Topology as Fluid Geometry: Two-Dimensional Spaces, Volume 2

This is the first of a three volume collection devoted to the geometry, topology, and curvature of 2-dimensional spaces. The collection provides a guided tour through a wide range of topics by one of the twentieth century's masters of geometric topology. The books are accessible to college and graduate students and provide perspective and insight to mathematicians at all levels who are interested in geometry and topology. The first volume begins with length measurement as dominated by the Pythagorean Theorem (three proofs) with application to number theory; areas measured by slicing and scaling, where Archimedes uses the physical weights and balances to calculate spherical volume and is led to the invention of calculus; areas by cut and paste, leading to the Bolyai-Gerwien theorem on squaring polygons; areas by counting, leading to the theory of continued fractions, the efficient rational approximation of real numbers, and Minkowski's theorem on convex bodies; straight-edge and compass constructions, giving complete proofs, including the transcendence of e and s, of the impossibility of squaring the circle, duplicating the cube, and trisecting the angle; and finally to a construction of the Hausdorff-Banach-Tarski paradox that shows some spherical sets are too complicated and cloudy to admit a well-defined notion of area. -- This is the second of a three volume collection devoted to the geometry, topology, and curvature of 2-dimensional spaces. The collection provides a guided tour through a wide range of topics by one of the twentieth century's masters of geometric topology. The books are accessible to college and graduate students and provide perspective and insight to mathematicians at all levels who are interested in geometry and topology. The second volume deals with the topology of 2-dimensional spaces. The attempts encountered in Volume 1 to understand length and area in the plane lead to examples most easily described by the methods of topology (fluid geometry): finite curves of infinite length, 1-dimensional curves of positive area, space-filling curves (Peano curves), 0-dimensional subsets of the plane through which no straight path can pass (Cantor sets), etc. Volume 2 describes such sets. All of the standard topological results about 2-dimensional spaces are then proved, such as the Fundamental Theorem of Algebra (two proofs), the No Retraction Theorem, the Brouwer Fixed Point Theorem, the Jordan Curve Theorem, the Open Mapping Theorem, the Riemann-Hurwitz Theorem, and the Classification Theorem for Compact 2-manifolds. Volume 2 also includes a number of theorems usually assumed without proof since their proofs are not readily available, for example, the Zippin Characterization Theorem for 2-dimensional spaces that are locally Euclidean, the Schoenflies Theorem characterizing the disk, the Triangulation Theorem for 2-manifolds, and the R. L. Moore's Decomposition Theorem so useful in understanding fractal sets. -- This is the final volume of a three volume collection devoted to the geometry, topology, and curvature of 2-dimensional spaces. The collection provides a guided tour through a wide range of topics by one of the twentieth century's masters of geometric topology. The books are accessible to college and graduate students and provide perspective and insight to mathematicians at all levels who are interested in geometry and topology. Einstein showed how to interpret gravity as the dynamic response to the curvature of space-time. Bill Thurston showed us that non-Euclidean geometries and curvature are essential to the understanding of low-dimensional spaces. This third and final volume aims to give the reader a firm intuitive understanding of these concepts in dimension 2. The volume first demonstrates a number of the most important properties of non-Euclidean geometry by means of simple infinite graphs that approximate that geometry. This is followed by a long chapter taken from lectures the author gave at MSRI, which explains a more classical view of hyperbolic non-Euclidean geometry in all dimensions. Finally, the author explains a natural intrinsic obstruction to flattening a triangulated polyhedral surface into the plane without distorting the constituent triangles. That obstruction extends intrinsically to smooth surfaces by approximation and is called curvature. Gauss's original definition of curvature is extrinsic rather than intrinsic. The final two chapters show that the book's intrinsic definition is equivalent to Gauss's extrinsic definition (Gauss's 2Theorema Egregium3 (2Great Theorem3)) Cover 1 Title page 4 Contents 6 Preface to the Three Volume Set 10 Preface to Volume 2 14 Chapter 1. The Fundamental Theorem of Algebra 16 1.1. Complex Arithmetic 17 1.2. First Proof of the Fundamental Theorem 20 1.3. Second Proof 22 1.4. Exercises 25 Chapter 2. The Brouwer Fixed Point Theorem 26 2.1. Statement of the Theorem 26 2.2. Introducing Extra Structure into a Problem 27 2.3. Two Elementary Problems 27 2.4. Three Advanced Problems 33 2.5. Exercises 42 Chapter 3. Tools 44 3.1. Polyhedral complexes 44 3.2. Urysohn’s Lemma and the Tietze Extension Theorem 46 3.3. Set Convergence 49 3.4. Exercises 51 Chapter 4. Lebesgue Covering Dimension 52 4.1. Definition of Covering Dimension 52 4.2. Euclidean n-Dimensional Space \Rn Has Covering Dimension n 53 4.3. Construction of Partitions of Unity 55 4.4. Techniques Needed in Higher Dimensions 56 4.5. Exercises 57 Chapter 5. Fat Curves and Peano Curves 60 5.1. The Constructions 60 5.2. The Topological Lemmas 64 5.3. The Analytical Lemmas 66 5.4. Characterization of Peano Curves 67 5.5. Exercises 69 Chapter 6. The Arc, the Simple Closed Curve, and the Cantor Set 72 6.1. Characterizing the Arc and Simple Closed Curve 72 6.2. The Cantor Set and Its Characterization 76 6.3. Interesting Cantor Sets 78 6.4. Cantor Sets in the Plane Are Tame 84 6.5. Exercises 88 Chapter 7. Algebraic Topology 90 7.1. Facts Assumed from Algebraic Topology 91 7.2. The Reduced Homology of a Sphere 92 7.3. The Homology of a Ball Complement 92 7.4. The Homology of a Sphere Complement 93 7.5. Proof of the Arc Non-Separation Theorem and the Jordan Curve Theorem 94 Chapter 8. Characterization of the 2-Sphere 96 8.1. Statement and Proof of the Characterization Theorem 96 8.2. Exercises 104 Chapter 9. 2-Manifolds 106 9.1. Definition and Examples 106 9.2. Exercises 106 Chapter 10. Arcs in \St Are Tame 110 10.1. Arcs in \St Are Tame 110 10.2. Disk Isotopies 112 10.3. Exercises 115 Chapter 11. R. L. Moore’s Decomposition Theorem 116 11.1. Examples and Applications 116 11.2. Decomposition Spaces 117 11.3. Proof of the Moore Decomposition Theorem 119 11.4. Exercises 122 Chapter 12. The Open Mapping Theorem 124 12.1. Tools 124 12.2. Two Lemmas 125 12.3. Proof of the Open Mapping Theorem 127 12.4. Exercise 127 Chapter 13. Triangulation of 2-Manifolds 128 13.1. Statement of the Triangulation Theorem 128 13.2. Tools 128 13.3. Proof of the Triangulation Theorem 130 13.4. Exercises 131 Chapter 14. Structure and Classification of 2-Manifolds 132 14.1. Statement of the Structure Theorem 132 14.2. Edge-pairings 133 14.3. Proof of the Structure Theorem 134 14.4. Statement and Proof of the Classification Theorem 138 14.5. Exercises 140 Chapter 15. The Torus 144 15.1. Lines and Arcs in the Plane 144 15.2. The Torus as a Euclidean Surface 146 15.3. Curve Straightening 149 15.4. Construction of the Simple Closed Curve with Slope k/l 151 15.5. Exercises 152 Chapter 16. Orientation and Euler Characteristic 154 16.1. Orientation 154 16.2. Euler Characteristic 156 16.3. Exercises 164 Chapter 17. The Riemann-Hurwitz Theorem 166 17.1. Setting 166 17.2. Elementary Facts from Trigonometry 166 17.3. Branched Maps of \St 169 17.4. Statement of the Riemann-Hurwitz Theorem 170 17.5. Proof of the Riemann-Hurwitz Theorem 170 17.6. Rational Maps 171 17.7. Exercises 172 Bibliography 174 Back Cover 181