Topology MATH-GA 2310 and MATH-GA 2320
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Topology MATH-GA 2310 and MATH-GA 2320 Sylvain Cappell Transcribed by Patrick Lin Figures transcribed by Ben Kraines Abstract. These notes are from a two-semester introductory sequence in Topology at the graduate level, as offered in the Fall 2013{Spring 2014 school year at the Courant Institute of Mathematical Sciences, a school of New York University. The primary lecturer for the course was Sylvain Cappell. Three lectures were given by Edward Miller during the Fall semester. Course Topics: Point-Set Topology (Metric spaces, Topological spaces). Homotopy (Fundamental Group, Covering Spaces). Manifolds (Smooth Maps, Degree of Maps). Homology (Cellular, Simplicial, Singular, Axiomatic) with Applications, Cohomology. Parts I and II were covered in MATH-GA 2310 Topology I; and Parts III and IV were covered in MATH-GA 2320 Topology II. The notes were transcribed live (with minor modifications) in LATEX by Patrick Lin. Ben Kraines provided the diagrams from his notes for the course. These notes are in a draft state, and thus there are likely many errors and inconsistencies. These are corrected as they are found. Revision: 21 Apr 2016 15:29. Contents Chapter 0. Introduction 1 Part I. Point-Set Topology 5 Chapter 1. Topological Spaces 7 1.1. Sets and Functions 7 1.2. Topological Spaces 8 1.3. Metric Spaces 8 1.4. Constructing Topologies from Existing Ones 9 Chapter 2. Properties of Topological Spaces 13 2.1. Continuity and Compactness 13 2.2. Hausdorff Spaces 15 2.3. Connectedness 15 Part II. The Fundamental Group 17 Chapter 3. Basic Notions of Homotopy 19 3.1. Homotopy of Paths 19 3.2. The Fundamental Group 21 3.3. Excursion: Basic Notions of Group Theory 23 3.4. Maps Between Spaces and Induced Homomorphisms 24 3.5. Homotopy of Maps and Spaces 25 3.6. Retractions 26 Chapter 4. Comparing Fundamental Groups of Spaces 29 4.1. Fundamental Groups of Product Spaces 29 4.2. Fundamental Group of Glued Spaces 30 4.3. Excursion: Combinatorial Group Theory 31 4.4. Van Kampen's Theorem 36 4.5. An Example from Knot Theory 37 Chapter 5. Covering Spaces 43 5.1. Covering Spaces 43 5.2. Covering Translations 46 5.3. Covering Spaces of Graphs and an Application to Algebra 47 5.4. Regular Covering Spaces 49 5.5. Construction of Universal Cover (of a path-connected space) 49 5.6. Generalization to Other Covering Spaces 51 5.7. Group Actions on Spaces 53 iii iv CONTENTS Part III. Manifolds 55 Chapter 6. Differentiable Manifolds and Smooth Functions 57 6.1. Topological and Differentiable Manifolds 57 6.2. Smooth Functions 59 6.3. Equivalence of Differentiable Manifolds 60 6.4. Excursion: Basic Facts from Analysis 60 Chapter 7. Tangent Spaces and Vector Bundles 63 7.1. Tangent Spaces 63 7.2. Vector Bundles 65 7.3. Derivatives over Manifolds 66 7.4. Manifolds with Boundary 68 Chapter 8. Degree of Maps 71 8.1. Degree of Maps (modulo 2) 71 8.2. Orientation 74 8.3. Degree of Maps for Oriented Manifolds 77 8.4. Applications 78 Part IV. Homology 83 Chapter 9. Homology Groups 85 9.1. Cellular Homology 85 9.2. Simplicial Homology 90 9.3. Singular Homology 92 Chapter 10. Comparing Homology Groups of Spaces 95 10.1. Induced Homomorphisms 95 10.2. Exact Sequences 98 10.3. Relative Homology 102 Chapter 11. A Discussion of the Axiomatic View of Homology 105 11.1. Axioms of Homology 105 11.2. H0 and H1 107 11.3. The Homotopy Axiom 110 Chapter 12. Miscellaneous Topics 113 12.1. Application: Lefschetz Fixed Point Theorem 113 12.2. The Mayer-Vietoris Theorem 116 12.3. Homology of a Product Space 116 12.4. Cohomology 118 12.5. Eilenberg Obstruction Theory 119 Index 121 CHAPTER 0 Introduction Before we begin to go through the theorems and proofs, we will go through, as an introduction, some problems and examples of Topology. We will start with the idea of a space. We won't define a topological space yet, but for now we can look at some examples: n Example 0.1. Euclidean spaces X ⊂ R = f(x1; : : : ; xn) j xi 2 Rg. We have the usual notion of Euclidean distance between two points in the space: 1 n ! 2 X 2 d ((x1; : : : ; xn); (y1; : : : ; yn)) = (xi − yi) : i=1 n n Example 0.2. The n-dimensional disk D = fv 2 R j kvk 6 1g. Similarly, the n-dimensional open disk D˚n = fv 2 Rn j kvk < 1g, and the (n − 1)-dimensional sphere Sn−1 = fv 2 Rn j kvk = 1g. Remark 0.3. S0 is comprised of the points ±1 on a line. We will also use the idea of continuity; we won't define this just yet, but for now we can perhaps talk about continuity using the ideas of Advanced Calculus. We now look at some classic questions in Topology. This is a classical questions that arises in many areas of mathematics: Question 0.4. Given a function f : X ! X, a point u 2 X is called a fixed point of f if f(u) = u. What can we say about whether or not f has fixed points? Example 0.5. If we have v 2 Rn, f : Rn ! Rn given by f(x) = x + v, we see that f has no fixed points if v 6= 0. On the other hand, if v = 0, then every x is fixed. This is an extreme example. Example 0.6. Let f : Rn ! Rn be a rotation around the origin. Then there is exactly one fixed point: the origin. Many other questions can, using a bit of cleverness, be turned into a question about having fixed points. As a result, there is a vast amount of literature dealing with this question. These results have had enormous impacts in many fields of mathematics. We will look at some famous examples of spaces and fixed points. Theorem 0.7 (Brouwer Fixed-Point Theorem). For any continuous function f : Dn ! Dn there exists a fixed point. That is a very strong and remarkably general statement. The analogue fails for Euclidean space, for open disks and for spheres. 1 2 0. INTRODUCTION Example 0.8. For open disks, push everything to the right, but dampen the effect as we get closer and closer to the right side. As a concrete example, on the open interval D1 = (−1; 1) take f(x) = (x + 1)=2. Example 0.9. For X = Sn, consider α(v) = −v. This is the antipodal map. Remark 0.10. Obviously the antipodal map does not work for X = Dn, as the origin remains fixed. However, there is good theory (by Hadamard) about which functions f : Sn ! Sn must have fixed points. We will now attempt a sketch of the proof for the Brouwer Fixed-Point Theo- rem. We will get stuck, because we have not yet developed some of the machinery needed yet (that we will later in the course). But we will reach a point that is hopefully somewhat intuitive. Proof-Sketch. We will use a proof by contradiction. Let f : Dn ! Dn. Assume that for all x 2 Dn, f(x) 6= x. Draw the line from f(x) through x to the edge Sn−1. Call g(x) the point where the line meets Sn−1. Thus we have defined the function g : Dn ! Sn−1. g(x)• • x •f(x) Figure 0.11. It is not hard to check that g is continuous. Note that if x 2 Sn−1, then x = g(x), so g does not move points on Sn−1. Consider the special case of n = 1. Then g sends values from [−1; 1] to {±1g, contradicts the Intermediate Value Theorem in Calculus, so g cannot be continuous. For n > 1, the story gets more complicated; we need further methods. As a way of looking at it intuitively, though, consider what the function g does: since it pulls the disk to its edge, it will need to rip it somewhere; as a result, g must be discontinuous somewhere. To show this rigorously, we will need to develop ways of measuring \holes" in n dimensions. Remark 0.12. This does not show how to find the fixed point (this is a con- sequence of argument by contradiction). Remark 0.13. Instead of focusing on f, we turned our attention to g. This is a nice feature of this argument: we turned from a question about points to one about the existence of continous maps between spaces. We won't get to this for weeks, but a basic strategy in Algebraic Topology is to turn questions about spaces and continuous maps between them into problems in Algebra, such as problems about groups/vector spaces and homomorphisms/linear maps. 0. INTRODUCTION 3 Another classic problem in Topology is one of classification: Question 0.14. Can we come up with a theory of classification for spaces? Well, there are so many spaces that we can't really classify them all, but we would very much like to classify all \nice" spaces, for example manifolds. These are spaces which are everywhere locally modeled on the same Euclidean space Rn. Example 0.15. S2, which everywhere locally looks like R2. Remark 0.16. The only 1-dimensional manifolds are S1 and D1, and combi- nations thereof. Example 0.17. An example of a 2-dimensional manifold is the torus (or donut), which has one hole. Figure 0.18. Surfaces similar to the torus but with more holes are also 2-dimensional mani- folds.