F- 1(b) will necessarily lead us at some moment to a qualitative surgery. Thus, if dim E = n - 1, then each chamber (i.e. each connected component of the set ]Rn \ E) is generally characterized by its unique set of Liouville tori. The number of these tori may be different for each chamber. This situation is most frequently encountered in concrete integrable systems in physics and mechanics. We shall therefore give some more attention to the case dim E = n - 1. We shall discuss a concrete mechanism responsible for the drift of real Liouville tori about a symplectic (phase) manifold. Consider a simple example - a well with the profile shown in Fig. 3.1.8. Suppose that a ball starts falling down along the well wall, the friction being now absent. Then the ball motion is completely determined by its initial position (the height), i.e. by its potential energy in the vertical gravitational field. In the absence of friction the ball will each time reach the initial height, fall down the well again, and so on. The process will be infmite. Now we "switch on" the frictional force. Let the friction be very small. Then we assume to a first approximation that the motion of the

3.1 Some Concepts of Hamiltonian Geometry

203

ball has not practically changed, and taking very small time intervals we may say that the system behaves as before. But on large time intervals the effect of small friction is telling appreciably. Clearly, in the course of time the ball will rise at an increasingly small height due to the energy loss (energy dissipation). Finally, the moment of qualitative transformation of the ball motion will come: it will for ever remain either in the left or in the right well (Fig. 3.1.8). The surgery proceeds at the moment when the level of the ball rise touches the saddle A. If above the point A the level line is homeomorphic to a circle, then at the moment the level line touchs the saddle A (the critical energy level), it (the level line) becomes a figure-of-eight, and as the energy goes on decreasing the level line splits into two separete circles which go down until they finally become the points of minimum [50]. Thus, the deformation of system's behaviour was caused here by the decrease in the ball energy. In the general case this situation can be described like this. Consider an integrable system with integrals H = II, h, ... ,In. Fix the values of all the integrals but one, namely, let the values of the functions h, ... , In be fixed and the value of the energy integral (i.e. H) vary. There starts the drift of the Liouville tori about the manifold, induced by the variation of the H value. In the course of this, the torus will sweep an (n + I)-dimensional surface which we denote by xn+l. Clearly, it is the common level surface of the integrals h, ... , In (the integral II is excluded). See Fig. 3.1.9. We may assume the surface X n+! to be the pre-image of the straight line lying in ]Rn and passing through the point a parallel to the coordinate axis H. The surface X may, accordingly, be considered outside the manifold M. Then H becomes a smooth function on the surface, i.e. H can be regarded as the map of X into a real straight line. As the energy increases, the Liouville torus or tori corresponding to the energy value a start deforming on X. Then, at a certain critical energy value c, the torus undergoes a surgery, i.e. splits into several tori, and the latter, carried away by the increasing energy, go on moving (Fig. 3.1.9). The question is how the surgeries on Liouville tori are classified. The answer for a rather large class of systems is given in the subse.ction to follow.

Motion of a ball and effect of small friction Decrease in the ball energy

General situation: the drift of Liouville tori induced by change of energy value

Fig.3.1.9

204

3. Visual Symplectic Topology and Visual Hamiltonian Mechanics

3.1.5 Visual Material Motion of a rigid body in an ideal fluid

Hamiltonian field with 1 degree of freedom is incompressible flow on 2-surface

Hamiltonian systems occur in may mechanical and physical problems. Figure 3.1.10 illustrates the inertial motion of a rigid body in an ideal incompressible liquid. The dynamics of such a system is described by Hamiltonioan equations. This permits the use of the methods of algebra, analysis and group theory in the study of such systems. In the case of 2-manifolds, a Hamiltonian field is depicted as a flow of incompressible liquid through a surface. In this case, incompressibility is equivalent to area conservation by figures on the surface, carried along by a flow of liquid (Figs. 3.1.1 I). The liquid may run off rather whimsically, but at each nonsingular point of the flow on the manifold the divergence of the flow is equal to zero. Figure 3.1.11 represents the front of moving flow. Figure 3.1.12 presents mathematical variations of close themes of optical effects, caustics, singularity theory [51]. These questions are also connected with the theory of Lagrange manifolds [18].

Page 205, Fig. 3.1.1 o. Motion of rigid body in an ideal fluid Page 206, Fig.3.1.11. Hamiltonian field on 2-surface as the flow of incompressible fluid. Atlantic myths: temple in the ocean Page 207, Fig.3.1.12. Smooth 2-surfaces as a mirrors; optical and reflection effects. To M. Bulgakov's "Master and Margaret": Roman soldier Mark controls the crucifixion of Jesus

208

3. Visual Symplectic Topology and Visual Hamiltonian Mechanics

3.2 QUalitative Questions of Geometric Integration of Some Differential Equations. Classification of Typical Surgeries of Liouville Tori of Integrable Systems with Bott Integrals Nondegenerate (= Bott) integrals

Integrable Hamiltonian systems with 2 degrees of freedom Handfindependent integrals Momentum mapping

\~: :..,., :~--~~ -:."% __ -j__________ :::

S

X

-; -d X2» (XI (XI)xI + (X2(X2)x2 - Al (XI) +>d X2)

,

where Al (x]) + A2(X2)f 0, (XI(X]) > 0, (X2(X2) > 0. This system has two integrals: H

/

= 1/2(AI + A2) ((XI xI + (X2X~) + ~: :

(

.2

.2)

K = 1 2(AI + A2) A2(X2 XI - AI(X2 X2 +

~:

'

'PI A2-'P2 AI Al + A2

Consider the projection p : T*]R2 ---+ ]R2, p(XI, X2, XI, X2) = (XI, X2). Here instead of the Poisson bracket we take the configuration space ]R2. From the conditions H = hand K = k it can be easily calculated that .2 2('Pi - Aih +(-I)ik) Xi = (Xi(AI + A2)2

'

Integrals of Liouville system Lissajous figures

Horizontal and vertical oscillations

i = 1,2 .

Hence, Uh,k = P(h,k) = {(XI, X2) E ]R21Ai :S 0; i = 1,2}, where Ai(xi) = 'Pi - Aih + (_I)i k. Thus, the most typical version has the form U(h,k) = [aI, bd x [a2, b2] C ]R2 (Fig. 3.2.12). For the interior points or'the rectangle we have four admissible velocities: Two on its sides and one (zero) at vertices. Trajectories in the rectangle (projections of phase trajectories)

Fig.3.2.12

are similar to Lissajous figures (the sum of independent horizontal and

verticale oscillations). The pre-image p-ICUh,k) is a torus with conditionally periodic motion. It can be shown that the DPM type changes in passing over (h,k), such that AI(xI) or A2(x2) has a multiple root. Let xi be the multiple root of Al on a segment [al,bJ] (Fig.3.2.13). Numerals in the figure indicate the number of admissible velocities. The corresponding h,k is the direct product of a figure-of-eight by a circle (Fig. 3.2.14). Its median line is the limiting cycle for the rest of the phase (integral) trajectories. In a Liouville system, the DPM type changes only for (h, k) E E. In the general case, the DPM type may change also for those (h, k) which are not critical values of the momentum map I=H x K.

Fig. 3.2.13

Fig. 3.2.14

220

3. Visual Symplectic Topology and Visual Hamiltonian Mechanics

3.2.6 The Liouville System on the Sphere Let a rigid body be fixed in the centre of mass and let it be in the central Newton field. Here JI(II) = (All, II) /2AIA2A3, and in addition to the energy and area integrals there also exists the integral K(w, II) = l/2(Aw,Aw) -l/2(A- I II,v). This integral was pointed out by Clebsch for the problem of the motion of a rigid body in a liquid. Consider the case G = O. We have:

TS2 = {(WII) E82 x 1R3IG(w, II) = O}

.

Now iI = HT• S 2 and K = KIT.S2 are the integrals of the system on T*8 2• Fig. 3.2.15 The bifurcation diagram of the momentum map I = iI x K : T*8 2 -+ 1R2 has the form shown in Fig. 3.2.15. The bifurcation set E consists ofthree rays and a parabola segment tangent to two of them. The set E splits the plane into five regions. For regions I, II, III we have h,k = 2T2 , for region IV - h,k = 4T2, and for the other points, not lying on E, we have h,k = 0. For the segments of E labelled by the numerall we have h,k = 28 1, for segment 2 - h,k = 481 and for segments 3 - h,k = R X8 1 (Fig. 3.2.16). For segments 4 we have h,k = (8 1 V 8 1) X 8 1 (Fig. 3.2.14). Next, the DPM for I has the form of two squares on opposite sides of the Poisson sphere (Fig. 3.2.17). Projections of the phase trajectories fill up the squares like Lissajous figures. Next, the DPM for type II has the shape of a ring (Fig. 3.2.18), and for type III it is another ring (Fig.3.2.19). On the ring, in both directions (since this is the image of two tori) there go trajectories touching by tum the ring boundaries. Next, the DPM for type IV consists of two rings (Fig. 3.2.20). Trajectories go Fig. 3.2.16 in both directions on each ring. The surgery in going over from region I to region II is shown in Fig. 3.2.21. Given this, the squares (DPM for region I) are glued with their sides and form a "horizontal" ring (DPM for region II). The surgery in going over from region II to region III is shown in Fig. 3.2.21. Here the horizontal ring (DPM for II) is glued

R.@

Fig. 3.2.19 )It

Pit-

Fig. 3.2.17

Vi

Fig. 3.2.18

Fig. 3.2.20

3.2 Qualitative Questions of Geometric Integration

221

along the segments of the cross-section V2 = 0, lying between the points PI, P2 and P3, P4, and tears along the segments P2P3 and P4PI, where PI, P2, P3, P4 are points on the cross-section V2 = 0 (Fig. 3.2.22). Here for the critical surface Lh,k the DPM is the entire Poisson sphere. The surgery in going over from region III to region IV is represented in Fig. 3.2.23. The ''vertical'' ring (DPM for III) breaks along the median line. Transition from region I to region IV causes the surgery on Liouville tori shown in Fig. 3.2.23. The corresponding events on the Poisson sphere are depicted in Fig.3.2.24. So, the upper and lower sides of the square are folded in two and glued, along the segment PI P4 there goes a break. We obtain one of the rings DPM IV (Fig. 3.2.24).

-

~

/\

Fig. 3.2.21

Fig. 3.2.22

yy

Fig. 3.2.23

Fig. 3.2.24

3.2.7 Inertial Motion of a Gyrostat Here the Euler-Poisson equations have the form

.AW +w X (Aw + A) = 0,

h

c;][ ,.D

11 ~r

iJ = v x w ,

i

,2.

where A is a constant vector in ~3. The integrals are of the form

H = 1/2(Aw, w),

G = (Aw + A), v),

K = (Aw + A), (Aw + A)) ;

I =H x G x K : 82 x R3

---+

h

R3 .

1) Let Al . A2 . A3 =t- O. The bifurcation set Eg = En {g = const} is presented in Fig. 3.2.25. Here E splits the space R3 into four regions: for region I we have h,g,k = T2, for II and III we have h,g,k = 2T2, and

Fig. 3.2.25

f


Fig. 3.2.27

;'=0

It

ii~_ .BL

,"

lI.

*

", JY '!::"':, '1*0

'"

3. Visual Symplectic Topology and Visual Hamiltonian Mechanics

for IV we have h,g,k = 0. The surgeries on the Liouville tori occurring along the straight line A are depicted in Fig. 3.2.26. It is now easy to describe to strucutre of Ih,g,k for points of the set E. For the reversal points we have h,g,k = 8 1 x 8;, where 81 is a circle with angular point (Fig. 3.2.27). 2) Let )'1 = ).2 = ).3 = O. This is the Euler case for an ordinary rigid body. Given this, Eg degenerate as shown in Fig. 3.2.28. Except for the trivial case 9 = h = k = 0, where h,g,k = 82 is a Poisson sphere filled with motionless points, non-empty sets are obtained when k > O. Then one can introduce parameters c =l/k and (1 = 2h/k and project the set E onto the plane (c, (1). This will give us the picture represented in Fig. 3.2.29. Here the DPM type changes not only in passing over the points (h, g, k) E E as in the case of Liouville tori in the preceding examples. There are 25 DPM types altogether corresponding to the 25 regions into which the image on the plane (c, (1) is split. We obtain three basic types of images of the Liouville tori on the Poisson sphere (Fig. 3.2.30, cases a, b, c). Cases a, b and c in Fig. 3.2.30 are distributed on the plane (c, (1) as shown in Fig. 3.2.31. These images of tori are stable in the sense of stability of smooth mappings. There may also exist two types of unstable images (Fig. 3.2.30, cases d, e) realized on the curves separating regions a, b, c.

Fig. 3.2.28 6"

iv

tfAa .' .- .....

.. :'. Iff. . : '. Ii

l~

: ....... ."..:.

1

......"" . "

I"

~t

•• '.",."

"N'"

."

ii e

Fig. 3.2.29

-1

Fig. 3.2.32

8)

Fig. 3.2.30

Fig. 3.2.31

3.2 Qualitative Questions of Geometric Integration

223

3.2.8 The Case of ChapJygin-Sretensky

Here Al = A2 = 4A3, Al = A2 = 0, A3 = >., IJ(v) = -VI. In addition to the energy integral H and the area integral G, there exists here a particular integral K = 2(W3 - A) (W[ + + 2W]/J3 on the set {G = O}. Consider the map 1= fI x K : T.8 2 -+ 8 2. The bifurcation set 17(A) is depicted in Fig. 3.2.32. The dashed lines do not belong to 17. Case "a" is the Goryachev-Chaplygin case of the motion of a rigid body. The set 17(A) splits JR3 = JR2(h, k)XJRI(A) into seven regions. Consider the richest case 4/3 < A2 < 4 (Fig. 3.2.32d). Here the plane JR2(h, k) is divided into seven regions which are the cross-sections of the corresponding seven regions in the space JR\h, k, A) (Fig. 3.2.33). The surgeries on Liouville tori along the arrows are given in Fig. 3.2.34. Here the surgery (H) is a composition of two surgeries of type (C) where both critical circles lie on one level. The critical level surface is the direct product of a circle by the curve obtained by glueing three copies of circles at two points. We now consider the Goryachev-Chaplygin case (A = 0). The bifurcation diagram is presented in Fig. 3.2.35. Figure 3.2.36 shows a splitting of the set {(h, k); k > O} c JR2(h, k) into ten regions, each corresponding to its own DPM type. The dashed lines do not separate regions. DPM for six cases J(a)- J(i) are depicted in Fig. 3.2.37 (see also Fig. 3.2.36). These are images of one Liouville toms. One can see stereographic projections from the pole (-1,0,0). Roman numerals indicate the number of admissible velocities. Figure 3.2.38 shows DPM for four regions lI(a) - 1I(d).

wD

Fig. 3.2.33

(A)=

l (C)1 (E)= III

(G)=tY (8)= f (~)= 111 (F)=lr l (H) =

f='JY

Fig. 3.2.34

Fig. 3.2.35

Fig. 3.2.36 Fig. 3.2.38

224

3. Visual Symplectic Topology and Visual Hamiltonian Mechanics

This is the image of two Liouville tori. The projection of each of these tori onto the Possion sphere is in this case symmetric about the plane 113 = O. Therefore, we depict here only the hemisphere 113 > O. The narrow band near the boundary of the ring in Fig. 3.2.38 is the image of one of the tori· which lies entirely in the image of the other torus.

3.2.9 The Case of Kovalevskaya Here Al = A2 = 2A3; >. = 0; II(v) = -VI. The additional integral has the form K = (wf - w~ + VI)2 + (2WIW2 T v2i. Consider the map I = H x G x K : 82 x ]R3 - t ]R3. The cross-sections of the bifurcation diagram Eg are presented in Fig. 3.2.39. a) g=O b) 0 < y2 < 1/2 c) g2 = 1/2

.',' ',' n.L ......

_71"

:-:'

Fig. 3.2.39

d) 1/2 < g2 < 4/3..;3 e) g2 =4/3..;3 f) 4/3..;3 < g2 < 1

3.2 Qualitative Questions of Geometric Integmtion

225

In region I we have h,g,k = T2, in II, III, IV we have h,g,k = 2T2, in V we have h,k =4T2. For the surgery e.g. in the case 0 < g2 < 1/2 see Fig. 3.2.40.

8

(A)=

1(BJ=ll (C)=Y

(~)i?(E)=1 ! (F)=tt Fig. 3.2.40

3.2.10 Visual Material

Figure 3.2.41 illustrates rotation of various tops differing from one another by their moments of inertia, i.e. roughly speaking, by the form. The whirligig is one of such tops. A rapidly spinning top is one 6f the principal parts of such wide-spread devices as gyroscopes. High demands are made of their symmetry and centring. If a top is non-symmetric, it can tear to pieces when it speeds up and the number of revolutions is large. Gyroscopes are used for aligning various apparatus (airplanes, ships, etc.). In modem mechanics of nonholonomic systems much attention is given to the study of solid bodies rolling along two-dimensional surfaces as well as various "skates" or a sharpened plate (blade) sliding on the ice or a similar surface. Integration of such equations of motion is a fascinating but difficult task. See e.g. the recent papers by V.V. Kozlov and Ya.V. Tatarinov. Figure 3.2.42 shows a Poisson spheres onto which

Moments of inertia Spinning top Gyroscopes

Nonho/onomic mechanical systems

226

Images of a Liouville tori on the Poisson sphere Vortex motion in the flow

3. Visual Symplectic Topology and Visual Hamiltonian Mechanics

a three-dimensional manifolds and the comprised in it Liouville tori of an integmble Hamiltonian system are mapped in the way described in Sect. 3.2.4. The properties of the bifurcation diagram of momentum mapping can be investigated by tracing out the behaviour of images of the Liouville torus on the Poisson sphere. In Fig.3.2.41 these images fill equatorial bands. Similar events are illustrated in Figs. 3.2.42. The images of tori can be depicted as fairly exotic sets. Beside that, Figs. 3.2.43,44 illustrate vortex motions in liquid flows, surgery on level surfaces of integrals of Hamiltonian systems. The inertial motion of a rigid body in an ideal incompressible liquid is described by Hamiltonian equations. But the appearance of turbulence immediately distorts and complicates the patterns of the lines of flow. Various geometrical images associated with such liquid flows (singular points of flows, vortices, sepamtrices of flows, etc.) are presented here.

Page 227, Fig. 3.2.41. Gyroscops as a rapidly spinning tops; rolling ball as a nonholonomic mechanical system Page 228, Fig. 3.2.42. Poisson spheres and Euler-Poisson equation of a motion of a rigid body. Ancient Egyptian priests and military court Page 229, Fig. 3.2.43. Vortex motion in a flow Page 230, Fig. 3.2.44. Boundary effects in the flow and analytic functions. Greek and Atlantic legend about the daughter of the god Poseidon

3.3 Three-Dimensional Manifolds and Visual Geometry

231

3.3 Three-Dimensional Manifolds and Visual Geometry of Isoenergy Surfaces of Integrable Systems 3.3.1 A One-Dimensional Graph as a Hamiltonian Diagram

We now consider constant-energy surfaces Q = Q3 (sometimes referred to as isoenergy surfaces), Le. surfaces described by the equa- Isoenergy 3-surfaces tion H = const. We investigate nonsingular isoenergy surfaces, that is, such surfaces on which the function H does not have critical points, i.e. gradHrO everywhere on Q. The implicit function theorem implies that in this case Q3 is a smooth 3-manifold in M4. Since M4 is always Isoenergy surfaces orientable (as a symplectic manifold), so are the 3-surfaces Q. The term are orientable "surface" should not be confused here with two-dimensional surfaces which we studied above. It is a known fact of mechanics that equilib- Equilibrium positions rium positions of a Hamiltonian system are given by the critical points of a Hamiltonian system of the Hamiltonian (i.e. by such points where grad H = 0). Therefore, when considering nonsingular (non-critic~l) constant-energy surfaces we thus assume them to have no equilibrium positions of the system. We also assume further on Q3 to be a compact closed manifold. Now restrict the second integral f to the surface Q, and not to introduce new notation denote the function obtained by the same letter f. Two-dimensional nonsingular level surfaces of the integral f on the manifold Q3 are obviously Liouville tori. Varying the f value, we make these tori drift along the Liouville tori inside manifold Q3. Clearly, the surgeries of Liouville tori (of Bott position) isoenergy 3-surface caused by variations of the value of the integral f on the surface Q3 are to be described in an absolutely the same manner as those caused by variations in the value of the Hamiltonian H on the surface X3 (see Sect. 3.2). We assume the integral f to be a Bott one on Q3. The Bott integrals Simple nondegenerate can be divided into two Classes: simple and complicated. We shall the (= Bott) integrals integral f simple if each connected component of the critical level surface rl(c) contains exactly one connected critical submanifold of the integral. The other Bott integrals will be called complicated. In other Complicated (= complex) words, the integral is complicated if on some connected component of nondegenerate integrals the level surface f-1(c), where c is the critical value of the integral f, there lie several critical submanifolds.

232 Simple nondegenerate integrals Almost all known physical integrable systems are nondegenerate Non-resonant systems

Graph as a diagram of an integrable system

___________ fR.1

~ _ -_

Gt

Fig. 3.3.1

3. Visual Symplectic Topology and Visual Hamiltonian Mechanics

We shall now, be concerned with simple Bott integrals. For the case of complicated integrals see the papers [60], [61] and the book [62] by the author. It turns out that the overwhelming majority of known physical and mechanical integrable systems possess Bott integrals, and are nonresonance on almost all nonsingular Liouville tori. A Hamiltonian system is called non-resonance (on an isoenergy. surface Q) if the Liouville tori on which the trajectories of the system form an irrational winding are everywhere dense in a given isoenergy manifold Q3. It turns out (for details see Ref. [62]) that if a Hamiltonian system is non-resonance (on some Q), its topology does not depend on the choice of the second integral f and is completely determined by the Hamiltonian H itself. We find it convenient to determine to topology of a manifold Q (and at the same time the behaviour on it of any second integral f independent of the Hamiltonian H) by means of a graph which we denote here by r(Q, H, f) or simply by r(Q, H) since, as mentioned above, in the nonresonance case this graph does not depend on the choice of the integral f. We describe the construction of this graph in the case of a simple integral f (more precisely, in the case of those Hamiltonian H which admit at least one simple integral f). The construction of a graph in the case of a Hamiltonian admitting complicated integrals is given in Ref. [62]. A graph r(Q, H) will be called the diagram ofan integrable Hamiltonian H on a given isoenergy surface Q = (H = const). The integral f can be viewed as a smooth map of the manifold Q into a real straight line lR 1• If a is a regular value in lR 1, then the preimage of the point a under the map f consists of a finite number of two-dimensional tori. Depict each torus as a point. Varying the value a we make these points move. As a result, they sweep the edges of some graph. If the value a has become critical, the edges can merge or, on the contrary, split into several edges, or run into a vertex. We therefore obtain a certain graph r(Q, H, f) which we call the diagram of the simple integral f on Q (Fig. 3.3.1). Clearly, this graph shows not only the topological structure of the manifold Q, but also the main details of the behaviour of the integral on Q since the merging and splitting of edges and other singular points (vertices) of the graph correspond to distinct critical manifolds of the function f on Q.

3.3 Three-Dimensional Manifolds and Visual Geometry

233

The theorems formulated in Sect. 3.2 allow us to say a good deal about the geometry of the graph r. The point is that we have in fact described the structure of all critical submanifolds of the simple Bott integral f and, therefore, we already know all the forms of the splitting and merging of the edges of the graph and all the types of its vertices.

Theorem 1. (see Refs. [54], [55]). Let Q3 be a compact non-singular isoenergy surface of an integrable system with a simple Bott integral f. Then all possible types of vertices of the graph r(Q, H, f) and all possible types of interaction of its edges are listed in Fig. 3.3.2. In particular, the vertices of the graph are divided into five types. All the above-mentioned types of vertices and interactions of edges are actually realized in real mechanical and physical integrable systems. Vertices of types 1, 2, 5 depict critical manifolds of the integral f on Q, which are minima or maxima of the integral. Such vertices and the corresponding submanifolds are called minimax. Vertices of types 3 and 4 depict the saddle critical submanifolds of the integral. We shall accordingly call them saddle vertices. Furthermore, one can say exactly which particular critical manifolds are depicted by these or those vertices of the graph.

1) The black-minimax vertices of the graph are minimax circles of the integral. 2) The white minimax vertices of the graph are minimax twodimensional tori. 3) The tripods (see type 3) are critical saddle circles of the integral with the property that when passing through them a Liouville torus splits into two tori (or, inversely, two tori merge into one as the direction of motion reverses). 4) The star with one incoming and one outgoing edges of the graph is the critical saddle circle of the integral in passing through which a Liouville torus again transforms into one torus. This transformation consists in two-fold winding round the torus. 5) The white minimax vertices of the graph with a point inside them are minimax Klein bottles.

r

Structure of the graph in the case of a simple nondegenerate system

Five types of "atoms" = vertices of graph r

Minimax and saddle atoms represent different types of transformations of Liouville tori

"Atoms" are represented as a 3-manifolds with boundary tori Atom A =minimax atom B = saddle = tripod atom A' = "star"

234

i

1

.:. 7t7

~'"

:.::,/;:.::. ..:.::* ........

'::. ':;"

3. Visual Symplectic Topology and Visual Hamiltonian Mechanics

Now we are in a position to demonstrate more clearly the processes where a torus "strikes" the critical energy level (these processes are described in Sect. 3.2). See Fig. 3.2.2. We can see that in cases I, 2, 5 the Liouville torus does not break into the critical level of the integral f for the simple reason that this level is either maximal or minimal. "Above" the maximum and "below" the minimum there are no points of the manifold Q at all, and therefore the torus has nowhere to move. As a result, in case I the. torus becomes a minimax circle and stops. In case 5 the torus transforms into a Klein bottle and stops. In case 2 two

Fig. 3.3.2

r

Graph is called molecule (constructed from atoms) /R.1

tori merge becoming a minimax torus and stop, or else we may assume the torus to rise from the depth of the manifold, to strike the minimax level of the integral and, reflected, to go again inside the manifold. It is only in cases 4 and 5 that the critical levels of the integral are inside the manifold Q. When passing through the type 3 level, the torus splits into two (or tori merge into one) and continues moving in the same direction. In case 4 the torus, on transforming, also continues moving in the same direction. In Fig. 3.3.3 we have schematically shown the graph r(Q, H, f) of the general form (for a simple integral) where all possible types of vertices are represented. From the viewpoint of the problems in question we may think that all the minima of the function f are at one level (e.g. f = 0). Similarly, all the maxima of the function f may be thought of as lying at one level, for instance, f = I (Fig. 3.3.3).

3.3.2 What Familiar Manifolds are Encountered Among Isoenergy Surfaces? In a three-dimensional space we consider a closed orientable 2-surface

M;, i.e. a sphere with ghandles. Suppose a heavy material point is

Fig. 3.3.3

moving along this sphere. There arises a mechanical system whose con-

3.3 Three-Dimensional Manifolds and Visual Geometry

235

figuration space (i.e. position space) is the surface M2. The phase space is a 4-manifold T*M 2• Its points are paris (x, ~), where x is a point of the surface and ~is a tangent vector to the surface at the point x (Fig. 3.3.4). An important example of a Hamiltonian system on the space T*M is a geodesic flow of some metric. To determine this system, a Riemannian metric 9ij should be introduced on the surface M. Then we can calculate the distance between any two points on the surface. On the surface there arise geodesic lines, i.e. lines minimizing the distance between any two of its sufficiently close points [63]. For example, such shortest (i.e. geodesic) lines on the Euclidean plane are straight lines. On a sphere embedded standardly in ]R3 (i.e. endowed with a standard invariant Riemannnian metric) the geodesics are the various equators, i.e. flat cross-sections of the sphere by planes passing throu~h the centre (Fig. 3.3.5). A geOdesic flow is a vector field on the space T*M defmed thus. Consider an arbitrary point (x,O on T*M. It defines uniquely the point x on M and the tangent vector ~ to M at this point. It is known that from any point of a manifold one can always send a single geodesic in an arbitrary direction. Consider a point sliding along the geodesic I (as a function of varying time t) with the velocity vector 'Y(t). We obtain the curve aCt) = 0, where c is sufficiently small. Since H(xo) = 0 and the point Xo is an isolated

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minimum, the equation H(x) = c describes a three-dimensional sphere Remark: isoenergy 3-surface 8 3 centred at the point Xo. Thus, the standard three-dimensional sphere of any geodesic flow on 8 3 belongs to the class (H). a 2-torus (not necessary Finally, mechanics gives us examples of integrable Hamiltonian sys- integrable) is diffeomorphic tems on n1anifolds M 4, whose Hamiltonian reaches either maximum or to a 3-torus minimum on the circle 8 1• Such are, for example, some systems describing the dynamics of a three-dimensional heavy rigid body (fixed at one point). As in the preceding example, we again consider the level surface of a Hamiltonian H, close to the minimax circle. Clearly, it is homeomorphic to the direct product of the circle by the sphere 8 2• The point is that at each point x of the critical circle, the disc normal to the circle is a 3-manifold. Consequently, the equation H = const determines the 2-sphere in this 3-manifold (= disc). Transporting the point along the 3-sphere as an "integrable" circle, we obtain the required direct product. Thus, the manifold 8 1 x 8 2 isoenergy 3-surface also belongs to the class (H). Let us sum up the results.

Proposition 1. To the class (H) oJisoenergy surfaces oJintegrable nondegenerate (BotO Hamiltonian systems there belong, in particular, the Jollowing manifolds: 83, lRp3, T3, 8 1 X 82. What unites these manifolds from the topological point of view? It is now difficult to answer this question because it is unclear in what terms it should be formulated. It turns out (and this will now be explained) that all these manifolds are represented in the form of glueing simple elementary "bricks". So, there arise some elementary manifolds which may subsequently be glued in an arbitrary order to form new 3-manifolds. All the 3-manifolds obtained by glueing these bricks appear to belong to the class (H) and, moreover, the class (H) contains no other manifolds (this is a nontrivial theorem, see Refs. [54], [55], [64]). We proceed to the study of the elementary bricks employed for construction of isoenergy surfaces of integrable systems.

Proposition 2. A1l3-manifolds enumerated in Proposition 1 are obtained by glueing two solid tori (through some diffeomorphisms oj their boundaries which are tori).

51

X

52 as an "integrable"

isoenergy 3-surface

Theorem: class (H) does not contain hyperbolic 3-manifolds, thus all hyperbolic manifolds lie outside of (H)

Class (H) coincides with a class of all 3-manifolds obtained by glueing of a solid tori and some number of copies of 51 x N2, where N2 is a disc with 2 holes

242

3. Visual Symplectic Topology and Visual Hamiltonian Mechanics

Proof. Consider two solid tori PI, P2 and let TI and T2 be their boundaries homeomorphic to a torus. Let I: TI ---t T2 be an arbitrary torusto-torus diffeomorphism. On each of these tori we determine standard parallels and meridians denoting them respectively by 0:1, (31 and 0:2, (32 (Fig. 3.3.10). The diffeomorphism maps the curves 0:1 and (31 into some smooth curves /(0:1) and /((31) on the torus T2. In the preceding chapter we got acquainted with deformation of closed curves on a torus. Recall that by deforming continuously the curves /(0:1) and /((31) we can make them become compositions of the basis curves 0:2 and (32. For example, the curve /(0:1) is always homotopic to a curve of the form 0:~(3~, i.e. one should first go b times round the meridian f3z and then a times round the parallel 0:2. Similarly, the curve 1((31) can be deformed into a curve of Fig.3.3.10

Group of unimodular integer-valued matrices is isomorphic to the group of al/ homotopic classes of diffeomorphisms for the torus

the form

o:~(3r There arises a square integer-valued 1* = (~

we consider only those diffeomorphisms which preserve torus orientation (i.e. those fixed beforehand), this matrix is unimodular, i.e. thas a unit determinant, ad - be = 1. So, each diffeomorphism 1 of tori is assigned a matrix 1* ..Inversely, with respect to each unimodular integer-valued matrix A one can construct such a diffeomorphism 1oftori that 1* coincides with the matrix A. Glueing two solid tori via the diffeomorphism 1 of their boundary tori, we obtain a closed compact 3-manifold. Since a diffeomorphism is determined by a matrix, it follows that each unimodular integer-valued matrix A is assigned a certain 3-manifold M = M(A). As an example we consider the matrices ( _

3-manifolds: 53, lR'p3, r 3, 51 x 52 are obtained by glueing of two solid tori

:). If

~ ~)

and

(~ ~) .

Glueing them by means of a solid torus, one can readily make sure that this gives respectively the sphere 8 3 and the manifold 8 1 x 8 2• We leave it to the reader as an exercise to construct matrices generating lRp3 and T3. Thus, the above-mentioned isoenergy surfaces are obtained by glueing two copies of one elementary brick - a solid torus (via a diffeomorphism of the boundaries).

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3.3.3 The Simplest Isoenergy Surfaces (with Boundary)

Consider the following five three-dimensional manifolds with boundary. Type I A solid torus, i.e. the direct product 8 1 x D2. Its boundary is one torns (Fig. 3.3.11). Typell The direct product T2 x DI will be called a cylinder. Its boundary consists of two tori (Fig. 3.3.11). A cylinder can be realized as a region in 1.3 (Fig. 3.3.11). To this end one should drill from a solid torns a thin solid torns concentric to the initial one. Type III The direct product A3 = N 2 X 8 1 will be called "trousers" or an oriented saddle, where N 2 is a disc with two holes (Fig. 3.3.11). To realize N. 2 x 8 1 in 1.3, one should drill from a solid torns two thin solid tori. The boundary N 2 x 8 1 consists of three tori. Type IV. Consider a solid torns in 1.3 and drill from it a thin solid torns going twice round the axis of the initial one (Fig. 3.3.11). This gives a 3-manifold whose boundary consists of two tori. We call it a non-oriented saddle (non-oriented trousers). Type V. Take a Klein bottle immersed in 1.3 (Fig. 3.3.11) and consider its tubular neighbourhood K3. As we have already found out, the boundary of this neighbourhood isa torns immersed in 1.3. The tubular neighbourhood is formed by small segments orthogonal to the Klein bottle and having the centre at its points. This gives the 3-manifold K3 immersed in 1.3. To remove its self-intersections, it suffices to pass over from 1.3 to 1.4 . Owing to the appearance of another dimension, one can remove by slight stirring all self-intersections of the Klein bottle (recall that the set of its points of self-intersection forms a circle) and, therefore, remove self-intersections of the 3-manifold K3. So, K3 can be smoothly embedded in 1.4 . We call K3 a non-orientable cylinder. Lemma 3. (see Refs. [54], [55]). Only two of those five manifolds are topologically independent. Manifolds IL IV and V are glueings of manifolds I and III Namely, it can be formally written that IV = I + I I I andY = I + IV, i.e. V = 2· I + III. This means that A3 is obtained by glueing a solid torus and trousers, while K3 is obtained by glueing two solid tori and trousers. Finally, I I = I + I I I.

Fig.3.3.11

244

Fig. 3.3.12

Topological classification of isoenergy 3-surfaces of integrable non-degenerate systems

Liouville foliation of a symplectic manifold M4 does not depend on the choice of a second integral f

3. Visual Symplectic Topology and Visual Hamiltonian Mechanics

The ''plus'' sign indicates here the glueing of elementary bricks via some diffeomorphisms of their boundary tori. We shall prove, for example, the ftrst relation. Consider manifold IV realized in 1.3• Consider the centre of a 2-disc (with two holes) and a circle h which passes through this centre and is the axis of the large solid torus (form which we have drilled a thin solid torus going twice along the axis). From the large solid torus we discard a small tubular neighbourhood of the circle h, i.e. drill a thin solid torus concentric to the large solid torus. We ftbre the remaining part R into circles (the dashed lines in Fig. 3.3.12) which go twice along the axis of the large solid torus. On the disc we draw two radii and label them by the letter a with arrows (Fig. 3.3.12). It is readily seen that R becomes the direct product of the disc with two holes by the circle, i.e. is trousers. Similar arguments should be applied to manifold V. We omit the proof and leave it to the reader. It can be shown that all the five manifolds enumerated above are realized as pieces of some isoenergy surfaces of classical mechanical integrable systems. Therefore we have called them isoenergy surfaces with boundary.

3.3.4 Any Isoenergy Surface of an Integrable Nondegenerate System Falls into the Sum of Five (or Two) Types of Elementary Bricks Theorem 3. (Fomenko [54], [55]). Let M4 be a smooth symplectic (phase, compact or noncompacV manifold and let v be a Hamiltonian system with Hamiltonian H, Liouville-integrable on one non-singular compact three-dimensional isoenergy surface Q, by means of a Bott integral f. Then the critical submanifolds of the integral f on Q can be only as follows: 1) minimax circles, 2) minimax two-dimensional tori, 3) saddle circles with orientable separatrix diagram, 4) saddle circles with non-orientable separatrix diagram, 5) minimal two-dimensional Klein bottles. Let m,p, q, s, r be respectively the numbers of these critical manifolds. Then the manifold is represented in the form of glueing (via some difJeomorphisms of boundary tori) of the following elementary bricks:

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245

Q = mI +pII +qIII +sIV +rV; the elementary 3-manifolds I, II, III, IV and V are described above.

This theorem provides a visual topological classification of isoenergy surfaces of integrable Bott systems. The numbers m,p, q, s, r have a clear interpretation - they point out how many critical manifolds of each type the integral f has on Q. If we forget for some time about a concrete form of the integral and conern ourselves with the simplest topological representation of the isoenery surface, the answer will be given by the following theorem.

Theorem 4. (see [54], [55]). Let Q be a compact nonsingular isoenergy surface of a Hamiltonian system integrable by means of a Bott integral f. Then Q can be represented in the form Q = m' I +q' II I, where m' and q' are some non-negative integers related to the numbers from Theorem 2 by the formulas: m' = m + s + 2r + p, q' = q + s + r + p.

Theorem: any compact orientable smooth 3-manifold is an isoenergy surface for some Hamiltonian system (not necessary integrable)

So, each isoenergy surface of an integrable system admits two decompositions: Hamiltonian (into the sum of five types of bricks) and topological (into the sum of two types of bricks). The first decomposition is a more "detailed" one, it "remembers" the structure and the number of critical manifolds of the integral f. The topological decomposition is more rough, though it is simpler (as far as notation is concerned). It ignores minute properties of the integral. Clearly, the Hamiltonian decomposition is uniquely restored from th~ diagram r(Q, H, f) of the simple integral f. Consider the various closed compact 3-manifolds obtained by glueings of elementary bricks of types I and III. Denote this class of 3manifolds by (Q). The theorems formulated above show that (H) C (Q). The inverse inclusion is valid too. It turns out (Brailov and Fomenko [64]) that the classes (H) and (Q) coincide, i.e. any 3-manifold obtained by glueing solid tori and trousers can be realized as a compact isoenergy surface of an integrable (by means of a Bott integral) Hamiltonian system on an appropriate symplectic manifold M4 (may be noncompact).

Hamiltonian decomposition and topological decomposition of isoenergy surface

Classes (H) and (a) coincide Connected some of two "integrable" isoenergy 3-surfaces is again an isoenergy 3-surface for some integrable system

246

3. Visual Symplectic Topology and Visual Hamiltonian Mechanics

3.3.5 New Topological Properties of the Isoenergy Surfaces Class Recall that a manifold M3 is called topologically reducible if it is representable as a connected sum M = M, UM2, where M, and M2 are not nomeomorphic to the standard sphere 8 3. Otherwise the manifold is called irreducible. The connected sum is defined as follows. From each of the manifolds M, and M2 we discard a ball. This gives manifolds with 2-sphere as boundary. Glueing these two spheres via an identity diffeomorphism, we obtain a new manifold denoted by M, UM2 (Fig. 3.3.13).

Fig.3.3.13 If v and v' are two integrable systems on isoenergy surfaces Q and QI, then we can always construct a new integrable system v Uv' on the connected sum Q UQI

New topological obstacles to integrability of a Hamiltonian dynamical system

Class (H) of all "integrable" isoenergy 3-surfaces is less than class (M) of all compact orientable 3-manifolds

Theorem 5. (A.T. Fomenko, H. Zjeschang). Let Q, and Q2 be isoenergy surfaces from the class (H) = (Q). Then their connected sum Q, UQ2 again belongs to this class. Inversely, if the isoenergy surface Qfrom the class (H) = (Q) is represented as the connected sum Q, UQ2 of some manifolds Q, and Q2, then each of them belongs necessarily to the same class (H) = (Q). It is not only the manifolds themselves but also the integrals on them that decompose here into a connected sum. Thus, the class of isoenergy surfaces of integrable Bott systems is closed with respect to the operation of taking a connected s~. Now we are ready to answer the question whether there exist purely topological obstacles for integrability of a Hamiltonian system on an individual isoenergy surface. Since we are dealing with smooth systems it is not excluded that the system may be integrable on one isoenergy surface and nonintegrableon another. It would be desirable to have invariants forbidding (or allowing) the existence of a Bott integral only on a single isolated isoenergy surface irrespective of the behaviour of the system on other isoenergy surfaces (including the closest ones). It turns out that such topological obstacles do exist. We shall not go into detail here and restrict ourselves to visual geometrical interpretation. For details we refer the reader to Refs. [54], [55].

Theorem 6. (see Refs. [54], [55]). The class of isoenergy surfaces of integrable (by means of a Bott integralj systems does not coincide with the class of all three-dimensional compact closed orientable manifolds.

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247

Corollary. Infinite subclass of3-manifolds cannot be isoenergy surfaces ofBott integrable systems, i.e. there exist topological obstacles forbidding integrability of a Hamiltonian system if a given isoenergy surface does not belong to the class (H) = (Q). To solve efficiently the question of whether the Hamiltonian system is integrable on a given isoenergy surface, one should be able to calculate efficiently the topological obstacles briefly described above. To this end it is useful to know as much as possible about the topology of the class of integrable isoenergy surfaces. We shall describe some of their interesting properties. Suppose a smooth 2-surface M is homeomorphic to a sphere and endowed with a smooth Riemannian metric ofgeneral position, i.e. on this surface there is not a single stable closed geodesic. Then the geodesic flow corresponding to this metric is non integrable on each individual nonsingular isoenergy 3-surface in the class of Bott integrals [54], [55]. We shall discuss the concept of stability of a periodic solution of a system v on an isoenergy 3-surface. The periodic solution is depicted as a closed integral trajectory 1 (Fig. 3.3.14). We consider a tubular neighbourhood of the trajectory, i.e. a solid torus whose boundary is a 2-torus with the axis I' We say that the trajectory 1 is stable if the solid torus fibres wholely (without "gaps") into concentric 2-tori with the circle 1 as their common axis, all these tori being invariant with respect to the system v. This means that all integral trajectories of the system which are close to 1 lie on the indicated 2-tori. The property of the system to have stable periodic solutions appears

to be closely connected with its integrability. There holds the following assertion: it on an isoenergy surface the system has not more than one stable periodic solution and moreover if the one-dimensional integervalued homology group HI (Q, Z) is a finite cyclic group (i.e. has the form Zq), then the system is non integrable on this isoenergy surface (it cannot have a smooth Bott integral). Roughly speaking, if on the isoenergy surface there are many nonzero cycles offinite order and few stable periodic solutions, the system is necessarily nonintegrable. See also [65]. So, the very fact of the existence of a Bott integral f in the system imposes strong restrictions upon the topology of the isoenergy surfaces. It

If a Hamiltonian system v has an isoenergy surface which does not belong to the class (H). then v is nonintegrable; for example: any Hamiltonian system is non-integrable on hyperbolic closed 3-manifold

Fig. 3.3.14

For any analytic integrable system (M4, H, f) always there exists a small smooth perturbation such that a new integrable system ( M4 , A, f) is Bott system on a given Q3 (V. V. Kalashnikov, jn.)

248 Set of Bott systems on a fixed isoenergy 3-surface is of the first category in the set of all smooth integrable systems; consequently form a thin set in weak topology

Bottsystems are dense in the set of all integrable smooth systems which have critical circles only and exactly one such circle on each critical level for H

Four faces of the remarkable class (H) of "integrable" isoenergy 3-surfaces: (HJo=(QJo=(SJo=(RJo and (H)=(Q)=(S) If "integrable" isoenergy surface is a connected sum of some manifolds K and N, then both K and N are "integrable"

3. Visual Symplectic Topology and Visual Hamiltonian Mechanics

is natural to ask whether these restrictions remain if the requirements on the integral are weakened. For instance, we require from the function f "less" than in the preceding cases. Consider the class (8) of 3-manifolds with the property that on them there exists a smooth function 9 in which all the critical points are organized into nondegenerate critical circles and all the nonsingular level surfaces are unions of 2-tori. In other words, the manifold belongs to (8) if and only if on this manifold there exists a Bott function in which all the critical manifolds are circles and all the nonsingular level surfaces are tori. Given this, the function does not already have necessarily to be an integral of some integrable system. Nevertheless, the class (8) appears to coincide with the class (Q) (S.V. Matveev) [66]. Thus, if on the 3-manifold M there exists a Bott function whose critical manifolds are circles and the non-critical level surfaces consist of tori, then on this manifold there necessarily exists an integral f of some integrable Hamiltonian system and the manifold is realized as its isoenergy surface. Thus, (H) = (Q) = (8). In some situations the requirements on the function f can be weakened still more. Consider the class (R) of 3-manifolds on which there exists a smooth Bott function h all of whose critical manifolds are circles. As distinct from manifolds of the class (8), it is not required here that the nonsingular level surfaces of the function be tori. In the literature such functions are called round Morse functions (Thurston, Azimov and others) [67]. See also the paper by Matveev, Fomenko and Sharko [68]. Clearly, (R) :J (H), i.e. any isoenergy surface of an integrable system admits a round Morse function. It can be proved that the class (R) is strictly larger than the class (H), i.e. there exist manifolds which admit a round Morse function but are not realized as isoenergy surfaces. The distinction between the classes (R) and (H) is due to their different behaviour under the operation of taking the connected sum of manifolds. According to Ref. [67], if M is any 3-manifold (we do not specify each time that we work in the class of compact connected closed orientable manifolds), then while considering its connected sum with a sufficient number of copies of the manifold 8 1 X 8 2 we finally necessarily get to the class (R). In other words, if m is sufficiently large, we have M ~ (~f,!1 8 1 X8 2) E (R) . As has already been mentioned, the class (H) possesses the remarkable property that if M = MI ~ M2 E (H) then the manifolds MI and M2 belong necessarily to (H). In terms of the

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249

class (8) this property has beeen proved by S. V. Matveev. From this it immediately follows that the class (R) is strictly larger than the class (H). Indeed, it suffices to take the manifold Mo which does not belong to the class (H) (such manifolds are known to exist). Taking its connected sum with a sufficiently large number of copies of the manifold 8 1 x 8 2, we get to the class (R). The manifold M obtained does not however lie in (H) since otherwise the initial manifold Mo would belong to the class (H), which contradicts the choice of Mo. Thus, ME (R), but does not belong to (H). The picture changes radically if we restrict our consideration to the class of irreducible 3-manifolds. It turns out that inside this class there holds the remarkable identity: (H)o = (Q)o = (8)0 = (R)o, where "0" indicates the set of irreducible manifolds. This means that given an irreducible 3-manifold with a round Morse function h on it, it is stated that on the same manifold there necessarily exists also a Bott integral of an appropriate integrable Hamiltonian system, and the manifold will be an hmenergy surface of this system. The topological theory of integrable systems, which we have briefly described here, was further developed by the author, in particular, in Refs. [60], [61], [62]. For example, the author discovered the topological invariant classifying the integrable systems up to rough topological equivalence. The fme classification (up to fine topological equivalence) was completed by the author in collaboration with H. Zieschang, S.V. Matveev and A.V. Bolsinov [69], [70]. It turns out that many interesting aspects of three-dimensional topology [71], [72] are closely connected with Hamiltonian mechanics [73].

3.3.6 One Example of it Computer Use in Symplectic Topology

The orbital (continuous) classification of integrable systems with two degrees of freedom was obtained by Bolsinov and Fomenko in 1993 (see Matem. Sbornik, in print)

Bolsinov and Fomenko in 1993 obtained the exact (continuous) classification of a smooth Hamiltonian systems on 2-surfaces (1 degree of freedom)

We have got acquainted with the remarkable class (H) of isoenergy 3manifolds of integrable Bott Hamiltonian systems. According to Chap. 2 (see Sect. 2.5), 3-manifolds can be ordered in the increasing order of their complexity, and there exists only a finite number of irreducible manifolds of a given complexity. This fact permits computer use for studying the Computers in 3-topology topology of "sufficiently simple" 3-manifolds. Developing further the and in symplectic geometry

250

3. Visual Symplectic Topology and Visual Hamiltonian Mechanics

investigations carried out by the author [54], [55], S.V. Matveev obtained the following result. Any closed orientable 3-manifold of complexity :::; 8 is an "integrable" isoenergy 3-surface

Theorem. All closed 3-manifolds of complexity not exceeding 8 belong to the class (H), i. e. are isoenergy surfaces of integrable nondegenerate (Bott) Hamiltonian systems. This fact has been proved using computer and is based on the results of Chap. 2 and on the classification theorem for isoenergy surfaces of integrable systems, proved in Refs. [54]; [55]. At first Matveev proved this theorem for the manifolds of complexity ~ 5. Let us demonstrate the rough idea of this proof. We shall analyze the belonging of all closed manifolds of complexity ~ 5 and almost all manifolds of complexity 6 to the class (H).

Toric 3-manifolds

Complexity of a 3-manifold is the number of vertices in its minimal almost special spine

Filtration of 3-manifolds with respect of complexity

I) A 3-manifold M will be called toric if its boundary consists of several tori or is empty. Let M be a toric 3-manifold of complexity k, i.e. let M have an almost special spine with k vertices. Using simplification technique (see Ref.[34]) one can prove that there exist toric manifolds Mi such that: 1. Mi has a special spine with ~ k vertices; 2. M is obtained from Mi by means of the followings operations: glueing the trousers N 2 X 8 1, taking sums connected with one another and with D2 x 8 1, 82 X 8 1 or RP3. Making these operations, we do not overstep the limits of the class (H). Therefore, if all toric manifolds with special spines with ~ k vertices lie in (H), so do all toric manifolds of complexity ~ k. II) All toric maI\ifolds having special spines with zero or one vertex lie in (H). There exist three such manifolds. From Ref. [46] it follows that all toric manifolds of complexity ~ 1 lie in (H). Note· that there exists a toric manifold of complexity 2 not lying in (H) - this is the complement of figure-of-eight. III) Let P be a special spine of a closed 3-manifold M and let 0: be one of its 2-components. Puncture it, i.e. discard an open 2-disc from it. After a collapse we obtain an almost special spine PI of a toric manifold MI, from which M is obtained by glueing a solid torus. The number of vertices kl of the spine PI is less than the number of vertices k of the

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251

spine P since in the cource of collapse the vertices on the boundary of the component a and, perhaps, some other vertices vanish. For example, if a adjoins twice a certain edge, there vanishes another 2-component adjoining the same edge. We call the 2-component a large if k ::; l. IV) Looking through all the special spines with::; 5 vertices, yielded by computer, shows that all of them have large 2-components. Not all of the special spines with 6 vertices have been counted. But those that were, also had large 2-components. The presence of a large 2-component guarantees the belonging of the manifold to the class (H), since it is obtained by glueing a solid torus to a manifold from (H). Let us summarize the results of this computer experiment. There exists a natural filtration of 3-manifolds by complexity, i.e. by the number of vertices in the minimal almost special spine [46]. For each k there exists only a fmite number of irreducible 3-manifold of complexity ::; k. The analysis of the results of enumeration of 3manifolds of complexity ::; 5 carried out, using computer, by Matveev & Savvateev [46] and then by Matveev for complexity ::; 10 shows that all closed orientable 3-manifolds of complexity ::; 8 lie in the class (H). Closed orientable 3-manifolds which do not belong to the class (H) appearfor the first time in complexity = 9. We shall denote them by Q, and Q2. The first of them appeared to be known to Thurston [74], [75], and the second was discovered independently by J. Weeks (76] and by S.V. Matveev & A.T. Fomenko [48]. Both these remarkable manifolds proved to be hyperbolic (i.e. manifolds of constant negative curvature), the second (the one discovered by Weeks, Matveev and Fomenko) having the smallest volume of all closed hyperbolic 3-manifolds known today. Matveev and Fomenko hypothesized (and grounded their hypothesis) that this manifold Q2 is actually a manifold of the smallest hyperbolic volume in the class of all closed hyperbolic 3-manifolds. For the details of this investigation see S.V. Matveev and A.T. Forrienko "Constant Energy Surfaces of Hamiltonian Systems, Enumeration of Three-Dimensional Manifolds in the Increasing Order of their Complexity and Calculation of the Volumes of Closed Hyperbolic Manifolds" published in Russian Math. Surveys 1988, Vol.43, issue 1, pp.3-24.

Thurston'S conjecture: the volume of hyperbolic 3-manifold 01 is minimal in the class of all hyperbolic closed 3-manifolds; V91 01 ~ 0.98 ... but it turned out that the volume of O:! is less! vol O:! ~ 0.94 ...

New conjecture: O:! has the smallest volume in class of all closed compact hyperbolic 3-manifolds Two remarkable hyperbolic 3-manifolds 01 and O:!

252 New theorem: geodesic flows of two ellipsoids in]R3 are orbitally (continuous) equivalent if and only if their semiaxes are proportional (Bolsinov, Fomenko)

3. Visual Symplectic Topology and Visual Hamiltonian Mechanics

Another remarkable property of these two manifolds is that if a Hamiltonian system has an isoenergy 3-surface homeomorphic either to Q1 or to Q2, this system is not integrable on this 3-surface (in the class of Bott integrals). This reveals new topological obstructions for integrability, that lie in the topology of the manifolds Q. These obstructions are essentially different from all familiar topological obstructions for integrability of Hamiltonian systems. It turns out that the following striking result is true (see [48]): If the 3-manifold Q3 is hyperbolic, then any Hamiltonian system is non-integrable (in the class of Bott integrals) on Q3. In other word, the class (H) of isoenergy 3-surfaces of integrable Hamiltonian systems does not contain hyperbolic manifolds. 3.3.7 Visual Material

One ofthe elements in Fig. 3.3.15 presents an oriented separatrix diagram of the critical circle of a Bott integral. It is formed by segments of the integral trajectories stuck to the circle. Two versions are possible: the diagram is homeomorphic either to a ring cut along its axis by a critical circle (Fig. 3.3.15) or to a Mobius strip. These two rings (strips) move apart and along (up and down) the manifold. Figure 3.3.16 is basically aimed at providing psychological perception of geometrical infinity.

Page 253, Fig. 3.3.15. Separatrix diagram of the critical manifolds for dynamical systems. German and Scandinavian legends; death of Alberich and punishment of the gods Page 254, Fig.3.3.16. Mathematical infinity and singular points. Meditation and dreams of ancient priests

4. Visual Images in Some Other Fields of Geometry and in Its Applications

4.1 Visual Geometry of Soap Films. Minimal Surfaces

Soap films

4.1.1 Boundaries Between Physical Media. Minimal Surfaces

Minimal surfaces

Suppose in 1.3 there are several adjoining but not mixing physical media, for instance, in a large vessel there are several immiscible fluids. Suppose, the whole system is in equilibrium. Since the media are immiscible, the boundaries (interfaces) between them are determined. These interfaces can be tought of (in the first approximation) as two-dimensional piecewise smooth surfaces separating the adjoining media. We consider for simplicity the case of two media which we denote by Al and A2• Let the pressures in the media be respectively equal to PI and P2. The equilibrium condition for the media proves to impose a strong restriction upon the geometry of their interface. To formulate this restriction, we require an important concept of local differential geometry, namely, the concept of mean curvature of the surface. Let P be a point on a smooth two-dimensional surface M2 in 1.3. At the point P consider a unit normal n, i.e. the vector of unit length orthogonal to the tangent plane to the surface at the point P (Fig. 4.1.1). Draw a plane II through this normal. It crosses the surface along some smooth curve 'Y. We are here interested only in a small neighbourhood of the point P. So, we obtain a flat curve lying in the plane II. It is a known fact that at each point of a curve its curvature can be calculated. Geometrically, the curvature is expressed as the inverse quantity to the radius of the so-called adjoining circle, i.e. the circle tangent to the curve

Interfaces between phYSical media

Fig.4.1.1

256

Function of curvature

Principal curvatures >'1 and >'2 are equal to the maximal and minimal values for the curvature of a normal cross-section

Mean curvature = >'1 + >'2, Gauss curvature =>'1 . >'2

Laplace-Poisson theorem

4. Visual Images in Some Other Fields of Geometry

at the given point and approximating the curve to within small quantities of second order (second-order tangency). The curvature can also be interpreted from the mechanical point of view. Along a curve line we send a material point of unit mass with a velocity constant in the absolute value. The direction of the veloctiy vector will, of course, change (it is directed along the tangent of the curve), but we assume its length to be constant. We thus obtain a mechanical system whose state is determined by the shape of the curve. When moving along the curve, the material point has the change the direction of motion, due to which it accelerates. This acceleration is depicted as the vector orthogonal to the curve (more precisely, to the tangent to the curve at the given point). The magnitude of the acceleration is just the curvature of the curve (at a given point). The curvature (acceleration) may change from point to point. We obtain a smooth function called the function of curvature of a flat curve. The curvature of a spatial curve is defined in a similar way. The curvature of a straight line is identically zero: a material point moves along a straight line uniformly and rectilinearly without acceleration. The curvature of a circle of radius R is equal to 1/ R. The acceleration is here constant in the absolute value. Let us return to the surface M. To each position of the plane II (passing through the normal to the surface) there corresponds its own intersection curve·, and, therefore, its own value of the curvature k(,) of this curve at the point P. Rotating the plane II (around the normal), we change the value k(,). Consider the maximal value Al of this curvature and its minimal value A2. We obtain two numbers. They may, in particular, coincide. From the definition we see that the curvature k{[) is constant in this case, i.e. independent of the angle of rotation of the plane. Consider the general case. The numbers Al and A2 are called the principal curvatures of the surface at a given point. These numbers change from point to point, i.e. Al and A2 are functions of the points. The sum of the principal curvatures, i.e. the expression Al + A2 is called the mean curvature H(P) of the surface at a given point.

The Laplace-Poisson theorem. Let M be a two-dimensional boundary between two media in ]R3, which are in equilibrium. Let PI and P2 be pressures in the media. Then at each regular point P of the surface its

4.1 Visual Geometry of Soap Films. Minimal Surfaces

257

mean curvature H(P) is equal to ",(PI - P2), where", is a constant independent of the point, and PI - P2 is the pressure difference in adjoining media. In particular, the mean curvature of the interface is. constant (at all regular points).

The constant", has an important physical meaning. The number 1/ '" is called the surface tension coefficient (on the interface); it characterizes the properties of the media. Thus, the mean curvature of the interface is constant. Consider two cases:

Surface tension coefficient

1) the mean curvature is equal to zero (this takes place if and only if the pressures in the neighbouring media are equal); 2) the mean curvature is nonzero; in this case it is strictly positive (sign reversal can be achieved by reversing the normal to the· surface).

Zero mean CUNature +-+ minimal surfaces Nonzero mean CUNature +-+ soap bubbles

Both cases are modelled well in the physical experiment familiar to every reder. This is the experiment with soap films. We take an aqueous soap solution, plunge a closed wire contour into it and take it off. Then on the contour there forms an iridescent soap film. It can be interpreted as a boundary between two adjoining gas media. Indeed, on each side of the film there is the air with equal pressures PI and P2 since the film is non-closed, and therefore far from the film both gas media adjoin and interpenetrate, diffuse into each other. Thus, we have realized case 1 (see above). Case 2 is realized as follows. Into a soap solution we plunge a thin tube and then blow through it a soap bubble. It will separate from the tube and fall down smoothly, acquiring the shape of a sphere. We have here two media: the interior. volume of the air with the pressure PI, and the exterior volume of the air with the pressure P2. It is clear that PI > P2 and that the system is in equilibrium only due to the fact that the surface tension forces on the soap film compensate the excess of internal pressure as compared with external. Thus, the mean curvature is here strictly positive (and constant). Let us concentrate on the first case, i.e. on soap films with boundary. The mean curvature of such films is zero. This class of surfaces can be shown to admit another, equivalent description. Namely, the mean curvature of a two-dimensional surface is equal to zero if and only if it is locally minimal, i. e. any sufficiently small

Physical experiments Wire contours and soap films

Soap films with boundary = minimal surfaces

258

Locally minimal 2-surfaces are exactly the surfaces of zero mean curvature

Fig.4.1.2

Minimal surfaces have non-positive Gauss curvature

4. Visual Images in Some Other Fields of Geometry

perturbation of the surface (concentrated in a small neighbourhood of an arbitrary poinlj does not diminish its area. On this ground, soap films of the first type (i.e. those possessing zero mean curvature) are called minimal surfaces. Thus, an equilibrium boundary between two physical media with equal pressures is always a minimal surface. Since the theory of minimal surfaces is very closely connected with boundaries between physical media, with optimal shapes of membranes, etc., it is in the centre of constant attention of a great number od modem researches originating from scientific works of the XVII~XVIIIth centuries. For the review of some of these trends see Refs. [33], [29]. These investigations are one of the most important branches of variational calculus [77], [78]. Here we shall only demonstrate some visual properties of minimal surfaces underlying profound mathematical studies. What is the geometrical meaning of equality to zero of the mean curvature? From the definition it immediately follows that at the point where the mean curvature is zero both the principal curvatures are equal in the abs9lute value and opposite in sign since AI + A2 = O. From the definition of the numbers AI and A2 it is seen that locally, near the point P, the surface is organized like a saddle (FigA.1.2). In one direction the surface goes up (relative to the normal). This direction corresponds to the positive principal curvature, for example, AI. In the other direction (it can be proved that it is always orthogonal to the first one) the surface goes down.·This direction corresponds to the negative principal curvature A2. So, near the point of zero mean curvature the surface is organized like a mountain pass, a saddle. Thus, all the points of a minimal surface are saddle points. In a particular case, a saddle may degenerate into a flat surface. Clearly, the mean curvature of an ordinary plane is equal to zero. Here both principal curvatures are zero.

4.1.2 Some Examples of Minimal Surfaces Catenoid, helicoid, Schwarz surface

We shall give some classical examples of minimal surfaces which have been investigated by mathematicians and physicists (from different points of view) beginning with Laplace, Lagrange, Plateau.

4.1 Visual Geometry of Soap Films. Minimal Surfaces

259

1) The catenoid (Fig. 4.1.3). It is formed by rotation around the axis of the curve given by the graph y = a ch x/a, where a = const. This curve is also referred to as a chain line with the shape of a sagging heavy chain (in the vertical gravitational field) fixed at two points. The boundary of the catenoid in Fig.4.1.3 consists of two coaxial circles (generally speaking, of distinct radii). 2) The helicoid (Fig.4.1.4). It is formed by uniform rotation of the straight line I intersecting the vertical z-axis and going uniformly up, so that each point of the straight line I draws a helical line. 3) The Schwarz surface (Fig. 4.1.5). Its boundary contour is formed by four edges of a regular tetrahedron. Then the Schwarz surface is a saddle surface bounded by four rectilinear segments. The Schwarz surface is interesting not only by itself but also for the fact that it permits construction of new minimal surfaces. It is relevant to formulate here the principles met by minimal surfaces. Property 1. If a minimal surface has a free boundary 'Y which is allowed to slide freely along a fixed two-dimensional surface M in ~.3, then the minimal surface intersects the surface M along the curve 'Y at a right angle 90° to the surface. Property 2. Three smooth pieces of minimal surfaces which form together a stable minimal surface and intersect along a common smooth curve will necessarily make with one another equal angles 271"/3 = 120°. Only four such singular edges (along each of which three sheets of the minimal surface intersect) can meet at one isolated singular point. The angles between any two singular edges coming to this point are equal at this point to 109°28'16". Property 3. If some portion of the boundary of the minimal surface M is contained in a certain straight line, then the reflection of the surface M' relative to this line is also a minimal surface, and the union of M and M' forms a smooth minimal surface without a break on the rectilinear region of the boundary. Property 4. If the minimal surface M meets a plane at a right angle, its mirror reflection M' relative to this plane is also a minimal surface and the union of M and M' forms a smooth minimal surface. Applying Properties 3 and 4 to the Schwarz surface, we can construct new, so-called periodic minimal surfaces. This fact is now widely used

-~ tJ)

-+-+-

Fig.4.1.3

Fig.4.1.4

f'V

Fig.4.1.5 Reflection of the minimal surface Periodic minimal surfaces

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4. Visual Images in Some Other Fields of Geometry

in architecture. For example, the famous Olympic Stadium in Munich (built in 1972) was designed using minimal surfaces. Its flexible roof was raised on special rods and was somehow similar to the Schwarz surface, i.e. proved to be optimal from the point of view of area and stability. Enneper surface

Sherk surface

4) The Enneper surface is given in JR.3 by means of the radius vector a(r,8) = (r cos 8 - ~ cos 38, -r sin 8 - ~ sin 38, r2 cos 28). Try to construct is graph. 5) The Sherk surface is given by the equation 1/a In cos ay / (cos ax: =Z, where a =const; or eza =cos ay / cos ax. Construct the graph. 4.1.3 Visual Material

Vortex-type motion

Flow in capillaries Theory of meniscus Drop shape Drop separation

The vortex motion frequently arising at the boundary between two media, when the system is not in equilibrium, is presented in Fig.4.1.6. Shown here are vortices in the atmosphere arising near the earth surface. In Fig.4.1,7 there are objects closely connected with the theory of surface tension of liquid. These are capillaries filled with liquid and drops separating from the moistened surface by gravity. The theory of flow in capillaries, the theory of meniscus and drop shape constitute one of the most interesting fieldS of application of the concepts of differential geometry [79]-[81]. Within the theory of minimal surfaces one can describe not only the process of drop separation, but also the process of transformation of an unstable cylindric liquid column into a set of individual drops, i.e. stringing out on a common horiiontal thread - the axis of the initial liquid .cylinder. Figure 4.1.8 shows the boundaries between media, vortex-type motions in the atmosphere or in the boundary liquid or gas layer.

Page 261, Fig. 4.1.6. Boundary between two media; Laplace-Poisson theorem. J.S. Bach: "Matthew's Passion"; the last talk with Jesus Page 262, Fig. 4.1.7. Capillares and surface tension of fluid Page 263, Fig. 4.1.8. Analytic functions and modelling of a vortex-type motion in the liquid. Egyptian priests and military judges in the court

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4. Visual Images in Some Other Fields of Geometry

4.2 Fractal Geometry and Homeomorphisms 4.2.1 Various Concepts of Dimension

Concept of dimension

There are different concepts of dimensions

For "nice spaces" al/ concepts of dimensions coincide Strange attractors

Fractal geometry

In the preceding chapters we mainly dealt with symplicial complexes, polyhedra and smooth manifolds for which the concept of dimension is intuitively clear and mathematically easily dermed in a natural way because it reduces to the concept of dimension of a Euclidean space. We believe that the concept of dimension ofll~n is known from elementary geometry and linear algebra. The dimension of a manifold is the dimension of Euclidean balls (domains) of which the manifold is glued. The dimension of a polyhedrol!. is the maximum dimension of simplexes entering in its composition. But in mathematics, mechanics and physics some manifolds are encountered (and particularly often in recent years) for which the concept of dimension needs special discussion and, moreover, not one but several distinct dimensions can be defined for them in a natural way. These dimensions may not coincide. It is intuitively clear that we are speaking of sets whose local organization is "substantially worse" than that of open regions in a Euclidean space. Strictly speaking, different concepts of dimension can be defined for an arbitrary topological space. But for "good spaces", to which manifolds and finite polyhedra belong, all these numbers (dimensions) coincide. But as soon as we proceed to the consideration of more complicated exotic (and sometimes in a sense "pathological") objects, different concepts of dimension lead us, generally, to different numbers. It was believed earlier that this occurs mainly for the class of spaces encountered seldom in practice, for instance, in physics, etc. But it has recently become clear and is widely known already that such anomalous (from the point of view of dimension) objects are encountered in classical fields of mathematics connected with concrete physical applications, for example, in the theory of differential equations, dynamical systems (the so-called "strange attractors"), etc. In this connection, the interest, in particular, in the analysis of different concepts of dimension has become lively again. The concepts of "fractal" and "fractal geometry" have been formulated (see the well-known book by Mandelbrot [10]). A whole complex of questions arising in the study of "locally complicated" objects is investigated in the framework of this

4.2 Fractal Geometry and Homeomorphisms

265

scientific direction. We shall only briefly touch upon this rich topic. We start our discussion with the concept of dimension. Recall that the family of sets Ui of a topological space X is called the covering (of the space X) if each point x from X belongs to a certain set Ui. An open covering consists of open sets, while a closed covering consits of closed sets. We now consider only finite coverings. The multiplicity of a covering is the largest of the integers n, such that there exist n elements of the covering (i.e. sets) Ui having a non-empty intersection. A family of sets Vj is called inscribed in the family of sets Uj provided that each element of the family Vj is contained at least in one elements of the family Uj. We shall now formulate the concept of topological dimension which goes back to the works by Brouwer, Lebesgue, Menger, Uryson. For details see e.g. Refs. [10], [82]. For simplicity we consider the class of compacts (compact sets), i.e. such that from among any of their open coverings one can always choose a finite covering.

Definition. The topological dimension dim X of a compact X is the smallest of the integers n, such that in any open covering of the space X one can inscribe a closed covering of multiplicity ~ n + 1.

Covering of a space Open, closed, and finite coverings Multiplicity of a covering

Topological dimension dim X

If there are no such numbers n, the dimension is assumed to be equal to infmity. The visual meaning of this definition is fairly simple. For example, when n = 2 it states that any two-dimensional "ground" (compact) can be tiled with arbitrarily small stones (closed sets), so that the stones adjoin one another not more than by three. At the same time, this ground cannot be tiled by arbitrarily small stones so that they adjoin only by two. Therefore, the statement that, for example, a square has two dimensions is sometimes called "The theorem on tilings" when streets The theorem on tilings or squares are paved with cobblestone, stones always adjoin by three, and one can avoid their adjoining by four. In case a three-dimensional volume is filled with sufficiently small stones (e.g. brick-work in a large cavity) there necessarily arise adjoinings by four. A great achievement of mathematical thought was the discovery of the fact that this number of adjoinings (the multiplicity of coverings) naturally contains the concept of dimension. This contribution is due to

266

4. Visual Images in Some Other Fields of Geometry

Lebesgue (1911). Figure 4.2.1 shows a small covering of multiplicity one of a one-dimensional segment by closed sets. It also shows the covering of multiplicity two of a two-dimensional square. It is characterized by the fact that to each knot (vertex) there comes exactly three one-dimensional edges whose union is the boundary of the elements of covering. Spherical Hausdorff measure I

I

I

,

I I

I

Fig. 4.2.1

Hausdorff dimension

Another approach to the concept of dimension is the idea expressed by Hausdorff (1919) and .later developed by Bezikovich. This idea was, in fact, first expressed by Caratheodory (1914). To formulate this concept of dimension, we need the spherical Hausdorff measure. Let X be a compact subset in a metric space, for instance, in a Riemannian manifold or a Euclidean space. Consider a finite covering of this compact by mdimensional balls Dr(ei) with radii ei. Let "1m be an m-dimensional volume of a standard m-dimensional unit ball (in a Euclidean space ]Rm). The number "1m can be calculated explicitly, but we do not need this formula here, and therefore it is omitted. Then the volume of the ball of radius ei will be written as "Imer. From now on we do not assume the number m to be an integer. Let m ~ 0 be an arbitrary non-negative real number. Calculate the sum L:i 'Ymer. Taking another covering of the compact X with balls, we shall obtain another value of this sum. Consider all possible coverings of the compact X with balls of radii not exceeding a fixed number {! and calculate hI! = inf!i D.

D(X) ~ dim X for any X Definition. The number D is called the Hausdorff dimension of the compact X (or the Hausdorff-Bezikovich dimension).

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267

In the indicated sense the number D = D(X) is the critical dimension. As distinct from the topological dimension dim X, the dimension D(X) need not necessarily be an integer. In particular, these two dimensions may not coincide. They are related by the inequality D ;::: dim, i.e. D(X) ;::: dim X for any X. For "good spaces" these dimensions coincide. At the same time there are examples of complex spaces for which

D>dim. Definition. (see Ref. [10]). The fractal is a set for which D > dim.

- - tt - •• • •

t

Definition of (ractal

In particular, any set for which the dimension is not an integer is a fractal. For details see the book by Mandelbrot [10].

4.2.2 Fractals Consider the so-called Cantor set(called so after George Cantor). Its simplest version is constructed as follows. Consider a unit segment from oto 1 on a real axis. Divide it into three equal parts after which discard the middle third, i.e. the open inverval from 1/3 to 2/3 (Fig.4.2.2). Each of the remaining two thirds also divide into three equal parts, after which discard their middle (open) parts. Repeating this procedure infinitely, we obtain a closed set K on the unit segment. It is intuitively clear (and can be proved exactly) that dim K = O. At the same time, the Hausdorff dimension of the Cantor set is strictly positive and equal to D(K) = log 2/log 3 : : : 0.6309. The construction of the Cantor set can be transformed in such a way (we leave it to the reader to guess in what particular way) that its Hausdorff dimension will be equal to any aforehand given number >. in the interval between 0 and 1. The equality dim K = 0 will hold as before. Thus, the Cantor set is a fractal. An example of the fractal is also the so-called Serpinsky carpet - a closed set on a Euclidean plane. Its construction is pratically similar to that of the Cantor set, but one should start with a flat square, step by step discarding from it increasingly small squares. Fractals naturally appear also in the theory of complex transformations of a plane. On a plane we introduce a complex variable z = x +iy and consider a complex polynomial fez) of degree k. It can be inter-

Fig. 4.2.2

Cantor set Serpinsky carpet

•• ••

268

4. Visual Images in Some Other Fields of Geometry

preted as a complex transformation of the plane, as a map f: C - C given by the formula w = f(z). This map can be iterated by considering the infinite sequence of its powers fn = where fn(z) = f (fn-I(Z)) = f(f( . .. f(z)) . .. ) (in times). For example, fl (z) = f(z), h(z) = f(f(z)), etc. Let z be an arbitrary point of the plane. Applying to it the inifinite sequence of transformations we obtain the inifinite sequence of points z, f(z), f2(z), ... , r(z) . .... In other words, we consider the orbit of the point z under the action of iterations of the map f. It is convenient to consider a complex compactified plane as a two-dimensional sphere. To this end one should add an infinitely remote point to a Euclidean plane. Then the infinite sequence of points r(z) necessarily has limiting points on the sphere. We say that a subset Q on the sphere (or on the plane) is invariant under the map f if it is carried into itself under the action of all iterations n = 1,2, .. . . Such sets often appear to be fractals [10]. Such objects appeared for the first time in well-known works by P. Fatou (1878-1929) and G. Julia (1893-1978). Since f is a complex polynomial, the mapw = f(z) always has fIXed points. They are defmed as roots of the equations f(z) = z, i.e. f(z) - z = O. If the degree of the polynomial f is equal to k, this equation has k roots. Some of them may coincide, i.e. to be multiple roots. The roots are represented by points Zl, .•• , Zk on the compactified plane C = ]R2 U00. Suppose at some of these points Zi there holds the inequality Id/dz f(zi)1 < 1. Then the map f is "contracting" at this point, i.e. the root Zi is the point of attraction of some other points of the plane. Next we may consider on the plane the domain of attraction of the root Zi, i.e. the set of points Z for which limn-too r(z) = Zi. Such sets also often appear to be fractals. They may have an exceptionally complicat~d and at the same time beautiful form [10]. We may also consider the following set of points {z : limn-too I r(z) 1= oo}, i.e. the domain of attraction of an infinitely remote point. These sets are also frequently referred to as fractals.

r,

Fractals in the theory of complex transformations of a plane

r,

r,

Fatou, Julia, Mandelbrot

Domains of attraction

4.2.3 Homeomorphisms

Fig. 4.2.3

Homeomorphism is one of the basic concepts in topology. Homeomorphism, along with the whole topology, is in a sence the basis of spatial preception. When we look at an object, we see, say, a telephone receiver

4.2 Fractal Geometry and Homeomorphisms

269

or a ring-shaped roll and first of all pay attention to the geometrical shape (although we do not concentrate on it specially) - an oblong figure thickened at the ends or a round rim with a large hole in the middle. Even if we deliberately concentrate on the shape of the object and forget about its practical application, we do not yet "see" the essence of the shape. The point is that oblongness, roundness, etc. are metric properties of the object. The topology of the form lies "beyond them". It is of interest that we do not need any special training to learn to perceive this essence of the shape because it is an inbom ability of our sight to see holes and connections, regions and lines, adjoinings and intersections, cIosedness, etc. We see all these visual properties since we are born, when the formal thinking is not yet formed, and therefore it is difficult to pay special attention to the properties of the object laid in our perception. It is perhaps just for this reason that the science investigating these properties, i.e. topology, developed later than the other fields of geometry, in fact, only in the XX-th century. In mathematics these "basic properties" are defined as invariants oj homeomorphism. Homeomorphism is a superposition of some figures upon others, but a more free superposition than a rigid matching of triangles in Greek geometry, in the proof of their equality (or inequality). It is useful to imagine figures or bodies made of caoutchouc, but much more elestic than it actually is. This allows us to expand, contract, twist the object. Only breaks and glueings forbidden. For example, if we melt a roll made of wax, this will be an unlawful transformation because at some moment glueings and breaks will appear. Although the transformations admitted by homeomorphism can be stubstantial, something still remains

unchanged. This "something" is just the subject matter of topology. For example, this is the number of connected pieces or the number of holes. The two problems often to be solved in topology are how to establish that two figures are homeomorphic (if at all) or how to show that they are not homeomorphic. To solve the first poblem, one should point out a more or less explicit geometric construction (most often to present a homeomorphism); to solve the second one, one should indicate the property which is distinct for these ojects, i.e. the invariant which is more convenient to represent in an algebraically calculate form. In classical Greek geometry there exist two kinds of equalities. Two triangles can be matches either by moving them continuously along the

Invariants of homeomorphism

270

I-+-u,m-u

4. Visual Images in Some Other Fields of Geometry

plane or (if they are axissymmetric) only by taking them out of the plane and changing their position in space. In the latter case this is the matter of orientation. More complicated cases are possible too. Take a "comb" Fig. 4.2.4 (Fig. 4.2.4) and bend aside one of its teeth. Doing so, assume the comb to be a one-dimensional curve. The figures obtained are, of coUrse, homeomorphic. But the establish the homeomorphism, one should not pay attention to the enveloping plane. Neither of them can be mapped into another via homeomorphism (prove itl). When considering the homeomorphism problem as depending on the position of the figure in the Problem of embedding enveloping space, we are led to the problem of embedding fonnulated as follows. Let in ]Rm (it is only for the sake of simplicity that we take a Euclidean space) two figures A and B be given of which we know that Setofa/l self-homeomorphisms ~eyare homeomorphic. Is the possible to construct a homeomorphism of Euclidean space h: 1 m - t ]Rm such that h(A) = B? In a specified fonn, we are given a homeomorphism y: A - t B and it is required that h be so constructed that for points x from A we have h(x) = y(x), i.e. that h might continue y from A to the whole of]Rm. It would seen reasonable that before turning to these general problems, .we should first investigate the set of all self-homeomorphisms of Klein's the space ]Rm. This would correspond to the well-known F. Klein's ErErlangen program langen program - the geometry is created by the transformation group for Group of homeomorphisms ]Rm; in this case this is the group of homeomorphisms as the widest of all groups of continuous transformations. But investigation of the group of homeomorphisms of]Rm (denoted here by H(]Rm)) has been delayed up to recent decades and is not yet quite completed now. We shall briefly touch upon some of the results obtained approximately after 1960. We shall begin with one construction proposed by an outstanding topologist of the XX-th century 1. Alexander. It is extremely simple. Consider homeomorphisms motionless on a ball D;' of radius p and centre at O. This means that h(x) = x if x lies in D;'. It turns out that for such h one can easily construct a one-parameter family of homeomorphisms or an isotopy to join h with an identity homeomorphism e (where e(x) = x for all x fonn ]Rm), i.e. such a family ht, where t varies fonn 0 to I and hi = e, ho = h and ht(x) depends continuously on the arguments x and t. Moreover, each ht is a homeomorphism. To prove this, we consider the family of homotheties kt : ]Rm - t ]Rm, i.e. similarity transformations.

IIIIIII

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(expansions, contractions) of the space ]Rm with coefficients t: x --? tx. Then it suffices to put ht = kt I hkt for t f 0 and separately ho = e. This construction admits many modifications. For example, the substitution c(x) = x/lxl2 (inversion; 0 is discarded) readily gives a similar result for homeomorphisms identical outside the ball Dm , or simple for homeomorphisms of Dm identical on its boundary. As a result, we can obtain a homeomorphic correspondence between the space of homeomorphisms preserving direction at the origin and the space of homeomorphisms preserving direction at infinity. The latter condition can be defined in essentially different ways, and we invite the reader to decide what the needed defmition should look like. This contribution is due to 1. Kister. For direction-preserving homeomorphisms we incidentally obtain a technical result: each of them is representable as a composition of two homeomorphisms each of which is identical on some ball. The condition is stronger (from the formal point of view) than the possibility of constructing an isotopy joining the homeomorphism with the identity transformation. Therefore, for a long time attempts have been made to show that each orientation-preserving homeomorphism (this condition is, of course, necessary) is representable as a finite composition (which can always be reduced to two) of homeomorphisms identical each on its own ball. Such homeomorphisms are called stable and the problem itself the stability of homeomorphisms. It is now solved in all dimensions except four, and its solution appeared to be linked with the most fundamental questions of the topology of manifolds, the answers to which have been found in recent decades. The next step made by R. Kirby [83] in 1968 looks like a rather simple extension of the methods based the Alexander's remark (see above). But it is precisely this remark that has ultimately led to the discovery of connection of different results obtained before in isolated fields of the topology of manifolds and, in the end, to the solution of the indicated problems. Consider the homeomorphisms of a torus Tm. Represent it as a quotient manifold by the action of]Rm of a group of integer-valued linear transformations corresponding to translations into linear combinations of m independent vectors which form an orthogonal frame. Let p be a natural projection of the space]Rm onto Tm. It maps into one point a whole

Stable homeomorphism

on

General homeomorphisms of a torus

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4. Visual Images in Some Other Fields of Geometry

orbit of this group, i.e. the set x + L:k Ckek (where Ck are integers). To each homeomorphism h of the torus there correspond homeomorphisms h of the space ~m which cover this space. As h(x), where x is a point from ~m, it is natural to take one of the points lying over the point h(P(x», and it suffices to make the choice for one of them because for the rest it is uniquely determined from the continuity condition. So, we have the condition ph(x) = hp(x) which can be written as the commuta-

~m~

tivity condition for the diagram

Stable homeomorphisms of the torus

Covering homeomorphism

~m p1 p1 Tm~ T m

We obtain the map HT(~m) - t G(Tm), where HT(~m) denotes the group of homeomorphisms mapping each set of points lying over one and the same point of the torus (the orbit) into another similar set over another point of the torus. In this case the homeomorphisms can be said to be permutative with the action of the indicated group. Each homeomorphism h of the torus generates a linear transformation A with an integer-valued matrix. It is most easy to define A as the transformation of the fundamental group of the torus induced by homeomorphism. An extensive definition will be following. Let the point 0 be motionless under the action of covering homeomorphism. This condition does not violate generality. Then this covering ~omeomorphism preserves the whole grid. (lattice) of points L: Ckek with integer coefficients Ck and determines on it the required linear transformation. Consider torus homeomorphisms generating an identity transformation (an identity matrix) in ~m. The group of these homeomorphisms will be denoted by Ho(Tm). Any homeomorphism can be transformed to Ho through the transformation h - t z-l h, where z is a standard homeomorphism generating the same linear transformation as h. Homeomorphisms from Ho are direction-preserving at infinity since they displace each point not more than by the size of the fundamental region (in our case, by the diameter of a unit cube). In particular, they are stable. Since the stability is obviously local, the corresponding homeomorphism of the torus, which we have taken in fact arbitrarily, is stable too. The arbitrariness was restricted by the fact that the homeomorphism was taken from Ho(rm), but any homeomorphism of a torus is a composition of such a

4.2 Fractal Geometry and Homeomorphisms

273

homeomorphism and a standard one. A standard homeomorphism is, of course, stable if it is orientation-preserving (and this we have assumed). Thus, the important step made by Kirby appeared to be possible because he overcame the inertia of thinking, and in the obviously local problem, which therefore obviously required consideration only of a local model of manifolds, i.e. a Euclidean space, he considered a manifold different from ~m and from the sphere (to which the attention of researchers was mostly given at that time), namely, the torus. The homeomorphisms of the torus all appeared to be stable without exception. To make further use of this remark, Kirby has to apply the technique to be described below. We shall first mention one consideration, also a remote descendant of Alexander's construction, with the help of which the isotopy joining a homeomorphism with the identity homeomorphism is proved to exist for any homeomorphism sufficiently close to the identity one. This isotopy may actually be constructed (see below) as continuously depending on the homeomorphism. In the topological language this property is called the local contractibility of the homeomorphism group. This theorem, formulated by A.V. Chemavsky, is one of the basic theorems in the abovementioned development in the topology of groups of homeomorphisms of manifolds [84]. The crucial point of the consideration can be formulated as a lemma. It consists in the fact that a small topological displacement of a cube in ~k can be covered by the topological perturbation of the whole of~k. Lemma. Let Iff be a cube with side 2p and centre at the origin. There exists c > 0, such that for each homeomorphism (embedding) g: If-+ ~k displacing points not less than by c (i.e. for all x we have: the

distance from x to g(x) is less than c) one can construct the isotopy ht : ~k -+ ~k which joins on map 9 with the identity map, i.e. ho = 1, hi = 9 on If. Given this, ht depends continuously on 9 and is identical for the identity embedding g.

If

The latter condition, in particular, means that the less the map 9 displaces the points, the closer the unknown isotopy is to identity. The required covering is constructed only for the cube If, and the embedding

Local contractibility of the homeomorphism group

274

4. Visual Images in Some Other Fields of Geometry

is determined on the larger cube If. This is necessary for the proof and is no obstacle for using the lemma. See Ref. [84]. The general result reported by Chernavsky [84] consists (if formulated very briefly) in local contractibility of the homeomorphism group of any manifold. For a noncompact manifold one should specify the topology of homeomorphism space, but we do not discuss it here. As a simple application of this result, the reader may answer the question which was formulated by D.V. Anosov and gave impetus to the proof of the theorem. Suppose a homeomorphism, which differs little from the identity one, is given on the boundary of a manifold. Can it be continued to the homeomorphism of the whole manifold? Returning to stable homeomorphisms we note that the above-menStability of tioned results make it possible to establish that homeomorphisms differing torus homeomorphisms little form the identity one are stable. Furthermore, this consideration has not dimensional restrictions, which isa pleasant exception to the basic results of the modem topology of manifolds. The arguments needed to translate the Kriby's proof of stability of torus homeomorphisms into the proof of stability of homeomorphisms ofl.m run into rather complicated facts of the topology of manifolds proved only for m f 4, and therefore the case 1.4 requires special consideration. It should be noted that from the torus one can easily go over to the manifold T x I., where T denotes the torus Tk-I. It turns out that each homeomorphism M x I., where M is a compact manifold (or generally a compact), is isotopic to the homeomorphism covering the homeomorphism M x 8 1• The concept of a stable homeomorphism admits an extension which we introduce without claiming a complete generality. We call a homeomorphism of a region in I.k onto a region in I.k stable if it becomes linear in the neighbourhood of a certain point by means of a composition with a homeomorphism identical outside some neighbourhood of the image of this point. This may be any arbitrary point of the given region. This means locality of the stability. Let now T x I. be represented as a subset (subregion) in ]Rk. This is obviously possible for k = 2, and for larger k is obtained by a successive Fig. 4.2.5 construction of "surfaces of rotation" (Fig. 4.2.5). Suppose we are given a homeomorphism h: I.k -+ I.k • What can be said about h(T x I.)? This

4.2 Fractal Geometry and Homeomorphisms

275

is an open subset in I.k and is, therefore, a manifold, a triangulatable one (as any open subset in I.k ). In the technical language of the topology of manifolds one can say that a piecewise linear structure of this manifold is defined. At this point the problem of homeomorphisms ofl.k first meets with the problem of structures on manifolds, which is one of the basic problems in modem totopolgy of manifolds. The contiguity of these problems turned out to be progress made in the development of the topology of manifolds since the 1970s. The piecewise linear structure is understood as a complete set of triangulations of a given manifold, all of them being connected with one another so that each two have a common subdivion. Moreover,the triangulations themselves should be locally isomorphic (perhaps, after the subdivion) to triangulations of a region in I.k , which is of course fulfilled in our case. The fundamental problem conventionally called Hauptvermutung consists in the statement of uniqueness of such a structure. Namely, ifa homeomorphism is established between two manifolds with piecewise linear structures, then between them one can also establish a piecewise linear homeomorphism, i.e. a homeomorphism linear on each simplex of a certain subdivision. If this problem were solved at least for T x I. (said Kirby in 1969), everything would be all right: we would have a homeomorphism h: T x I. -+ h(t x I.) linear on simplexes, and since the homeomorphism h-1h is stable (as a homeomorphism ofT x I.), so will be h at least on T x I.; by virute of locality of this property, h will be stable everywhere. The question of uniqueness of structures on T x lR. appeared to be rather mature by that time. By the way, it soon became clear that this question is generally solved negatively, but Siebenmann then made an important remark. The point is that there exists an obstacle (a homology class with coefficients in the group Z2) responsible for non-uniqueness of the structures. It was unknown at that time whether the required class is equal to zero. But it was known that it becomes zero in passing over to some finite-sheeted covering which is again homeomorphic to T x I.. By virtue of locality of our problem, it is sufficient that we establish stability of the homeomorphism covering the given one. This solution of the problem of stable homeomorphisms (for k 4) would in itself be a rather peripheral fact of the topology of manifolds

r

Piecewise linear structure

Problem of stable homeomorphisms

276

4. Visual Images in Some Other Fields of Geometry

and would be of interest mainly as an example of establishing an unexpected contiguity between two branches of the study of manifolds. In the course of this, however, profound properties of the various structures on manifolds were discovered. The discussion of different visual properties of homeomorphisms are contained also, in particular, in [86]-[90].

4.2.4 Visual Material Fractals Random processes Random sequences Complex dynamiCS

"From chaos to order"

Medieval scientific ideas

Fractals appear naturally in modelling the Brownian motion, in constructing sets resembling the Cantor set, in studying random processes. Examples of random sequences of numbers are decimal decompositions, for example, of numbers 7r and e. Figure 4.2.6 demonstrates typical pictures arising in fractal geometry in the study of the complex dynamics (see Ref. [10]) of complex transformation of a plane. Fractals arise in dynamic systems in the study of translations along the trajectories of the system. It is often useful to imagine the structure of fundamental regions of a given action of the group. Various elements of fractal geometry are presented in Figs. 4.2.7 and 8. Figure 4.2.7 (from the cycle "From chaos to order"), which has something in common with the ideas expressed by Shiryaev in his book [16], provides an abstract interpretation of the idea of the probability theory and statistical fractal geometry concerning a gradual discovery of harmony and order in a primarily seeming chaos of images. Figure 4.2.8 is an attempt to illustrate the idea of mathematical (geometrical) infinity closely connected with the idea of fractals. Infinite variations and the variety of one and the same habitual image resemble distortion of fundamental region under discrete action of an infinite group. Figures 4.2.9,10 and 11 are devoted to the attempt of modem geometrical interpretation of some scientific ideas which excited already medieval scholars and painters (in particular, Bosch, Bruegel, Diirer and others). All of them were interested in the geometrical theory of perspective, developed the principles of a correct drawing of objects, were pioneers in discovering mathematical mechanisms of optical illusions, etc. The creative activity of authors in those times was naturally under the influence of a specific (and very interesting) scientific world outlook which also found its expression in drawings and paintings of that epoch. Such is, for example, the well-known Bruegel's engraving "Alchemists".

Page 277, Fig. 4.2.6. Complex dynamics and transformation of the plane. The last battle with titans; ancient and medieval Greece

278

Medieval and modern concepts of mathematical infinity

4. Visual Images in Some Other Fields of Geometry

Figure 4.2.9 can be interpreted as a mathematical variation of this theme whose essence is visual modelling of the idea of mathematical infinity which was the subject of lively interest and excitation for many scholars beginning from Renaissance. In this sense Figs. 4.2.9, 10 and 11 are an attempt to imagine how medieval geometricians would have depicted the ideas, legends and other things that stirred their minds if they had commanded the concept of mathematical infinity to the extent accessible to a mathematician nowadays. Of course, many scielltific concepts of authors of the Middle Ages have now lost their topicality and seem rather strange to the contemporary reader and viewer, but many modem geometrical ideas have naturally grown from the soil fertilized by them. Therefore, we think it interesting to synthesize in several graphical works somewhat strange medieval images and modem mathematical concepts of infinity, deformation, continuity. Figures 4.2.9,10 and 11 present both many symbols and images of medieval science (geometrical and astronomical devices, chemical materials and equipment, etc.) and purely modem objects (optical lenses, apparatus, etc.). But this mixture is not mechanical, it is hereditary because it is viewed by the eyes of today' s spectator who feels the relation between thorough mathematical investigations of our time developed for example from the attempts to prove the Fermat theorem (see the fragment in Fig.4.2.l1) and the peculiar scientific thinking which populated the world of that time with strange fantastic images.

Page 279,. Fig. 4.2.7. Idea of mathematical infinity and fractal dimension. From cycle: "From chaos to order". Fanatics Page 280, Fig.4.2.8. Idea of mathematical infinity. Prediction of a future by ancient god Page 281, Fig.4.2.9. Comparison of medieval and modern concepts of mathematical infinity. Variation of the theme of Bruegel's "Alchimists". From the cycle "Conversation with medieval authors" Page 282, Fig. 4.2.1 0. Modern mathematical variation on the theme of well-known biblical book "Revelation" (St. John); from the cycle "Conversation with medieval authors" Page 283, Fig.4.2.11. Modern mathematical variation on the theme of well-known medieval legend "Temptation of Saint Anthony"; from the cycle "Conversation with medieval authors"

284 Visualization and computers in number theory

Visual computer geometry

4. Visual Images in Some Other Fields of Geometry

4.3 Visual Computer Geometry in the Number Theory At first glance the theory of numbers is deprived of any geometricity. But this is actually not the case. At the contemporary stage of development of computers it has become possible to explain to a wide range of readers that visual geometry helps not only to illustrate some abstract situations from the number theory, but sometimes also to solve new problems. Visual computer geometry is of help in finding a mathematical statement, after which there follows the stage of its logical proof. In such a sophisticated theory as number theory it is sometimes very difficult even to formulate a plausible hypothesis; Modern computers permit a correct choice of hypothesis after processing many experimental data and representing them in the graphic form. Much depends on the way of representation. The information should be aptly coded by visual images so that the pictures appearing in the computer display might help the researcher to fine the correct direction of subsequent steps in his study. We now consider the problem of representation of natural numbers n :::: 1 as the sums

n =ni +n2+"

Waring problem

'+n~

,

(1)

where all the numbers 1ti, 1 ::; i ::; s are non-negative integers. Let us fix arbitrary values r ~ 2 and s ~ 1. In this case, all natural numbers are divided into two classes. One class involves all natural numbers representable in the form (1) and the second class involves those which cannot be represented in the form (1) with given parameters r and s. There appears the general problem (the generalized Waring problem): hwo shall we describe each of the indicated classes? This problem is exceedingly complicated. We shall briefly describe both the results obtained earlier and the recent ones. Paricular attention will be given here to the results initially "groped for" using computer drawings (after which they were proved mathematically). This work was done by A.A. Zenkin [91], [92]. We shall describe some of these results. We agree that the natural numbers not representable in the form (1) will be black while those representable will be white. In the original experiments [91], [92] the colours were more spectacular, they were

4.3 Visual Computer Geometry in the Number Theory

285

respectively red and green. Now consider one of the possible ways to present a natural series on a computer display. Take an infinite strip and mark it into equal squares in which write natural numbers successsively (Fig.4.3.l). Fix an integer d and divide this infmite strip into pieces of length d. This number will be called the image modulus. Recall how the picture is formed on a television screen. An electron beam runs through the first horizontal line, jumps over to the beginning of the second line, runs it through, etc. As a result, the beam sweeps the whole of the square screen creating a picture. We shall do the same. We lay the first segment of length d onto the first line of the display, the second segment onto the second line, etc. Then the display gets filled with natural numbers form I to a certain N determined by the display size. Since we are primarily interested in the property of natural numbers to be "representable or non-representable" (in the form (I)), we do not need the absolute value of the numbers. If needed, this information can be restored by counting the number of squares on the display. So, marking non-representable numbers in black and representable in white, we see on the computer display a spotty black-and-white carpet consisting of black and white squares (for given r, s, d). This carpet can be exceedingly sophisticated. Since our basic problem is a visual determination of regularities in the spotty carpet of numbers, its solution can sometimes be sped up by adding sound information. The display of each black square was accompanied by the sound signal "C" and that of each white square by the signal "G". As a result, the computer not only drew a certain set, but also played ''music''. Its rhythmics is evidently some invariant of the represented set and (what is important) does not depend on the number d. Fix r = 2 and start increasing the parameter s = 1,2,3, .... Fix the value of d. So, we are studying the question of representability of numbers in the form a) of the square of a certain number, b) of the sum of two squares, c) of the sum of three squares, etc. Let us look at the character of picture variation on the display (Fig. 4.3.2). At first (for s = I) almost the whole of the display is black. Here and there one can see white squares. One can also see how few numbers are squares. As s increases, the display starts "whitening". Finally, for s = 4 the whole display flares white. Black squares have vanished. For

1112.131416' 16 1; I Fig. 4.3.1

Visual determination of regularities in the "carpet of numbers"

Any natural number is the sum of four squares (Lagrange)

286

4. Visual Images in Some Other Fields of Geometry

s = 5 the picture no longer changes: the display remains white. Clearly, we have "seen" a certain regularity.

Computer hypothesis 1. The sum offour squares (i.e. s = 4) suffices to represent any natural number in the form (1). Visualization of Lagrange theorem

It turns out that we have seen on the display the well-known Lagrange theorem which he proved in 1740.

The Lagrange theorem. Any natural number is representable in the form of the sum offour squares. Every mathematician knows and can give many examples from his scientific work when it appears much more difficult to feel or "see" a correct hypothesis than later to prove it. Visual images are particUlarly often used in geometry and topology where one has to work with multidimensional objects which, in principle, do not always admit picturing in a three-dimensional space. By Fig. 4.3.2 the computer informed us about something more. For s = 2 the carpet surely has a simple structure: we clearly see six vertical black columns, i.e. columns entirely consisting of black squares. This is also a well-known theorem. The Euler theorem (1749). Let a natural number n have the form n == 3,6, 7(mod8). Then all numbers of the form n·4 k , where k = 0, 1,2, ... , are not representable in the form (1), i.e. n· 4k f nI + n~. We haven chosen mod d so well that the numbers indicated in the Euler theorem which are not representable in the form (1) are positioned exactly one under another on the display to form the black columns. For s = 3, there remain not many black squares in Fig.4.3.2. The remaining white field is clearly seen to be organized into several oblique strips concentrating along white segments and having the same slope angle. Clearly, this is again not accidentally. But for s = 3, the carpet in Fig.4.3.2 is not yet well pronounced. Let us try to change slightly the picture so as to make it more "completed". We shall change modd. The result is shown in Fig. 4.3.3, where we can see pictures for d = 22,24. It

4.3 Visual Computer Geometry in the Number Theory [

0

287

r

Euler theorem !=2.

r

n

(L.ELlfe'l.)

n

c

c Gauss theorem Lagrange theorem S:::

3 (Gauss)

is quite clear that when d = 24, the picture acquires its "final" form. The columns have become vertical. Of course, there are black squares outside columns, but we are now primarily interested in those black squares which constitute the vertical columns. This is a no less famous Gauss theorem.

The Gauss theorem (IS01). Natural numbers are not representable as the sum of three squares if and only if they have the form n = Sk +7, k =

Fig. 4.3.2

White-black carpets of numbers

288 I"

I ••

I,

,

• I

J' I ,

JI.

'--

'-',

•

'IIIIIL -I'

'.JI. J'.. II,,;'".

..... .

'."" I JI' " -. ... JlI. " " JI, ,__

IIIIIL

"

.

1,'-

-1",

.,.,

JI •

...J' •• '.'", J' '.'- I,

" . JI •• ". JI-. '..

mod 22.

mo« 2.3 n

•

•

t

t

• • •

.•

~

0,1,2, ... , (i.e. n == 7(mod8)) and if they are obtained from these ones via mulitplication by any power of the numeral 4, i.e. if they have the form n· 41, where l = 1,2,3, ....

All details of the Gauss theorem are clearly seen on the display (Fig.4.3.3). Note that in any k-th column on the display (modd) there stand numbers of the form n == k(mod d), i.e. numbers which when divided by d give one and the same residue k. The main series of columns (continuously black) are visually distinguished in Fig.4.3.3. Therefore, they consist of numbers which when divided by 8 give a residue equal to 7 (numbers == 7(mod8)), i.e. numbers of the form n = 8k + 7, k = 0,1,2, .... Employing the sweepingout procedure, which in any set of numbers finds all subsequences of the type of arithmetic and geometric progresssions, one can readily make sure that for d = 24 all the rest of the black squares of the picture in Fig. 4.3.3 are elements of geometric progressions of the for n· 41,1 = 1,2,3, ... , where n is the element of one of the columns of the principal series. So, the intrinsic harmony of the visual image, our intuitive ideas concerning the regularity, symmetry and order prove here to be effective control parameters of the process of visual determination of mathematical regularities. The problem next in complicacy is representability of numbers as the sum of cubes. If we dealt not wich integers but with rational numbers, the answer to the question would be the following.

n

~

• . • • • . • • • • •• , • • , . •

•

4. Visual Images in Some Other Fields of Geometry

III "II

• ~

•

Theorem. (T. Raily 1825, Richmond 1930). Any rational number a is representable as the sum of three cubes of rational numbers, namely: a3 -3 6 -a3 +3 5a+3 6 )3 ( a2 +34a )3 3 ( ) ( a = 32a2 + 34a +36 + 32a2 +34a +36 + 32a2 +34a +36

•

We now proceed to the classical Waring problem. We fix r = 3 and • vary s, i.e. investigate the possibility of representing numbers n as the ~ sum of s cubes: = nf + ... + We again display the information t I • • • in the form of a carpet. In Fig.4.3.4 there are pictures (for s = 1,7,8) J. (Gltl.lr) successively appearing with increasing s = 1,2,3,4,5,6,7,8,9. This Fig. 4.3.3 In 0 2lr S material would generally be more convenient to illustrate by a cartoon t

n

n;.

4.3 Visual Computer Geometry in the Number Theory

289

shot directly from the computer display. The first shots will be almost all black, then the white field begins increasing. Finally, for s = 8 in the white field there remain only two black squares which vanish for s = 9.So for s = 9 the whole display is white! By the way, the last two black squares in the eighth shot are numbers 23 and 239.

Computer hypothesis 2. Any natural number is representable as the sum of nine cubes. This is the well-known theorem by the German mathematician A. Wiferich which he proves in 1909.

Computer hypothesis 3. At first (jar small s) there is very much black colour (and the set of black squares is obviously infinite), then there becomes much less black colour (and the set of black squares becomes obviously finite). And finally black colour vanishes altogether (the set of black squares is obviously empty).

1=1r

(

n

•

•

•

.• •. •

• ••

•

From the pictures obtained one can hypothesize that this situation ( may also hold for the other powers r = 4,5,6, .... The schematic expression of this hypothesis is presented in Fig. 4.3.5. Here s is the number of summands in the sum (1), r 2: 2 is fixed exponent, N is the number L 4= 1 of black (i.e. non-representable) natural numbers. Let r be fixed. Since the number of summands s increases monotonly (Fig. 4.3.5), there exists!+r _ _ _un _----. a certain value s = G(r) (of course, r-dependent) such that for s < G(r) the black set is inifmite and for s 2: G(r) finite. In other words, G(r) is the smallest number of summands in the sums (1) for which the black set becomes finite for the first time. [ A similar boundary arises with a further increase of s, i.e. when the number of elements of the black set goes over from finite to zero. This second boundary we denote by g(r). So, g(r) is the smallest number of summands in the sum (1) for which the black set appears empty for the C first time~ To say it differently, for s 2: G(r) it is only afinite amount of natural numbers that is not decomposed into the sum nI + ... + n~, while for 1= 8 Fig. 4.3.4 s > g(r) all natural numbers are representable in the form nJ + ... +n~.

• •

•

290

4. Visual Images in Some Other Fields of Geometry

Hilbert functions

These two functions, G(r) and g(r), are called Hilbert functions. We have presented our arguments, prompted by computer drawing, in favour of the hypothesis that the number g(r) is finite. This does not, of course, follow from its defmition.

Waring's conjecture

Waring's hypothesis. For any power r ~ 2 there exists (and is finite) a smallest number of summands g(r) such that for all s > g(r) the set of natural numbers non-representable in the form nj +... +n~ is empty. Here nj are non-negative integers.

Waring formulated this hypothesis in 1770 on the basis of the Lagrange theorem and of the empirical assumptions of his predecessors that any natural number can be represented as a sum of nine cubes and nineteen fourth powers. There was no Wiferich theorem (1909) at that time, but the corresponding empirical hypothesis on the sums of cubes did exist. o So, we have the equailities g(2) =4, g(3) = 9, g(4) = 19. The function g(r) is seen to increase rapidly, and therefore it is a priori not excluded that for a certain ro it will appear to be equal to infmity. This would U 3· .. G('t) ... 1('T.) mean that for any arbitrarily large number of summands s there exists a Fig. 4.3.5 sufficiently large natural n non-representable as the sum n = nro+...+n~o. As a result of considerable attempts made by many mathematicians. Hilbert-Waring theorem Waring's hypothesis was finally proved by D. Hilbert in 1909, and since then it has been known as the Hilbert-Waring theorem. The classical Waring's hypothesis was further on extended in different directions. We shall not go into detail of these extensions, but only mention the names of some mathematicians who obtained here some further results: I.M. Vinogradov (1938, the Goldbach-Waring problem), Hua Lo-gen (1937), Hua Lo-gen and R.E. Haston (1938), E.M. Right (1934), B.I. Segal (1933), K.F. Rot (1951),V.I. Nechaev (1953), A.A. Karatsuba (1962) and others. There exists, however, one natural extension which we shall touch upon here. In the course of computer experiments (A.A. Zenkin) on the classical Waring problem, an interesting fact was unexpectedly discovered. If in the sums (1) the condition "ni are non-negative integers" is replaced by the condition "ni are natural (i.e. positive) integers", the situation changes. It is of interest that this substitution of conditions was Empirical hypothesis

4.3 Visual Computer Geometry in the Number Theory

291

primarily made by computer itself because it always understands too literally the desire of the experimenter to economize its memory. It began discarding zero summands, i.e. the cases where ni = 0, which in its opinion provided little information. It turned out that already in the simplest case of the Lagrange theorem, i.e. for r = 2, instead of the Lagrange theorem there appeared the cartoon shown in Fig.4.3.6. Ths distinction is that as the number of summands s increases; the black field decreases to the value s = 6, but with a further increase of s the number of black squares remains unchanged! There remains a residue of seven numbers of the form s+{1,2,4,5,7,10,13} for all s ~ 6. This visual empirical result (obtained due to an accidental misunderstanding between the experimenter and the computer) has made the following contribution.

Fig. 4.3.6

Computer hypothesis 4. For r = 2 andfor all s ~ 6, any natural number n ~ 1 is representable as the sums (1) with the restriction nj ~ I except the numbers 1, 2, ... , s - 1 and numbers of the form s + z, where Z is one of the numerals {1,2,4, 5, 7,10, 13}. An additional and purposeful analysis of scientific literature dis-

covered the theorem of an American mathematician G. Poll which he proved as far back as 1933 and which coincides litera111y with computer hypothesis 4.

292

4. Visual Images in Some Other Fields of Geometry

The question naturally arose: What a strange set did the computer distingush following Poll? It was decided to see, using computer, the corresponding cartoon for cubes, i.e. for r = 3, which is n analogue of the Wiferich theorem, but with the restriction on the summands of the form nj ~ 1 in the sums (1). The result is demonstrated in Fig.4.3.7.

Computer hypothesis 5. For r = 3 and for all s ~ 14, any natural number n is representable as the sum (1), where all nj ~ 1, except the numbers for the following natural numbers: 1,2, ... , s - 1 and s +74 numbers of the form s + z, where Z are elements of the "strange" set {1-6, 8-13, 15-20, 22-25, 27, 29-32, 34, 36-39, 41, 43-46, 50, 51, 53, 55, 57, 58, 60, 62, 64, 65, 67, 69, 71, 72, 74, 76, 79, 81, 83, 86, 88, 90, 93, 95, 97, 100, 102, 107, 109, 114, 116, 121, 123, 128, 135, 142, 149}.

As we see, there appears here a strange set which (and similar ones) 1. Lagarius called the exceptional set. Computer hypothesis 5 was then given a rigorous mathematical proof in Ref. [91]. This is a new mathematical result. We deal with an example of a successful use of computer drawing for the formulation of a correct mathematical hypothesis. Later on, many interesting properties of such sets were investigated using computer, in particular, the hypothesis of their finiteness was first formulated and then rigorously proved.

14

FiQ.4.3.7

Appendix 1. Visual Geometry of Some Natural and Nonholonomic Systems

1.1 On Projection of Liouville Tori in Systems with Separation of Variables For simplicity we consider systems with two degrees of freedom, systems actually arising in nature, i.e. having Lagrangian of second degree in velocities: L=

n

n

i,j=1

j=1

Natural mechanical systems

L ajj(q)(iiQj - II(q) + L aj(q)qj

There are usually no linear summands (e.g. for the motion of a point along a motionless surface) and then the system is called natural. There is no conventional term to define the presence of linear terms. In some cases we distinguish between reversible and i"eversible natural systems. Reversibility implies that if q(t) is a solution, so is q(-t). The Hamiltonian corresponding to our Lagrangian has the form

Consider completely integrable systems with Hamiltonian of particular form for n = 2 (which means two degrees of freedom). We omit some details referring the reader to the works by Tatarinov [93] and [94]. It is known that in the phase space of completely integrable systems there appears fibre bundle (with singularities) of tori carrying conditionally periodic windings, and if in the phase space their behaviour is more or less the same, on the position manifold (configuration space) it is

Reversible and irreversible natural systems

Liouville foliation for the integrable system

294

~ .: . . :.1) . . .:.: . . .:. :.':":::';':":~':":""::"". '.

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.

. : :, ", .. :: "

.:/::

.:.:: :....:

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Fig.A.1.1

Fig.A.1.2

AI. Visual Geometry of Some Natural and Nonholonomic Systems

rather varied because the phase tori and their windings can be projected onto this manifold differently ([93], p.292). We mean here the projection (PI, P2, ql , q2) --+ (ql, q2). Let there be given a natural system with a cyclic coordinate (ql = r, q2 = cp).

r.,

1 2 r a(r)

H=-p2+~+II(r)

.

Then the torus is projected in a particularly simple way. One should put it onto the table and press it from above. The meridians cp = const go over into segments orthogonal to the parallels r = const. The behaviour of the trajectories of motion may vary as shown in Fig. ALl. The appearance of a linear term of the form Q(r )P

0 or '1'3 < 0 (over a truncated rectangle) we shall see a torus which will also be truncated. One can make sure that in the situation shown in Fig. A.1.12 (when one angle is cut out of the rectangle) a disc is cut of the torus (Fig.ALl3). By virtue of the above-mentioned symmetry, I c1c2 consists of two symmetric pieces which are glued together along the boundary (Fig. A.1.14). We obtain a pretzel! (And not a torus). It can readily be verified that in other versions of intersectij)n of a rectangle with a circle we obtain manifolds shown in Fig. AI.IS. This problem has been studied in detail by Okuneva [95], [96] who gave an exhaustive description of the qualitative picture of the behaviour of trajectories on integral manifolds.

Fig.A.1.12

Fig.A.1.13

.€D'.

Fig.A.1.14

::':'. ':.::

"

••

"

:~ '0:' '0 . :" I' •.

......... .

. ' 00 ...) ...J "

2

"

O· Q' :"

4,.,~.j

....f

~

Fig.A.1.15

Appendix 2. Visual Hyperbolic Geometry

-=

-

2.1 Discrete Groups and Their Fundamental Region

In this Appendix we consider the discrete groups of motion of Euclidean spaces, spheres and Lobachevskian spaces generated by reflections. The · .A.2. 1 groups of motions of a space are called discrete if the images of any point FIg form a discrete set under the action of all elements of the group. Discrete groups are characterized by the fact that they have a fundamental region i.e. a set whose images cover under the action of the group the entire space and do not intersect except, perhaps, at the points of the boundary. Examples of discrete groups may be the following:

Fig. A.2.2

Fig. A.2.3

1. The group oftranslations ofa Euclidean plane by integer-valued vectors in a fixed Cartesian coordinate system. A fundamental region of this group is the square 0 ~ x ~ 1, 0 ~ y ~ 1 (Fig.A.2.1). The hexagon (Fig. A.2.2), the curvilinear figure (Fig. A.2.3) are also fundamental regions of the same group. 2. The group of rotations of a circle through angles multiple by 21'i / n, where n is a fixed integer. In this case any arc of the circle, equal to 21'i / n, is a fundamental region. 3. The group of motions of a ring, consisting of two elements: identical and central symmetry motions. Any semicircle, or one of the yin-yang figures (Fig. A.2A), is a fundamental region.

These examples show that a fundamental region of a discrete group can be chosen in more than one way, and may be rather multiform. But for the groups of motion of a plane (space) a fundamental region can always be chosen in the form of a convex polygon (polyhedron) ([97],

2.2 Discrete Groups Generated by Reflections in the Plane

301

Chap.4, Sect. 1). A convex polygon (polyhedron) is understood as an intersection of a family of half planes (half spaces) which in our case is everywhere finite. The general definition allows the polygon (polyhedron) to have an infinite number of sides (faces), but in this case some natural restrictions should be imposed upon the family of half planes (half spaces).

2.2 Discrete Groups Generated by Reflections in the Plane We shall understand reflection as axial symmetry; the axis of symmetry will be called a mirror. The discrete groups of motions of a plane, generated by reflection, are distinguished from among all other discrete groups of motions for simplicity of their geometrical description. More precisely, a fundamental region of such a group can be chosen in the form of a convex polygon all of whose angles are integer parts of 7r and vice versa, each such polygon is a fundamental region for a certain discrete group generated by reflections. We shall describe how, knowing the group generated by reflections, one can find its fundamental polygon of the indicated form. To this end, we consider the mirrows of all reflections belonging to the group (Fig. A.2.5). They divide the plane into convex polygons which we shall call cells. We shall prove that any cell is a fundamental region of the group and has angles equal to an integer part of 7r. To begin with we shall show that the image of a mirror under reflection from another mirror is also a mirror (the mirror is understood here only as the mirror of reflection belonging to a given group). We use here the following notation; for any straight line II, reflection relative to II is denoted by RIl. Let II] and II2 be two mirrors. Then RRIJl (II2) = RIll' RIl2 . RIll (Fig. A.2.6) obviously belong to the given group since RIll and RIl2 belong to this group. Thus, mirrors and, therefore, cells exchange places under the action of the group. Any cell is carried to the neighbouring one via reflection relative to their common side which is a mirror. By means of a composition of such reflections, which is the element of the group, any cell can be carried into any other one. More accurate arguments ([98], Chap. 5, Sect. 3, Subsect. 3) show that a cell can be carried into

Fig.A.2.4

Fig. A.2.S

Fig. A.2.6

302

Fundamental region

Mi"ors

Group generated by reflections Coxeter polygons

A2. Visual Hyperbolic Geometry

itself only by a unit element of the group. The latter two statements just imply that a cell is a fundamental region. We shall now show that the angles of the cell are integer parts of II. Consider an arbitrary angle of a cell P, limited to straight lines IIo and III. Consider the sequence of straight lines given by the recurrent formula IIn+1 = Riin (lIn-I) (n ~ 1). According to what has been proved above, all IIn are mirrors. If we assume the initial angle not to be an integer part of II, then at some moment we obtain a mirror intersecting a cell (Fig. A.2.7), which contradicts the definition of a cell. Therefore, the angle under consideration is an integer part of 7r. On the other hand, any polygon with angles equal to integer parts of 7r can be shown [99] to be a fundamental region for the group generated by reflections relative to the sides of this polygon. Thus, the classification of reflection-generated discrete groups of motion of a plane is reduced to the classification of polygons with angles of the form 7r / n. Such polygons are called Coxeter polygons. All Coxeter polygons can be directly enumerated. All of them are presented in Fig.A.2.8. To solve an analogous problem in a three-dimensional space and in spaces of higher dimensions, it is useful to introduce some new concepts.

Fig.A.2.7

1M1OP1L~~ (It-)

(1 )

(2 )

(3) Fig.A.2.8

~ (1)

(5)

(6)

~. (8 )

(9 )

i.iai pfane

2.3 The Gram Matrix and the Coxeter Scheme

303

2.3 The Gram Matrix and the Coxeter Scheme Coxeter polygons can be defined by the Gram matrix of a set of normalized vectors orthogonal to the sides and directed outwards. This matrix will be called the Gram matrix of a polygon. Another way to define Coxeter polygons is the Coxeter scheme. The Coxeter scheme is a graph with vertices corresponding to the sides of the polyhon. Two vertices are not joined if the corresponding sides are perpendicular. If the sides intersect at an angle 7r In, the vertices are joined by an edge of multiplicity n - 2 (or by a simple edge labelled n). If the sides are parallel, the vertices of the scheme are joined by a bold-faced edge (or by an edge labelled (0). Obviously, a Coxeter polyhedron is uniquely defined, with an accuracy to the motions of the plane, by the Gram matrix or by the Coxeter scheme. The Gram matrices of the polygons depicted in Fig. A.2.8 are presented in Table A.2.2 gives their Coxeter schemes. Naturally, to the case of Fig. A.2.8(9), i.e. to a polygon having not a single side there corresponds a zero-dimensional Gram matrix and an empty Coxeter scheme. Table A.2.1

o (1)

~

(2)

Table A.2.2

(1) (I )

()IIIIIIO

(3 )

0

L (6 )

~~ (:1-: o

o

"

()

Gram matrix of a polygon Coxeter scheme

Coxeter polyhedron

~

(8)

l

0 i -1 o 0 -1 1 (j )

2.4 Reflection-Generated Discrete Groups in Space

Reflection-generated discrete groups

Reflection in space will be understood as symmetry about a plane. Similarly to the case ofthe groups on a plane, the reflection-generated discrete

Reflection in space

304

Coxeter polyhedra

Direct product of Coxeter polyhedra Decomposable Coxeter polyhedra

Classification of Coxeter polyhedra

Orthogonal subsets

Subschemes in a Coxeter scheme

A2. Visual Hyperbolic Geometry

groups of motions of a space are defmed by their fundamental region which may be any convex polyhedron with dihedral angles of the form 7r/n. Such polyhedra are called Coxeter polyhedra. The concepts of the Coxeter scheme and the Gram matrix are extended, without any essential changes, to Coxeter polyhedra of any dimension. One-dimensional Coxeter polyhedra are, naturally, a segment, a ray, a straight line. Their Coxeter schemes will be 0- - - -0, 0 and an empty scheme, respectively. A direct product of Coxeter polyhedra is, obviously, a Coxeter polyhedron. Inversely, if a Coxeter polyhedron is represented as the direct product of two polyhedra of lower dimension, the latter are also Coxeter polyhedra. A polyhedron representable in the form of the direct product of polyhedra of lower dimensions will be called decomposable. Among the Coxeter polyhedra of Fig. A.2.8, decomposable are 8(1), 8(2), 8(3), 8(4) for n = 2; 8(5) and 8(9). The definitions of the Coxeter scheme, the Gram matrix and the direct product of polyhedra imply equivalence of the following conditions: 1) The Coxeter polyhedron falls into the direct product of Coxeter polyhedra of lower dimensions. 2) The set of vectors orthogonal to the faces of a polyhedron falls into a union of pairwise orthogonal subsets. 3) The Gram matrix ofa polyhedron has a cell-diagonal form under a permutation of rows and the same permutation of columns. 4) The Coxeter scheme of a polyhedron falls into a union of pairwise disconnected subschemes. The classification of Coxeter polyhedra reduces, obviously, to the classification of undecomposalbe Coxeter polyhedra. Exercise 1. Suppose in an n-dimensional Euclidean space we are given a set of vectors forming pairwise non-acute angles, which does not fall into a union of pairwise orthogonal subsets. Then the vectors of the set may have no more than one linear deprendence. In particular, the set contains no more than n + 1 vectors. All the coefficients of the linear dependence, if it does exist, are positive.

2.4 Reflection-Generated Discrete Groups in Space

305

Since the angles between the vectors orthogonal to the faces of a Coxeter polyhedron and directed outwards are equal to 7r-7r In, They are not acute, and the statement of Exercise 1 implies that undecomposable Coxeter polyhedra are simplexes or simplicial cones. To the simplexes there correspond non-negative definite degenerate Gram matrices whose rank is smaller than the matrix dimension by unity. To the simplicial cones there correspond positive definite Gram matrices. Note also that all the diagonal elements of the Gram matrix of a Coxeter polyhedron are equal to 1, while the non-diagonal elements are either equal to -lor have the form - cos 7rI n. All the matrices satisfying the above-mentioned conditions are classified [99], and thus there exists a complete description of Coxeter polyhedra in Euclidean spaces of any dimension. The schemes of undecomposable Coxeter polyhedra in Euclidean spaces of dimension not higher than three are presented in Table A.2.3.

Undecomposable (irreducible) Coxeter polyhedron

Complete classification of Coxeter polyhedra in Euclidean space ]Rn

Table A.2.3 Si mpl'ic"q'[

n

Simpi4r.fS ill Ell

.{

~

2

\lo=c:::oCHIiD

3

n >='

CMfS

(si"'pR*~ ill

in E It

$")

0

/'I. 0--0

0-0-0 o-a:;::D ()-(E3)

~

Reflection-generated discrete groups with positive definite Gram matrices have the direct product of simplicial cones as their fundamental region. These groups have a motionless (fixed) point and can be interpreted as reflection-generated discrete groups of motions of a sphere (with centre at the fixed point). The Coxeter schemes of these groups are given in the right-hand column in Table A.2.3.

Positive definite Gram matrices Simplicial cones

306

A2. Visual Hyperbolic Geometry

Exercise 2. Prove that the discrete group of motions of a Euclidean space has a (common) ftxed point if and only if it (this group) is ftnite. Figure A.2.9 shows all Coxeter polyhedra on a two-dimensional sphere, their Coxeter schemes and sphere subdivision into cells under the corresponding discrete group. Exercise 3. Count the number of cells in each case.

o--!!:--o

0

Fig.A.2.9

Lobachevskian or hyperbolic plane

2.5 A Model of the Lobachevskian Plane We shall consider a three-dimensional pseudo-Euclidean space V 2,1 with the scalar product defined by the quadratic form (x, x) = -x~ + xI + x~ .

Model

The set c = {x E V2,1 : (x,x) < 0 is an open cone which consists of two connected components C+ and C- (Fig. A.2.1 0). A set of rays emanating from zero and lying in c+ is a model of the Lobachevskian plane A2.

2.5 A Model of the Lobachevskian Plane

307

Fig.A.2.10

The rays lying on the boundary of C+ are called infinitely remote points of the Lobachevskian plane. Straight lines in this model consists of rays lying in the intersection of a certain two-dimensional subspace of the space V 2,1 with c+ if this subspace is non-empty. Each straight line in A2 can be defined by the vector of the space V 2,1 orthogonal to this subspace. Given this, the condition that the intersection of this subspace with C+ is non-empty is equivalent to the fact that the restriction of the quadratic form to it has a signature (1,1), which is in turn equivalent to the fact that the orthogonal complement to it is spanned by a vector with a positive scalar square. Thus, each vector x E V 2,1, such that (x, x) > 0 defines a straight line IIx C A2. It is convenient to draw the cross-section of the cone C+ which intersects each ray lying in the cone at exactly one point. Such a crosssection can also be interpreted as a model of the Lobachevskian space. An example of such a cross-section is that by the plane Xo = 1. In this case, the unit circle is a model of the Lobachevskian plane and the chords of this circle are the straight lines. This model is called the Klein model. It shows very well three possible cases of mutual disposition of two straight lines on a Lobachevskian plane. They can intersect each other (Fig.A.2.l1(1», be parallel (Fig. A. 2.1 1(2» or diverge (Fig.A.2.11(3». In these cases, the lines of intersection of two-dimensional planes of the space V2,1 specifying these straight lines, lie in C+, lie on the boundary of the space and do not lie in its closure, respectively. We shall also say that two straight lines on a Lobachevskian plane intersect at a proper point, "at infinity" or "behind infinity" depending on which of the three cases is realized.

Infinitely remote points

Fig.A.2.11

Klein model of a hyperbolic plane Intersection "at infinity" or "behind infinity"

308 Motions of a hyperbolic plane

Pseudo-orthogonal transformations

A2. Visual Hyperbolic Geometry

The motions of the Lobachevskian plane in the model described are tranformations corresponding to the c+ -preserving pseudoorthogonal transformations of the space V 2,1. The reflections (relative to straight lines) are motions corresponding to pseudoorthogonal reflections relative to the two-dimensional c+intersecting subspaces of the space V2, 1• Such a reflection Re relative to the two-dimensional subspace orthogonal to the vector e is given by the formula

Rex = x _ 2(e, x) e . (e, e)

The invariants of relative position of straight lines on a Lobachevskian plane are the angle between them in case they intersect and the distance between them in case they diverge. The case of parallel straight lines is limiting 'for both these cases. The angle between parallel lines and the distance between them are assumed to be equal to zero. On the other hand, from the description of motions in the model in question it follows that such an invariant is the scalar product of the vectors of the space V 2,1, which define straight lines (these vectors are assumed to be normalized and agreeably oriented). The relation between these invariants is given as follows:

Fig.A.2.12

Note that the first formula coincides with that for the case of Euclidean plane. The minus sign appears because (eJ,e2) = 7r 2 )'

(IIe;,rre

Convex polygons on the hyperbolic plane

2.6 Convex Polygons on the Lobachevskian Plane The definition of a convex polygon on the Lobachevskian plane does not differ from the case of Euclidean plane. Each convex polygon in A2 can be given by the set of vectors in V2,1 specifying the sides of the polygon. In order that this set of vectors can be uniquely restored from the polygon, we assume the vectors to be normalized and directed outward the cone

2.7 Coxeter Polygons on the Lobachevskian Plane

309

constructed in V 2,1 over the given polygon (Fig. A.2.12). The polygon is defined, up to the motions, by the Gram matrix of this set of vectors. The sum of the angles of a polygon on the Lobachevskian plane is less than the sum of the angles of a Euclidean polygon with the same number of sides. The difference between these sums of angles is equal to the area of the polygon on the Lobachevskian plane [97]. Therefore, as distinguished from the Euclidean plane, the area of a polygon on the Lobachevskian plane is fmite if some of its adjoint sides are parallel, in other words, if some of the vertices lie at infinity (Fig.A.2.13). However, if at least one vertex lies behind infmity, the area of the polygon is infinite. Indeed, in the latter case one can construct a straight line lying entirely in the polygon so that the polygon contains the whole of the half plane restricted by this straight line (Fig. A.2.l4). Infinity of the area of the polygon now follows from infinity of the area of the whole Lobachevskian plane (the latter follows, for example, from the existence of an infinite diescrete group of motions on the Lobachevskian plane). Thus, a convex polygon has a finite area on the Lobachevskian plane if and only if it is a convex hull of a finite number of points which are either proper or lie at infmity.

Polygons of a finite area and of an infinite area

2.7 Coxeter Polygons on the Lobachevskian Plane As in the Euclidean plane, discrete reflection-generated groups of motions of the Lobachevskian plane are in one-to-one correspondence with Coxeter polygons. Coxeter polygons on the Lobachevskian plane can also be defined by the Gram matrix or the Coxeter scheme. The Coxeter scheme is somewhat modified as compared with the Euclidean case. Namely, we shall join by a bold-faced edge only vertices corresponding to parallel sides. If the sides diverge, the corresponding vertices will be joined by a dashed edge. In case it is necessary to reconstruct the polygon from the Coxeter scheme, we shall label the dashed edge the same as the corresponding element of the Gram matrix.

Exercise 4. Prove that if the two straight lines, on which some sides of a Coxeter polygon lie, intersect then their intersection is a vertex. In other words, continuations of non-adjoint sides do not intersect.

Discrete reflection-generated groups of motions of a hyperbolic plane

310

Instruction. Make use of the fact that all the angles of a Coxeter polygon are non-obtuse and that the sum of the angles of an arbitrary triangle on the Lobachevskian plane is less than 7r. It is now obvious that to a pair of adjoint sides of a Coxeter polygon there corresponds a subscheme of its Coxeter scheme of the form 0 or 0 n 0 (where n ~ 3), to a pair of sides intersection at an infinitely remote vertex there corresponds a subscheme 0 and to a pair of non-adjoint sides there corresponds a subscheme

D

o o

*3

2~~., ~., ~J LOO

i+!..f..J:{i .fc, ~, 'i

~4

[}

(3)

(S)

The scheme ofFig.A.2.l8(J) is an infinite-volume tetrahedron since the subschema is indefinite, and therefore the corresponding vertex lies behind infinity. But if we add another face to the tetrahedron (which will cut the vertex), we shall obtain the scheme of a restricted trigonal prism (Fig. A.2.l8(2)). In Fig. A.2.l8(3}-(6) one can see other examples of restricted trigonal prisms. The vertices joined by a dashed edge correspond to the bases of the prisms. Makarov [101 ]-[ 104] has proposed some geometric constructions which help in building many examples of Coxeter polyhedra in Lobachevskian space. Note that for all the schemes of Fig.A.2.l8 the marks on dashed edges are uniquely determined by the condition imposed on the rank of the Gram matrix. The same is valid for Coxeter polyhedra of another combinatorial structure. E.M. Andreev has proved that an acute-angled finite-volume polyhedron in Lobachevskian space is uniquely defined

(4)

(6)

Fig.A.2.18 Restricted trigonal prisms

Dihedral angles of acute-angled finite-volume polyhedron

314

A2. Visual Hyperbolic Geometry

by its dihedral angles. For a polyhedron to exist, its angles should satisfy some natural restrictions in the form of inequalities [105], [106]. Andreev's result can be interpreted as the classification of acute-angled polyhedra (and, in particular, Coxeter polyhedra) of fmite volume in Lobachevskian space. This classification does not, however, provide an explicit representation of these polyhedra (an explicit representation is understood here as the defmition of a polyhedron by a set of vectors in V 3,J). Indeed, to find all the elements of the Gram matrix of a polyhedron from its bihedral angles, it suffices to solve a system of algebraic equations. One of the ways to find an explicit representation of some Coxeter polyhedra is discussed in the subsection to follow. Discrete groups of motions of hyperbolic space Integer-valued automorphisms of hyperbolic quadratic forms

Fundamental polyhedron

Bianchi groups Integer-valued quadratic forms

2.9 Discrete Groups of Motions of Lobachevskian Space and Groups of Integer-Valued Automorphisms of Hyperbolic Quadratic Forms We consider an arbitrary integer-valued quadratic form of the signature (3,1). Over the field of real numbers this form is equivalent to the form -xij + XI + x~ + x~, and therefore the group of its automorphisms can be interpreted as the group of motions of a Lobachevskian space. The subgroup of integer-valued automorphisms is discrete, it contains a maximal reflection-generated subgroup. Vinberg [107] has found the algorithm for constructing the fundamental polyhedron of the maximal reflection-generated subgroup in an arbitrary discrete group of motions. This algorithm can be applied to groups of integer-valued automorphisms of hyperbolic quadratic forms. Shaikheev [108] has found a series of Coxeter polyhedra of a rather complicated combinatorial structure. M.K. Shaikheev has considered extended Bianchi groups which can be defined as the groups of integer-valued automorphisms of quadratic forms -2XJX2 +2x~ +2mx~, -2XJX2 +2xJ +2X3X4 +

m;J x~,

for m == 1,2(mod4) and for m == 3(mod4) ,

where m is a positive integer free of squares. He has shown that for m ~ 30 the fundamental polyhedron of this group has a finite volume if and only if m f 22, 23, 26, 29. This fundamental polyhedron has been explicity found in all the cases.

2.10 Reflection-Generated Discrete Groups

315

We shall give another example reported by E.B. Vinberg. It comes out of restriction of the quadratic form

to the hyperplane Xo + Xl + X2 + X3 = O. The fundamental polyhedron of the maximal reflection-generated subgroup in the group of integer-valued automorphisms of this form is combinatorially organized as a simplex with two cut-off vertices (Fig. A.2.19; the figure also presents its Coxeter scheme). The vectors defining this polyhedron are as follows: v) V3 Vs

= -e) +2e2 = -e4 ; = eo + 3e4 ;

e3; V2 V4 V6

Maximal reflection-generated subgroup

~ ,

= -e2 +e3 ; = eo +4e) -

2e2 - 2e3 ; = eo + 2e) - 2e3 + 2e4 ,

'0--