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Two-Dimensional Spaces, 2 AS 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 Subject Classification. Primary 57-01, 57M20.

For additional and updates on this book, visit www.ams.org/bookpages/mbk-109

Library of Congress Cataloging-in-Publication 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, . | Non-. | 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

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Preface to the Three Volume ix

Preface to Volume 2 xiii

Chapter 1. The Fundamental of 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 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 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 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 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. 75 7.1. Facts Assumed from Algebraic Topology 76 7.2. The Reduced of a 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- 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 129 15.1. Lines and Arcs in the Plane 129 CONTENTS vii

15.2. The Torus as a Euclidean 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 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 =,metry = ). 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 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 , algebra and theory, the development of with its elaboration in analysis and differential , our notions of , , and volume, motion, , 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 ? 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 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 and : areas by , by cut and paste, by counting, by slicing. Applications to the understanding of the real numbers, algebra, , and the development of calculus. Limitations imposed on the measurement of 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 and the hyperbolic trig functions? Why is non-Euclidean geometry called hyperbolic?Whatarethe gross intuitive differences between Euclidean and ? The approach to curvature is backwards to that of Gauss, with definitions that are obviously under bending, with the intent that curvature should obviously the degree to which a surface cannot be flattened into the plane. Gauss’s Theorema Egregium then comes at the 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. A student with the rudiments of topology—open and closed sets, continuous functions, compact sets and uniform continuity—should be able to understand almost all of the second volume with the exeption of a little bit of algebraic topology used to prove results that are intuitively reasonable and can be assumed if necessary. The final volume should be well within the reach of someone who is comfortable with integration and change of variables. We will make an attempt in many places to review the tools needed. Comments on exercises: Most exercises are interlaced with the text in those places where the development suggests them. They are an essential part of the text, and the reader should at least make note of their content. Exercise sections which appear at the end of most chapters refer back to these exercises, sometimes with hints, occasionally with solutions, and sometimes add additional exercises. Readers should try as many exercises as attract them, first without looking at hints or solutions. Comments on difficulty: Typically, sections and chapters become more diffi- cult toward the end. Don’t be afraid to quit a chapter when it becomes too difficult. Digest as much as interests you and move on to the next chapter or section. Comments on the bibliography: The book was written with very little direct reference to sources, and many of the proofs may therefore differ from the standard ones. But there are many wonderful books and wonderful teachers that we can learn from. I have therefore collected an annotated bibliography that you may want to explore. I particularly recommend [1,G.H.Hardy,A Mathematician’s Apology], [2,G.P´olya, How to Solve It], and [3,T.W.K¨orner, ThePleasureof Counting], just for fun, light reading. For a bit of hero worship, I also recommend the biographical references [21,E.T.Bell,Men of Mathematics], [22,C.Henrion, Women of Mathematics], and [23, W. Dunham, Journey Through Genius]. And I have to thank my particular heroes: my brother Larry, who taught me about uncountable sets, space-filling curves, and mathematical induction; Georg P´olya, who invited me into his home and showed me his mathematical notebooks; my ad- visor C. E. Burgess, who introduced me to the wonders of Texas-style mathematics; R. H. Bing, whose Sling, Dogbone Space, Hooked Rug, Baseball Move, epslums and PREFACETOTHETHREEVOLUMESET xi deltas, and Crumpled added color and wonder to the study of topology; and W. P. Thurston, who often made me feel like Gary Larson’s character of little brain (“Stop, professor, my brain is full.”) They were all kind and encouraging to me. And then there are those whom I only know from their writing: especially Euclid, , Gauss, Hilbert, and Poincar´e. Finally, I must thank Bill Floyd and Walter Parry for more than three decades of mathematical fun. When we would get together, we would work hard every morning, then talk mathematics for the rest of the day as we hiked the cities, coun- trysides, mountains, and woods of Utah, Virginia, Michigan, Minnesota, England, France, and any other place we could manage to get together. And special thanks to Bill for cleaning up and improving almost all of those figures in these books which he had not himself originally drawn.

Preface to Volume 2

The first of three volumes in this set was devoted to the measurement of lengths and areas, and to some of the consequences that study had in number theory, algebra, and analysis. Euclid was able to solve quadratic equations by geometric construction. But when mathematicians tried to extend those results to equations of higher degree and to differential equations, a number of fascinating difficulties arose, all involving limits and continuity, best modelled by topology. In this second volume we assume that the reader has had a first course in topology and is comfortable with open and closed sets, connected sets, compact sets, limits, and continuity. Two good references are W. S. Massey [25]andJ.R. Munkres [24]. The following discoveries led to the topics of this second volume. (1) The solution of cubic and quartic equations required serious consideration of complex numbers, thought at first to be mysterious. But the mystery disappeared when it was seen that complex numbers simply model the Euclidean plane. Abel and Galois proved that equations of degrees 5 and higher could not be solved in the relatively simple manner by formula as had sufficed in equations of degrees 1 through 4. But Gauss, without giving explicit solutions, managed to prove the Fundamental Theorem of Algebra that ensured that complex numbers sufficed for their solution. Gauss gave proofs involving the geometry and topology of the plane. (2) Newton showed that the study of motion could be greatly simplified if, in- stead of examining standard equations, one examined differential equations. Prov- ing the existence of solutions to rather general differential equations led to problems in topology. One of the standard proof techniques involves Brouwer’s Fixed Point Theorem. This volume proves that theorem in dimension 2 and outlines the proof in general dimensions. (3) Descartes demonstrated that mechanical devices other than straight edge and can construct curves of very high degree. Once curves of very general form are accepted as interesting, further delicate questions of length and area arise: finite curves of infinite length, finite curves of positive area, space filling curves, disks whose interiors have smaller areas than their closures, 0-dimensional sets through which no light rays can penetrate, continuous functions that are nowhere differentiable, sets of fractional dimension. This volume gives examples of many of these phenomena. (4) The study of solutions to equations became more unified when all variables were considered to be complex variables. Riemann modelled complex curves by surfaces, which are 2-dimensional manifolds and are called Riemann surfaces.The analysis of 2-dimensional manifolds led naturally to notions, such as triangulation, genus, and Euler characteristic. These notions are explained in this volume.

xiii xiv PREFACE TO VOLUME 2

All of these considerations required the study of limits and continuity, and the abstract notion that models limits and continuity in their most general settings is the notion of topology. Henri Poincar´ewrote: As for me, all of the diverse paths which I have successively followed have led me to topology. I have needed the gifts of this science to pursue my studies of the curves defined by differential equations and for the generalization to differential equations of higher order, and, in particular, to those of the three body problem. I have needed topology for the study of nonuniform functions of two variables. I have needed it for the study of the periods of multiple and for the application of that study to the expansion of perturbed functions. Finally, I have glimpsed in topology a means to attack an important problem in the theory of groups, the search for discrete or finite groups contained in a given continuous .

Bibliography

Plain Fun (top recommendations for easy, but rewarding, pleasure).

[1] Hardy, G. H., A Mathematician’s Apology, Cambridge University Press, 2004 (eighth print- ing). [2] P´olya, G., How to Solve It, Princeton Univerity Press, 2004. [3] K¨orner, T. W., The Pleasures of Counting, Cambridge University Press, 1996.

-MoreFun

[4] Davis, P. J. and Hersh, R., The Mathematical Experience, Houghton Mifflin Company, 1981. [5] Rademacher, H., Higher Mathematics from an Elementary Point of View,Birkh¨auser, 1983. [6] Hilbert, D., and Cohn-Vossen, S., Geometry and the Imagination, (translated by P. Nemeyi), Chelsea Publishing Company, New York, 1952. [College level exposition of rich ideas from low-dimensional geometry, with many figures.] [7] D¨orrie, H., 100 Great Problems of : Their History and Solution, Dover Publications, Inc., 1965, pp. 108-112. [We learned our first proof of the fundamental theorem of algebra here.] [8] Courant, R. and Robbins, H., What is Mathematics?, Oxford University Press, 1941.

Classics (a chance to see the thinking of the very best, in chronological order).

[9] Euclid, The Thirteen Books of Euclid’s Elements, Vol. 1-3, 2nd Ed., (edited by T. L. Heath) Cambridge University Press, Cambridge, 1926. [Reprinted by Dover, New York, 1956.] [10] Archimedes, The Works of Archimedes, edited by T. L. Heath, Dover Publications, In., Mineola, New York, 2002. See also the exposition in P´olya, G., Mathematics and Plausible Reasoning, Vol. 1. Induction and Analogy in Mathematics, Chapter IX. Physical Mathemat- ics, pp. 155-158, Princeton University Press, 1954. [How Archimedes discovered the calculus.] [11] Wallis, J., in A Source Book in Mathematics, 1200-1800,editedbyD.J.Struik.,Harvard University Press, 1969, pp. 244-253. [Wallis’s formula for π.] [12] Gauss, K. F., General Investigations of Curved Surfaces of 1827 and 1825, Princeton Uni- versity Library, 1902. [Available online. Difficult reading.] [13] Fourier, J., The Analytical Theory of Heat, translated by Alexander Freeman, Cambridge University Press, 1878. [Available online, 508 pages. The introduction explains Fourier’s thoughts in approaching the problem of the mathematical treatment of heat. Chapter 3 ex- plains his discovery of Fourier series.] [14] Riemann, B., Collected Papers, edited by Roger Baker, Kendrick Press, Heber City, Utah, 2004. [English translation of Riemann’s wonderful papers.]

159 160 BIBLIOGRAPHY

[15] Poincar´e, H., Science and Method, Dover Publications, Inc., 2003. [Discusses the role of the subconscious in mathematical discovery.]Also,The Value of Science,translatedbyG.B. Halstead, Dover Publications, Inc., 1958. [16] Klein, F., Vorlesungenuber ¨ Nicht-Euklidische Geometrie, Verlag von Julius Springer, Berlin, 1928. [In German. An algebraic development of non-Euclidean geometry with respect to the Klein and projective models. Beautiful figures. Elegant exposition.] [17] Hilbert, D., Gesammelte Abhandlungen (Collected Works), 3 volumes, Springer-Verlag, 1970. [In German. The transcendence of e and π appears in Volume 1, pp. 1-4. Hilbert’s space-filling curve appears in Volume 3, pp. 1-2.] [18] Einstein, A., The Meaning of Relativity, Princeton University Press, 1956. [19] Thurston, W. P., Three-Dimensional Geometry and Topology, edited by Silvio Levy, Prince- University Press, 1997. [An intuitive introduction to dimension 3 by the foremost ge- ometer of our generation.] [20] W. P. Thurston’s theorems on surface diffeomorphisms as exposited in Fathi, A., and Lau- denbach, F., and Po´enaru, V., Travaux de Thurston sur les Surfaces,S´eminaire Orsay, Soci´et´eMath´ematique de France, 1991/1979. [In French.]

History (concentrating on famous mathematicians).

[21] Bell, E. T., Men of Mathematics, Simon and Schuster, Inc., 1937. [The book that convinced me that mathematics is exciting and romantic.] [22] Henrion, C., Women of Mathematics, Indiana University Press, 1997. [23] Dunham, W., Journey Through Genius, Penguin Books, 1991.

Supporting Textbooks

- Topology

[24] Munkres, J. R., Topology, a First Course, Prentice-Hall, Inc., 1975. [The early chapters explain the basics of topology that form the prerequisites for the latter half of this book. The later chapters contain rather different views of some of the later theorems in our book.] [25] Massey, W. S., Algebraic Topology: An Introduction. Springer-Verlag, New York - Heidelberg-Berlin, 1967 (Sixth printing: 1984), Chapter I, pp. 1-54. [Aparticularlynice introduction to covering spaces.] [26] Hatcher, A., Algebraic Topology , Cambridge University Press, 2001. [A very nice introduc- tion to algebraic topology, a bit of which we need in Volume 2.] [27] Munkres, J. R., Elements of Algebraic Topology, Addison-Wesley, 1984. [Another nice in- troduction.] [28] Alexandroff, P., Elementary Concepts of Topology, translated by Alan E. Farley, Dover Publications, Inc., 1932. [A wonderful small book.] [29] Alexandrov, P. S., Combinatorial Topology, 3 volumes, translated by Horace Komm, Gray- lock Press, Rochester, NY, 1956. [30] Seifert, H., and Threlfall, W., A Textbook of Topology, translated by Michael A. Goldman; and Seifert, H., Topology of 3-Dimensional Fibered Spaces, translated by Wolfgang Heil, Academic Press, 1980. [Available online.] [31] Hurewicz, W., and Wallman, H., Dimension Theory, Princeton University Press, 1941. [See Chapters 4, 5, and 6 of Volume 2.] BIBLIOGRAPHY 161

-Algebra

[32] Herstein, I. N., Abstract Algebra, third edition, John Wiley & Sons, Inc., 1999. [See our Chapter 6 of Volume 1.] [33] Dummit, D. S., and Foote, R. M., Abstract Algebra, third edition, John Wiley & Sons, Inc., 2004. [See our Chapter 6 of Volume 1.] [34] Hardy, G. H., and Wright, E. M., An Introduction to the Theory of Numbers,fourth edition, Oxford University Press, 1960. [See our Chapter 5 of Volume 1.]

-Analysis

[35] Apostol, T. M., , Addison-Wesley, 1957. [Good background for Rie- mannian metrics in Chapter 1 of Volume 1, and also the chapters of Volume 3.] [36] Lang, S., Real and Functional Analysis, third edition, Springer, 1993. [Chapter XIV gives the differentiable version of the open mapping theorem. The proof uses the contraction mapping principle. See our Chapter 12 of Volume 2 for the topological version of the open mapping theorem.] [37] Spivak, M., Calculus on Manifolds, W. A. Benjamin, Inc., New York, N. Y., 1965. [Good background for Riemannian metrics in Chapter 1 of Volume 1 and for Volume 3.] [38] J¨anich, K., Vector Analysis, translated by Leslie Kay, Springer, 2001. [Good background for Riemannian metrics in Chapter 1 of Volume 1 and for Volume 3.] [39] Saks, S., Theory of the Integral, second revised edition, translated by L. C. Young, Dover Publications, Inc., New York, 1964. [Wonderfully readable.] [40] H. L. Royden, H. L., Real Analysis, third edition, Macmillan, 1988. [The place where we first learned about nonmeasurable sets.]

References from our Paper on Hyperbolic Geometry in Flavors of Geometry (reprinted here as our Volume 3, Chapter 2)

[41] Flavors of Geometry, edited by Silvio Levy, Cambridge University Press, 1997. [42] Alonso, J. M., Brady, T., Cooper, D., Ferlini, V., Lustig, M., Mihalik, M., Shapiro, M., Short, H., Notes on word hyperbolic groups, from a Geometrical Viewpoint: 21 March — 6 April 1990, ICTP, Trieste, Italy, (E. Ghys, A. Haefliger, and A. Verjovsky, eds.), World Scientific, Singapore, 1991, pp. 3–63. [43] Benedetti, R., and Petronio, C., Lectures on Hyperbolic Geometry, Universitext, Springer- Verlag, Berlin, 1992. [Expounds many of the facts about hyperbolic geometry outlined in Thurston’s influential notes.] [44] Bolyai, W., and Bolyai, J., Geometrische Untersuchungen, B. G. Teubner, Leipzig and Berlin, 1913. (reprinted by Johnson Reprint Corp., New York and London, 1972) [Historical and biographical materials.] [45] Cannon, J. W., The combinatorial structure of cocompact discrete hyperbolic groups,Geom. Dedicata 16 (1984), 123–148. [46] Cannon, J. W., The theory of negatively curved spaces and groups, Ergodic Theory, Symbolic Dynamics and Hyperbolic Spaces, (T. Bedford, M. Keane, and C. Series, eds.) Oxford University Press, Oxford and New York, 1991, pp. 315–369. [47] Cannon, J. W., The combinatorial Riemann mapping theorem, Acta Mathematica 173 (1994), 155–234. [48] Cannon, J. W., Floyd, W. J., Parry, W. R., Squaring : the finite Riemann mapping theorem, The Mathematical Heritage of Wilhelm Magnus — Groups, Geometry & Special 162 BIBLIOGRAPHY

Functions, Contemporary Mathematics 169, American Mathematics Society, Providence, 1994, pp. 133–212. [49] Cannon, J. W., Floyd, W. J., Parry, W. R., Sufficiently rich families of planar rings, preprint. [50] Cannon, J. W., Swenson, E. L., Recognizing constant curvature groups in dimension 3, preprint. [51] Coornaert, M., Delzant, T., Papadopoulos, A., Geometrie et theorie des groupes: les groupes hyperboliques de Gromov, Lecture Notes 1441, Springer-Verlag, Berlin-Heidelberg-NewYork, 1990. [52] Euclid, The Thirteen Books of Euclid’s Elements, Vol. 1-3, 2nd Ed.,(T.L.Heath,ed.) Cambridge University Press, Cambridge, 1926 (reprinted by Dover, New York, 1956). [53] Gabai, D., Homotopy hyperbolic 3-manifolds are virtually hyperbolic,J.Amer.Math.Soc. 7 (1994), 193–198. [54] Gabai, D., On the geometric and topological rigidity of hyperbolic 3-manifolds, Bull. Amer. Math. Soc. 31 (1994), 228–232. [55] Ghys, E., de la Harpe, P., Sur les groupes hyperboliques d’apr`es Mikhael Gromov, Progress in Mathematics 83, Birkh¨auser, Boston, 1990. [56] Gromov, M., Hyperbolic groups, Essays in Group Theory, (S. Gersten, ed.), MSRI Publi- cation 8, Springer-Verlag, New York, 1987. [Perhaps the most influential recent paper in geometric group theory.] [57] Hilbert, D., Cohn-Vossen, S., Geometry and the Imagination, Chelsea Publishing Company, New York, 1952. [College level exposition of rich ideas from low-dimensional geometry with many figures.] [58] Iversen, B., Hyperbolic Geometry, London Mathematical Society Student Texts 25, Cam- bridge University Press, Cambridge, 1993. [Very clean algebraic approach to hyperbolic ge- ometry.] [59] Klein, F., Vorlesungenuber ¨ Nicht-Euklidische Geometrie, Verlag von Julius Springer, Berlin, 1928. [Mostly algebraic development of non-Euclidean geometry with respect to Klein and projective models. Beautiful figures. Elegant exposition.] [60] Kline, M. Mathematical Thought from Ancient to Modern Times, Oxford University Press, New York, 1972. [A 3-volume . Full of interesting material.] [61] Lobatschefskij, N. I., Zwei Geometrische Abhandlungen, B. G. Teubner, Leipzig and Berlin, 1898. (reprinted by Johnson Reprint Corp., New York and London, 1972) [Original papers.] [62] Mosher, L., Geometry of cubulated 3-manifolds, Topology 34 (1995), 789–814. [63] Mosher, L., Oertel, U., Spaces which are not negatively curved,preprint. [64] Mostow, G. D., Strong Rigidity of Locally Symmetric Spaces, Annals of Mathematics Studies 78, Princeton University Press, Princeton, 1973. [65] Poincar´e, H., Science and Method, Dover Publications, New York, 1952. [One of Poincar´e’s several popular expositions of science. Still worth reading after almost 100 years.] [66] Ratcliffe, J. G., Foundations of Hyperbolic Manifolds, Graduate Texts in Mathematics 149, Springer-Verlag, New York, 1994. [Fantastic bibliography, careful and unified exposition.] [67] Riemann, B., Collected Papers, Kendrick Press, Heber City, Utah, 2004. [English translation of Riemann’s wonderful papers] [68] Swenson, E. L., Negatively curved groups and related topics, Ph.D. dissertation, Brigham Young University, 1993. [69] Thurston, W. P., The Geometry and Topology of 3-Manifolds, lecture notes, Princeton Uni- versity, Princeton, 1979. [Reintroduced hyperbolic geometry to the topologist. Very exciting and difficult.] [70] Weyl, H., Space—Time—, Dover, New York, 1922. [Weyl’s exposition and develop- ment of relativity and gauge theory which begins at the beginning with motivation, philoso- phy, and elementary developments as well as advanced theory.] BIBLIOGRAPHY 163

Further Technical References (arranged by chapter)

For the entirety of Volume 2

[71] Newman, M. H. A., Elements of the Topology of Plane Sets of Points, Cambridge University Press, 1939.[A good alternative introduction to the topology of the plane.]

- Volume 1, Chapter 1

[72] Feynman, R., The Character of Physical Law, The M.I.T. Press, 1989, p. 47.[All of Feynman’s writing is fun and thought provoking.]

- Volume 1, Chapter 2

[73] Gilbert, W. J., and Vanstone, S. A., An Introduction to Mathematical Thinking,Pearson Prentice Hall, 2005. [The place where I learned the algorithmic calculations about the Euclidean . See our Chapter 2.]

- Volume 1, Chapter 3

- Volume 1, Chapter 4

[74] Reid, C., Hilbert, Springer Verlag, 1970. [A wonderful biography of Hilbert, with an extended discussion of the Hilbert address in which he stated the Hilbert problems. See our Chapter 4.]

- Volume 1, Chapter 5

[75] Apostol , T. M., Calculus , Volume 1, Blaisdell Publishing Company, New York, 1961. [The place where I first learned areas by counting. See our Chapter 5.]

- Volume 1, Chapter 6

[76] Hilton, P., and Pedersen, J., Approximating any regular by folding paper,Math. Mag. 56 (1983), 141-155. [Method for approximating many algorithmically by paper- folding.] [77] Hilton, P., and Pedersen, J., Folding regular star and number theory Math. Intel- ligencer 7 (1985), 15-26. [More paper-folding.] [78] Burkard Polster, Variations on a Theme in Paper Folding, Amer. Math. Monthly 111 (2004), 39-47. [More paper-folding approximations to angles. See Chapter 6 and the impossibility of trisecting an .] 164 BIBLIOGRAPHY

- Volume 1, Chapter 7

[79] Wagon, S., The Banach-Tarski Paradox, Cambridge University Press, 1994.[A wonderful exposition of the Hausdorff-Banach-Tarski paradox, without the emphasis on the graph of the . See our Chapter 7.]

- Volume 1, Chapter 7; Volume 3, Chapter 1.

[80] Coxeter, H. S. M., and Moser, W. O., Generators and Relations for Discrete Groups, second edition, Springer-Verlag, 1964. [The place where I learned that groups can be viewed as graphs (the or the Dehn Gruppenbild). See our Chapter 7 where we use the graph of the free group on two generators and Chapter 25 where we use graphs as approximations to non Euclidean geometry.]

- Volume 2, Chapter 13

[81] Peano, G. , Sur une courbe, qui remplit toute une aire plane, Mathematische Annalen 36 (1), 1890, pp. 157-160. [The first space-filling curve, described algebraically. See our Chapter 12.] [82] Peano, G., Selected works of Giuseppe Peano, edited by Kennedy, Hubert C., and translated. With a biographical sketch and bibliography, Allen & Unwin, London, 1973. [83] Hilbert, D., Uber¨ die stetige Abbildung einer auf ein Fl¨achenst¨uck, Mathematische Annalen 38 (3), 1891, pp. 459-460. [Hilbert gave the first pictures of a space-filling curve. See our Chapter 12.] [84] G. P´olya, Uber¨ eine Peanosche Kurve, Bull. Acad. Sci. Cracovie, A, 1913, pp. 305-313. [P´olya’s -filling curve. See our Chapter 12.] [85] Lax, P. D., The differentiability of P´olya’s , Adv. Math., 10, 1973, pp. 456-464. [Lax recommends the non-isosceles triangle in P´olya’s construction since it simplifies the description of the path followed to the point represented by a binary expansion. See our Chapter 12.]

- Volume 2, Chapter 6

[86] Mandelbrot, B., The Geometry of Nature, W. H . Freeman & Co, 1982. [Mandelbrot suggests the use of Hausdorff dimension as a means of recognizing sets that are locally complicated or chaotic. He defines these to be . See our Chapter 13.] [87] Falconer, K. J., The Geometry of Fractal Sets, Cambridge University Press, 1985. [See reference [86] and our Chapter 13.] [88] Devaney, R. L., Differential Equations, Dynamical Systems, and an Introduction to Chaos with Morris Hirsch and , 2nd edition, Academic Press, 2004; 3rd edition, Academic Press, 2013. [See reference [84] and our Chapter 13.]

- Volume 2, Chapter 8 and 11

[89] Moore, R. L., Concerning upper semi-continuous collections of continua, Trans. Amer. Math. Sc. 27 (1925), pp. 416-428. [Moore shows that his topological characterization of the plane or 2-sphere allows him to prove his theorem about decompositions of the 2-sphere. See our Volume 2, Chapters 8 and 11.] BIBLIOGRAPHY 165

[90] Wilder, R. L., Topology of Manifolds, American Mathematical Society, 1949 . [Our proof of the topological characterization of the sphere is primarily modelled on Wilder’s proof, with what we consider to be conceptual simplifications. See our Chapter 8.]

- Volume 2, Chapter 13

[91] Rad´o, T., Uber¨ den Begriff der Riemannschen Fl¨ache, Acts. Litt. Sci. Szeged 2 (1925), pp. 101-121. [The first proof that 2-manifolds can be triangulated. See our Chapter 20.]

- Volume 2, Chapter 14

[92] Andrews, Peter, The classification of surfaces, Amer. Math. Monthly 95 (1988), 861-867l [93] Armstrong, M. A., Basic Topology, McGraw-Hill, London, 1979. [94] Burgess, C. E., Classification of surfaces, Amer. Math. Monthly 92 (1985), 349-354. [95] Francis, George K., Weeks, Jeffrey R., Conway’s ZIP proof, Amer. Math. Monthly 106 (1999), 393-399.

- Volume 2, Chapter 15

[96] Rolfsen, D., Knots and Links, AMS Chelsea, vol 346, 2003. [See our Chapter 22.]

For the entirety of Volume 3, see the references above taken from our article in Flavors of Geometry, beginning with reference [41].

- Volume 3, Chapter 3

[97] Misner, C. W., and Thorne, K. S., and Wheeler, J. A., Gravitation,W.H.Freemanand Company, 1973.

- Volume 3, Chapters 4 and 5

[98] Abelson, H., and diSessa, A., Turtle Geometry, MIT Press, 1986. [The authors use the paths of a computer turtle to model straight paths on a curved surface.] 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 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.

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