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Papers on Topology Analysis Situs and Its Five Supplements Henri Poincaré Translated by John Stillwell American Mathematical Society London Mathematical Society Papers on Topology Analysis Situs and Its Five Supplements https://doi.org/10.1090/hmath/037 • SOURCES Volume 37 Papers on Topology Analysis Situs and Its Five Supplements Henri Poincaré Translated by John Stillwell Providence, Rhode Island London, England Editorial Board American Mathematical Society London Mathematical Society Joseph W. Dauben Jeremy J. Gray Peter Duren June Barrow-Green Robin Hartshorne Tony Mann, Chair Karen Parshall, Chair Edmund Robertson 2000 Mathematics Subject Classification. Primary 01–XX, 55–XX, 57–XX. Front cover photograph courtesy of Laboratoire d’Histoire des Sciences et de Philosophie—Archives Henri Poincar´e(CNRS—Nancy-Universit´e), http://poincare.univ-nancy2.fr. For additional information and updates on this book, visit www.ams.org/bookpages/hmath-37 Library of Congress Cataloging-in-Publication Data Poincar´e, Henri, 1854–1912. Papers on topology : analysis situs and its five supplements / Henri Poincar´e ; translated by John Stillwell. p. cm. — (History of mathematics ; v. 37) Includes bibliographical references and index. ISBN 978-0-8218-5234-7 (alk. paper) 1. Algebraic topology. I. Title. QA612 .P65 2010 514.2—22 2010014958 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 a chapter 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. Requests for such permission should be addressed to the Acquisitions Department, American Mathematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294 USA. Requests can also be made by e-mail to [email protected]. c 2010 by the American Mathematical Society. All rights reserved. Printed in the United States of America. The American Mathematical Society retains all rights except those granted to the United States Government. ∞ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. The London Mathematical Society is incorporated under Royal Charter and is registered with the Charity Commissioners. Visit the AMS home page at http://www.ams.org/ 10 9 8 7 6 5 4 3 2 1 15, 14, 13, 12, 11, 10 Contents Translator’s Introduction vii Topology before Poincar´evii Poincar´e before topology viii The Analysis situs paper x The five supplements xii The Poincar´e conjecture xiv Comments on terminology and notation xv Acknowledgements xvi Translator’s Bibliography xix On Analysis Situs 1 Analysis Situs 5 Introduction 5 §1. First definition of manifold 7 §2. Homeomorphism 9 §3. Second definition of manifold 10 §4. Oppositely oriented manifolds 13 §5. Homologies 15 §6. Betti numbers 15 §7. The use of integrals 16 §8. Orientable and nonorientable manifolds 19 §9. Intersection of two manifolds 23 §10. Geometric representation 31 §11. Representation by a discontinuous group 36 §12. The fundamental group 38 §13. Fundamental equivalences 41 §14. Conditions for homeomorphism 44 §15. Other modes of generation 53 §16. The theorem of Euler 61 §17. The case where p is odd 68 §18. Second proof 70 Supplement to Analysis Situs 75 §I. Introduction 75 §II. Schema of a polyhedron 78 §III. Reduced Betti numbers 81 §IV. Subdivision of polyhedra 84 §V. Influence of subdivision on reduced Betti numbers 85 v vi CONTENTS §VI. Return to the proofs of paragraph III 89 §VII. Reciprocal polyhedra 92 §VIII. Proof of the fundamental theorem 96 §IX. Various remarks 100 §X. Arithmetic proof of a theorem of paragraph VII 103 §XI. The possibility of subdivision 105 Second Supplement to Analysis Situs 111 Introduction. 111 §1. Review of the principal definitions 111 §2. Reduction of tables 114 § 3. Comparison of the tables Tq and Tq 117 §4. Application to some examples 120 §5. Generalization of a theorem in the first supplement 125 §6. Internal torsion of manifolds 131 On Certain Algebraic Surfaces; Third Supplement to Analysis Situs 135 Cycles on Algebraic Surfaces; Fourth Supplement to Analysis Situs 151 §1. Introduction 151 §2. Three-dimensional cycles 155 §3. Two-dimensional cycles 162 §4. One-dimensional cycles 169 §5. Various remarks 172 Fifth Supplement to Analysis Situs 179 §1 179 §2 179 §3 187 §4 200 §5 207 §6 215 Index 225 Translator’s Introduction Topology before Poincar´e Without much exaggeration, it can be said that only one important topological concept came to light before Poincar´e.This was the Euler characteristic of surfaces, whose name stems from the paper of Euler (1752) on what we now call the Euler polyhedron formula. When writing in English, one usually expresses the formula as V − E + F =2, where V = number of vertices, E = number of edges, F = number of faces, of a convex polyhedron or, more generally, of a subdivided surface homeomorphic to the two-dimensional sphere S2. In Poincar´e (for example, in §16 of his Analysis situs paper) one finds the French version S − A + F =2, where S stands sommets and A for arˆetes. The formula is usually proved by showing that the quantity V − E + F remains invariant under all possible changes from one subdivision to another. It follows that V − E + F is an invariant of any surface, not necessarily homeomorphic to S2.This invariant is now called the Euler characteristic.ThusS2 has Euler characteristic 2, whereas the torus has Euler characteristic 0. Between the 1820s and 1880s, several different lines of research were found to converge to the Euler characteristic. (1) The classification of polyhedra, following Euler (and even before him, Descartes). Here the terms edges and faces have their traditional meaning in Euclidean geometry. (2) The classification of surfaces of constant curvature, where the “edges” are now geodesic segments. Here one finds that surfaces of positive Euler char- acteristic have positive curvature, those of zero Euler characteristic have zero curvature, and those of negative Euler characteristic have negative curvature. (3) More generally, the average curvature of a smooth surface is positive for positive Euler characteristic, zero for zero Euler characteristic, and nega- tive for negative Euler characteristic. This follows from the Gauss–Bonnet theorem of Gauss (1827) and Bonnet (1848). (4) The study of algebraic curves, revolutionized by Riemann (1851) when he modelled each complex algebraic curve by a surface—its Riemann surface. vii viii TRANSLATOR’S INTRODUCTION Under this interpretation, a number that Abel (1841) called the genus of an algebraic curve turns out to depend on the Euler characteristic of its Riemann surface. Genus g is related to Euler characteristic χ by χ =2− 2g. Moreover, the genus g has a simple geometric interpretation as the number of holes in the surface. Thus S2 has genus 0 and the torus has genus 1. (5) The topological classification of surfaces, by M¨obius (1863). M¨obius stud- ied closed surfaces in R3 by slicing them into simple pieces by parallel planes. By this method he found that every such surface is homeomor- phic to a standard surface with g holes. Thus closed surfaces in R3—that is, all orientable surfaces—are classified by their genus, and hence by their Euler characteristic. (Despite his discovery of the nonorientable surface that bears his name, M¨obiusdid not classify nonorientable surfaces. This was done by Dyck (1888).) (6) The study of pits, peaks, and passes on surfaces in R3 by Cayley (1859) and Maxwell (1870). A family of parallel planes in R3 intersects a surface S in curves we may view as curves of constant height (contour lines) on S. If the planes are taken to be in general position, and the surface is smooth, then S has only finitely many pits, peaks, and passes relative to the height function. It turns out that number of peaks − number of passes + number of pits is precisely the Euler characteristic of S. All of these ideas admit generalizations to higher dimensions, but the only sub- stantial step towards topology in arbitrary dimensions before Poincar´ewas that of Betti (1871). Betti was inspired by Riemann’s concept of connectivity of surfaces to define connectivity numbers, now known as Betti numbers P1,P2,..., in all di- mensions. The connectivity number of a surface S may be defined as the maximum number of disjoint closed curves that can be drawn on S without separating it. This number P1 is equal to the genus of S, hence it is just the Euler characteristic in disguise. For a three-dimensional manifold M one can also consider the maximum num- ber P2 of disjoint closed surfaces in M that fail to separate M as the two-dimensional connectivity number of M. However, the idea of separation fails to explain the one-dimensional connectivity of M, since no finite set of curves can separate M. Instead, one takes the maximum number of curves that can lie in M without form- ing the boundary of a surface in M.(ForasurfaceM, this maximum is the same as Riemann’s connectivity number.) Betti defined Pm similarly, in a manifold M of arbitrary dimension, as the maximum number of m-dimensional pieces of M that do not form the boundary of a connected (m + 1)-dimensional piece of M. Thus Betti brought the concept of boundary into topology in order to generalize Riemann’s concept of connectivity. This was Poincar´e’s starting point, but he went much further, as we will see. Poincar´e before topology In the introduction to his first major topology paper, the Analysis situs, Poincar´e (1895) announced his goal of creating of creating an n-dimensional geometry.
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