Differential Topology Bjørn Ian Dundas January 10, 2013 2 Contents 1 Preface 7 2 Introduction 9 2.1 Arobot’sarm: ................................. 9 2.2 The configuration space of two electrons . ..... 14 2.3 Statespacesandfiberbundles . .. 15 2.4 Furtherexamples ................................ 17 3 Smooth manifolds 25 3.1 Topologicalmanifolds. .. 25 3.2 Smoothstructures ............................... 28 3.3 Maximalatlases................................. 33 3.4 Smoothmaps .................................. 37 3.5 Submanifolds .................................. 41 3.6 Productsandsums ............................... 45 4 The tangent space 49 4.1 Germs ...................................... 51 4.2 Thetangentspace ............................... 57 4.3 Thecotangentspace .............................. 62 4.4 Derivations ................................... 67 5 Regular values 73 5.1 Therank..................................... 73 5.2 Theinversefunctiontheorem . .. 76 5.3 Theranktheorem................................ 77 5.4 Regularvalues.................................. 80 5.5 Transversality .................................. 87 5.6 Sard’stheorem ................................. 88 5.7 Immersionsandimbeddings . 90 6 Vector bundles 95 6.1 Topologicalvectorbundles . ... 96 3 4 CONTENTS 6.2 Transitionfunctions. 100 6.3 Smoothvectorbundles ............................. 102 6.4 Pre-vectorbundles ............................... 105 6.5 Thetangentbundle............................... 107 6.6 Thecotangentbundle.............................. 113 7 Constructions on vector bundles 117 7.1 Subbundles and restrictions . 117 7.2 Theinducedbundle............................... 122 7.3 Whitneysumofbundles ............................ 124 7.4 More general linear algebra on bundles . ..... 126 7.5 Normalbundles ................................. 131 7.6 Orientations................................... 133 7.7 ThegeneralizedGaussmap . 134 8 Integrability 137 8.1 Flowsandvelocityfields . 138 8.2 Integrability: compactcase. .... 143 8.3 Localflows.................................... 147 8.4 Integrability................................... 149 8.5 Second order differential equations . ..... 150 9 Local phenomena that go global 153 9.1 Refinementsofcovers.............................. 153 9.2 Partitionofunity ................................ 155 9.3 Riemannianstructures . 159 9.4 Normalbundles ................................. 161 9.5 Ehresmann’sfibrationtheorem. 164 10 Appendix: Point set topology 171 10.1 Topologies: openandclosedsets. ..... 172 10.2Continuousmaps ................................ 173 10.3 Basesfortopologies. 174 10.4Separation ................................... 174 10.5Subspaces .................................... 175 10.6Quotientspaces ................................ 176 10.7Compactspaces ................................ 177 10.8Productspaces ................................. 178 10.9Connectedspaces ................................ 178 10.10Settheoreticalstuff . 179 11 Hints or solutions to the exercises 181 References 204 CONTENTS 5 Index 205 6 CONTENTS Chapter 1 Preface 1.0.1 What version is this, and how stable is it? The version you are looking at right now is a β-release resulting from the major revision on Kistrand, Northern Norway in June 2012. The last stable manuscript: August 2007. If you have any comments or suggestion, I will be more than happy to hear from you so that the next stable release of these notes will be maximally helpful. The plan is to keep the text available on the net, also in the future, and I have occa- sionally allowed myself to provide links to interesting sites. If any of these links are dead, please inform me so that I can change them in the next edition. 1.0.2 Acknowledgments First and foremost, I am indebted to the students who have used these notes and given me invaluable feedback. Special thanks go to Håvard Berland, Elise Klaveness and Karen Sofie Ronæss. I owe a couple of anonymous referees much for their diligent reading and many helpful comments. I am also grateful to the Department of Mathematics for allowing me to do the 2012 revision in an inspiring environment. 1.0.3 The history of manifolds Although a fairly young branch of mathematics, the history behind the theory of manifolds is rich and fascinating. The reader should take the opportunity to check up some of the biographies at The MacTutor History of Mathematics archive or at the Wikipedia of the mathematicians that actually are mentioned by name in the text (I have occasionally pro- vided direct links). There is also a page called History Topics: Geometry and Topology Index which is worthwhile spending some time with. Of printed books, I have found Jean Dieudonné’s book [4] especially helpful (although it is mainly concerned with topics beyond the scope of this book). 7 8 CHAPTER 1. PREFACE 1.0.4 Notation We let N = 0, 1, 2,... , Z = ..., 1, 0, 1,... , Q, R and C be the sets of natural numbers, integers,{ rational} numbers,{ real− numbers} and complex numbers. If X and Y are two sets, X Y is the set of ordered pairs (x, y) with x an element in X and y an element in Y . If n is a natural× number, we let Rn and Cn be the vector space of ordered n-tuples of real n 2n n or complex numbers. Occasionally we may identify C with R . If x =(x1,...,xn) R , 2 2 n ∈ n+1 we let x be the norm x1 + + xn. The sphere of dimension n is the subset S R | | n+1··· 0 1⊆ of all x = (x0,...,xn)q R with x = 1 (so that S = 1, 1 R, and S can be viewed as all the complex∈ numbers e|iθ |of unit length). Given{− functions} ⊆ f : X Y and g : Y Z, we write gf for the composite, and g f only if the notation is cluttered→ and → ◦ the improves readability. The constellation g f will occur in the situation where f and g are◦ functions with the same source and target,· and where multiplication makes sense in the target. 1.0.5 How to start reading The text proper starts with chapter 3 on smooth manifolds. If you are weak on point set topology, you will probably want to read the appendix 10 in parallel with chapter 3. The introduction 2 is not strictly necessary for highly motivated readers who can not wait to get to the theory, but provides some informal examples and discussions that may put the theme of these notes in some perspective. You should also be aware of the fact that chapter 6 and 5 are largely independent, and apart from a few exercises can be read in any order. Also, at the cost of removing some exercises and examples, the section on derivations 4.4, the section on the cotangent space/bundle 4.3/6.6 can be removed from the curriculum without disrupting the logical development of ideas. Do the exercises, and only peek at the hints if you really need to. Kistranda January 10, 2013 Chapter 2 Introduction The earth is round. This may at one point have been hard to believe, but we have grown accustomed to it even though our everyday experience is that the earth is (fairly) flat. Still, the most effective way to illustrate it is by means of maps: a globe is a very neat device, but its global(!) A globe character makes it less than practical if you want to represent fine details. This phenomenon is quite common: locally you can represent things by means of “charts”, but the global character can’t be represented by one single chart. You need an entire atlas, and you need to know how the charts are to be assembled, or even better: the charts overlap so that we know how they all fit together. The mathematical framework for working with such situations is manifold theory. These notes are about manifold the- ory, but before we start off with the details, let us take an informal look at some examples illustrating the basic structure. 2.1 A robot’s arm: To illustrate a few points which will be important later on, we discuss a concrete situation in some detail. The features that appear are special cases of general phenomena, and hopefully the example will provide the reader with some deja vue experiences later on, when things are somewhat more obscure. Consider a robot’s arm. For simplicity, assume that it moves in the plane, has three joints, with a telescopic middle arm (see figure). 9 10 CHAPTER 2. INTRODUCTION y x z Call the vector defining the inner arm x, the second arm y and the third arm z. Assume x = z = 1 and y [1, 5]. Then the robot can reach anywhere inside a circle of radius 7.| | But| most| of these| | ∈ positions can be reached in several different ways. In order to control the robot optimally, we need to understand the various configura- tions, and how they relate to each other. As an example, place the robot in the origin and consider all the possible positions of the arm that reach the point P = (3, 0) R2, i.e., look at the set T of all triples (x, y, z) R2 R2 R2 such that ∈ ∈ × × x + y + z = (3, 0), x = z =1, and y [1, 5]. | | | | | | ∈ We see that, under the restriction x = z = 1, x and z can be chosen arbitrarily, and | | | | determine y uniquely. So T is “the same as” the set (x, z) R2 R2 x = z =1 . { ∈ × || | | | } Seemingly, our space T of configurations resides in four-dimensional space R2 R2 = R4, × ∼ but that is an illusion – the space is two-dimensional and turns out to be a familiar shape. We can parametrize x and z by angles if we remember to identify the angles 0 and 2π. So T is what you get if you consider the square [0, 2π] [0, 2π] and identify the edges as in × the picture below. A B B A 2.1. A ROBOT’S ARM: 11 See http://www.it.brighton.ac.uk/staff/jt40/MapleAnimations/Torus.html for a nice animation of how the plane model gets glued. In other words: The set T of all positions such that the robot reaches P = (3, 0) may be identified with the torus. This is also true topologically in the sense that “close configurations” of the robot’s arm correspond to points close to each other on the torus. 2.1.1 Question What would the space S of positions look like if the telescope got stuck at y = 2? | | Partial answer to the question: since y = (3, 0) x z we could try to get an idea of what points of T satisfy y = 2 by means of inspection− − of the graph of y .