Two-Dimensional Spaces, Volume 2 TOPOLOGY AS FLUID GEOMETRY James W. Cannon 10.1090/mbk/109 Two-Dimensional Spaces, Volume 2 TOPOLOGY AS FLUID GEOMETRY Two-Dimensional Spaces, Volume 2 TOPOLOGY AS FLUID GEOMETRY James W. Cannon AMERICAN MATHEMATICAL SOCIETY Providence, Rhode Island 2010 Mathematics Subject Classification. Primary 57-01, 57M20. For additional information and updates on this book, visit www.ams.org/bookpages/mbk-109 Library of Congress Cataloging-in-Publication Data Names: Cannon, James W., author. Title: Two-dimensional spaces / James W. Cannon. Description: Providence, Rhode Island : American Mathematical Society, [2017] | Includes bibli- ographical references. Identifiers: LCCN 2017024690 | ISBN 9781470437145 (v. 1) | ISBN 9781470437152 (v. 2) | ISBN 9781470437169 (v. 3) Subjects: LCSH: Geometry. | Geometry, Plane. | Non-Euclidean geometry. | AMS: Geometry – Instructional exposition (textbooks, tutorial papers, etc.). msc Classification: LCC QA445 .C27 2017 | DDC 516–dc23 LC record available at https://lccn.loc.gov/2017024690 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy select pages for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. 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Visit the AMS home page at http://www.ams.org/ 10987654321 222120191817 Contents Preface to the Three Volume Set ix Preface to Volume 2 xiii Chapter 1. The Fundamental Theorem of Algebra 1 1.1. Complex Arithmetic 2 1.2. First Proof of the Fundamental Theorem 5 1.3. Second Proof 7 1.4. Exercises 10 Chapter 2. The Brouwer Fixed Point Theorem 11 2.1. Statement of the Theorem 11 2.2. Introducing Extra Structure into a Problem 12 2.3. Two Elementary Problems 12 2.4. Three Advanced Problems 18 2.5. Exercises 27 Chapter 3. Tools 29 3.1. Polyhedral complexes 29 3.2. Urysohn’s Lemma and the Tietze Extension Theorem 31 3.3. Set Convergence 34 3.4. Exercises 36 Chapter 4. Lebesgue Covering Dimension 37 4.1. Definition of Covering Dimension 37 4.2. Euclidean n-Dimensional Space Rn Has Covering Dimension n 38 4.3. Construction of Partitions of Unity 40 4.4. Techniques Needed in Higher Dimensions 41 4.5. Exercises 42 Chapter 5. Fat Curves and Peano Curves 45 5.1. The Constructions 45 5.2. The Topological Lemmas 49 5.3. The Analytical Lemmas 51 5.4. Characterization of Peano Curves 52 5.5. Exercises 54 Chapter 6. The Arc, the Simple Closed Curve, and the Cantor Set 57 6.1. Characterizing the Arc and Simple Closed Curve 57 6.2. The Cantor Set and Its Characterization 61 6.3. Interesting Cantor Sets 63 v vi CONTENTS 6.4. Cantor Sets in the Plane Are Tame 69 6.5. Exercises 73 Chapter 7. Algebraic Topology 75 7.1. Facts Assumed from Algebraic Topology 76 7.2. The Reduced Homology of a Sphere 77 7.3. The Homology of a Ball Complement 77 7.4. The Homology of a Sphere Complement 78 7.5. Proof of the Arc Non-Separation Theorem and the Jordan Curve Theorem 79 Chapter 8. Characterization of the 2-Sphere 81 8.1. Statement and Proof of the Characterization Theorem 81 8.2. Exercises 89 Chapter 9. 2-Manifolds 91 9.1. Definition and Examples 91 9.2. Exercises 91 Chapter 10. Arcs in S2 Are Tame 95 10.1. Arcs in S2 Are Tame 95 10.2. Disk Isotopies 97 10.3. Exercises 100 Chapter 11. R. L. Moore’s Decomposition Theorem 101 11.1. Examples and Applications 101 11.2. Decomposition Spaces 102 11.3. Proof of the Moore Decomposition Theorem 104 11.4. Exercises 107 Chapter 12. The Open Mapping Theorem 109 12.1. Tools 109 12.2. Two Lemmas 110 12.3. Proof of the Open Mapping Theorem 112 12.4. Exercise 112 Chapter 13. Triangulation of 2-Manifolds 113 13.1. Statement of the Triangulation Theorem 113 13.2. Tools 113 13.3. Proof of the Triangulation Theorem 115 13.4. Exercises 116 Chapter 14. Structure and Classification of 2-Manifolds 117 14.1. Statement of the Structure Theorem 117 14.2. Edge-pairings 118 14.3. Proof of the Structure Theorem 119 14.4. Statement and Proof of the Classification Theorem 123 14.5. Exercises 125 Chapter 15. The Torus 129 15.1. Lines and Arcs in the Plane 129 CONTENTS vii 15.2. The Torus as a Euclidean Surface 131 15.3. Curve Straightening 134 15.4. Construction of the Simple Closed Curve with Slope k/ 136 15.5. Exercises 137 Chapter 16. Orientation and Euler Characteristic 139 16.1. Orientation 139 16.2. Euler Characteristic 141 16.3. Exercises 149 Chapter 17. The Riemann-Hurwitz Theorem 151 17.1. Setting 151 17.2. Elementary Facts from Trigonometry 151 17.3. Branched Maps of S2 154 17.4. Statement of the Riemann-Hurwitz Theorem 155 17.5. Proof of the Riemann-Hurwitz Theorem 155 17.6. Rational Maps 156 17.7. Exercises 157 Bibliography 159 Preface to the Three Volume Set Geometry measures space (geo =earth,metry = measurement). Einstein’s theory of relativity measures space-time and might be called geochronometry (geo = space, chrono=time,metry = measurement). The arc of mathematical history that has led us from the geometry of the plane of Euclid and the Greeks after 2500 years to the physics of space-time of Einstein is an attractive mathematical story. Geometrical reasoning has proved instrumental in our understanding of the real and complex numbers, algebra and number theory, the development of calculus with its elaboration in analysis and differential equations, our notions of length, area, and volume, motion, symmetry, topology, and curvature. These three volumes form a very personal excursion through those parts of the mathematics of 1- and 2-dimensional geometry that I have found magical. In all cases, this point of view is the one most meaningful to me. Every section is designed around results that, as a student, I found interesting in themselves and not just as preparation for something to come later. Where is the magic? Why are these things true? Where is the tension? Every good theorem should have tension between hypothesis and conclusion. — Dennis Sullivan Where is the Sullivan tension in the statement and proofs of the theorems? What are the key ideas? Why is the given proof natural? Are the theorems almost false? Is there a nice picture? I am not interested in quoting results without proof. I am not afraid of a little algebra, or calculus, or linear algebra. I do not care about complete rigor. I want to understand. If every formula in a book cuts the readership in half, my audience is a small, elite audience. This book is for the student who likes the magic and wants to understand. A scientist is someone who is always a child, asking ‘Why? why? why?’. — Isidor Isaac Rabi, Nobel Prize in Physics 1944 Wir m¨ussen wissen, wir werden wissen. [We must know, we will know.] — David Hilbert The three volumes indicate three natural parts into which the material on 2- dimensional spaces may be divided: Volume 1: The geometry of the plane, with various historical attempts to understand lengths and areas: areas by similarity, by cut and paste, by counting, by slicing. Applications to the understanding of the real numbers, algebra, number theory, and the development of calculus. Limitations imposed on the measurement of size given by nonmeasurable sets and the wonderful Hausdorff-Banach-Tarski paradox. ix x PREFACE TO THE THREE VOLUME SET Volume 2: The topology of the plane, with all of the standard theorems of 1 and 2-dimensional topology, the Fundamental Theorem of Algebra, the Brouwer Fixed-Point Theorem, space-filling curves, curves of positive area, the Jordan Curve Theorem, the topological characterization of the plane, the Schoenflies Theorem, the R. L. Moore Decomposition Theorem, the Open Mapping Theorem, the trian- gulation of 2-manifolds, the classification of 2-manifolds via orientation and Euler characteristic, dimension theory. Volume 3: An introduction to non-Euclidean geometry and curvature. What is the analogy between the standard trigonometric functions and the hyperbolic trig functions? Why is non-Euclidean geometry called hyperbolic?Whatarethe gross intuitive differences between Euclidean and hyperbolic geometry? The approach to curvature is backwards to that of Gauss, with definitions that are obviously invariant under bending, with the intent that curvature should obviously measure the degree to which a surface cannot be flattened into the plane. Gauss’s Theorema Egregium then comes at the end of the discussion. Prerequisites: An undergraduate student with a reasonable memory of cal- culus and linear algebra, but with no fear of proofs, should be able to understand almost all of the first volume.