Papers on Topology: Analysis Situs and Its Five Supplements (History of Mathematics, 37)

John Stillwell was the recipient of the Chauvenet Prize for Mathematical Exposition in 2005. The papers in this book chronicle Henri Poincar???©'s journey in algebraic topology between 1892 and 1904, from his discovery of the fundamental group to his formulation of the Poincar???© conjecture. For the first time in English translation, one can follow every step (and occasional stumble) along the way, with the help of translator John Stillwell's introduction and editorial comments. Now that the Poincar???© conjecture has finally been proved, by Grigory Perelman, it seems timely to collect the papers that form the background to this famous conjecture. Poincar???©'s papers are in fact the first draft of algebraic topology, introducing its main subject matter (manifolds) and basic concepts (homotopy and homology). All mathematicians interested in topology and its history will enjoy this book. This volume is one of an informal sequence of works within the History of Mathematics series. Volumes in this subset, "Sources", are classical mathematical works that served as cornerstones for modern mathematical thought. The Papers In This Book Chronicle Henri Poincare's Journey In Algebraic Topology Between 1892 And 1904, From His Discovery Of The Fundamental Group To His Formulation Of The Poincare Conjecture. For The First Time In English Translation, One Can Follow Every Step (and Occasional Stumble) Along The Way, With The Help Of Translator John Stillwell's Introduction And Editorial Comments. Now That The Poincare Conjecture Has Finally Been Proved, By Grigory Perelman, It Seems Timely To Collect The Papers That From The Background To This Famous Conjecture. Poincare's Papers Are In Fact The First Draft Of Algebraic Topology, Introducing Its Main Subject Matter (manifolds) And Basic Concepts (homotopy And Homology). All Mathematicians Interested In Topology And Its History Will Enjoy This Book. These Famous Papers, With Their Characteristic Mixture Of Deep Insight And Inevitable Confusion, Are Here Presented Complete And In English For The First Time, With A Commentary By Their Translator, John Stillwell, That Guides The Reader Into The Beart Of The Subject. One Of The Finest Works Of One Of The Great Mathematicians Is Now Available Anew For Students And Experts Alike.--jeremy Gray. The Ams And John Stillwell Have Made An Important Contribution To The Mathematics Literature In This Translation Of Poincare. For Many Of Us, These Great Papers On The Foundations Of Topology Are Given Greater Clarity In English. Moreover, Reading Poincare Here Illustrates The Ultimate In Research By Successive Approximations (akin To My Own Way Of Mathematical Thinking)--stephen Smale. I Am A Proud Owner Of The Original Complete Works In Green Leather In French Bought For A Princely Sum In Paris Around 1975. I Have Read In Them Exten-sively, And Often During Topology Lectures I Refer To Parts Of These Works. I Am Happy That There Is Now The Option For My Students To Read Them In English--dennis Sullivan --book Jacket. Machine Generated Contents Note: On Analysis Situs -- Analysis Situs -- Introduction -- §1. First Definition Of Manifold -- §2. Homeomorphism -- §3. Second Definition Of Manifold -- §4. Oppositely Oriented Manifolds -- §5. Homologies -- §6. Betti Numbers -- §7. Use Of Integrals -- §8. Orientable And Nonorientable Manifolds -- §9. Intersection Of Two Manifolds -- §10. Geometric Representation -- §11. Representation By A Discontinuous Group -- §12. Fundamental Group -- §13. Fundamental Equivalences -- §14. Conditions For Homeomorphism -- §15. Other Modes Of Generation -- §16. Theorem Of Euler -- §17. Case Where P Is Odd -- §18. Second Proof -- Supplement To Analysis Situs -- §i. Introduction -- §ii. Schema Of A Polyhedron -- §iii. Reduced Betti Numbers -- §iv. Subdivision Of Polyhedra -- §v. Influence Of Subdivision On Reduced Betti Numbers -- §vi. Return To The Proofs Of Paragraph Iii -- §vii. Reciprocal Polyhedra -- §viii. Proof Of The Fundamental Theorem -- §ix. Various Remarks -- §x. Arithmetic Proof Of A Theorem Of Paragraph Vii -- §xi. Possibility Of Subdivision -- Second Supplement To Analysis Situs -- Introduction -- §1. Review Of The Principal Definitions -- §2. Reduction Of Tables -- §3. Comparison Of The Tables Tq And T'q -- §4. Application To Some Examples -- §5. Generalization Of A Theorem In The First Supplemnt -- §6. Internal Torsion Of Manifolds -- On Certain Algebraic Surfaces; Third Supplement To Analysis Situs -- Cycles On Algebraic Surfaces; Fourth Supplement To Analysis Situs -- §1. Introduction -- §2. Three-dimensional Cycles -- §3. Two-dimensional Cycles -- §4. One-dimensional Cycles -- §5. Various Remarks -- Fifth Supplement To Analysis Situs -- [§]1 -- [§]2 -- [§]3 -- [§]4 -- [§]5 -- [§]6. Henri Poincaré ; Translated By John Stillwell. Includes Bibliographical References And Index. These famous papers, with their characteristic mixture of deep insight and inevitable confusion, are here presented complete and in English for the first time, with a commentary by their translator, John Stillwell, that guides the reader into the heart of the subject. One of the finest works of one of the great mathematicians is now available anew for students and experts alike. —Jeremy Gray The AMS and John Stillwell have made an important contribution to the mathematics literature in this translation of Poincaré. For many of us, these great papers on the foundations of topology are given greater clarity in English. Moreover, reading Poincaré here illustrates the ultimate in research by successive approximations (akin to my own way of mathematical thinking). — Stephen Smale I am a proud owner of the original complete works in green leather in French bought for a princely sum in Paris around 1975. I have read them extensively, and often during topology lectures I refer to parts of these works. I am happy that there is now the option for my students to read them in English. —Dennis Sullivan The papers in this book chronicle Henri Poincaré's journey in algebraic topology between 1892 and 1904, from his discovery of the fundamental group to his formulation of the Poincaré conjecture. For the first time in English translation, one can follow every step (and occasional stumble) along the way, with the help of translator John Stillwell's introduction and editorial comments. Now that the Poincaré conjecture has finally been proved, by Grigory Perelman, it seems timely to collect the papers that form the background to this famous conjecture. Poincaré's papers are in fact the first draft of algebraic topology, introducing its main subject matter (manifolds) and basic concepts (homotopy and homology). All mathematicians interested in topology and its history will enjoy this book. This volume is one of an informal sequence of works within the History of Mathematics series. Volumes in this subset, “Sources”, are classical mathematical works that served as cornerstones for modern mathematical thought.