Papers on Group Theory and Topology Max Dehn
Papers on Group Theory and Topology
Translated and Introduced by John Stillwell
With 151 Illustrations
Springer-Verlag New York Berlin Heidelberg London Paris Tokyo John Stillwell Department of Mathematics Monash University, Clayton Victoria 3168, Australia
AMS Classification: 01-A75, 55-03, 20 F 05,20 F 32
Library of Congress Cataloging in Publication Data Dehn, Max, 1878-1952. Papers on group theory and topology. Includes bibliographies. I. Combinatorial group theory-Collected works. 2. Topology-Collected works. I. Title. QA17I.D393A25 1987 512'.22 87-317
© 1987 by Springer-Verlag New York Inc. Softcover reprint of the hardcover 1st edition 1987 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer-Verlag, 175 Fifth Avenue, New York, New York 10010, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use of general descriptive names, trade names, trademarks, etc. in this publication, even if the former are not especially identified, is not to be taken as a sign that such names, as understood by the Trade Marks and Merchandise Marks Act, may accordingly be used freely by anyone.
Printed and bounded by R.R. Donnelley & Sons, Harrisonburg, Virginia. Printed in the United States of America
987654321
ISBN-13: 978-1-4612-9107-7 e-ISBN-13: 978-1-4612-4668-8 DOl: 10.1007/978-1-4612-4668-8 PREFACE
The work of Max Dehn (1878-1952) has been quietly influential in mathematics since the beginning of the 20th century. In 1900 he became the first to solve one of the famous Hilbert problems (the third, on the decomposition of polyhedra), in 1907 he collaborated with Heegaard to produce the first survey of topology, and in 1910 he began publishing his own investigations in topology and combinatorial group theory. His influence is apparent in the terms Dehn's algorithm, Dehn's lemma and Dehn surgery
(and Dehnsche Gruppenbilder, generally known in English as Cayley diagrams), but direct access to his work has been difficult. No edition of his works has been produced, and some of his most important results were never published, at least not by him.
The present volume is a modest attempt to bring Dehn's work to a wider audience, particularly topologists and group theorists curious about the origins of their subject and interested in mining the sources for new ideas. It consists of English translations of eight works : five of
Dehn's major papers in topology and combinatorial group theory, and three unpublished works which illuminate the published papers and contain some results not available elsewhere. In addition, I have written a short introduction to each work, summarising its contents and trying to establish its place among related works of Dehn and others, and I have added an appendix on the Dehn-Nielsen theorem (often known simply as Nielsen's theorem) . The latter theorem was never published by Dehn, though Nielsen gives him credit for it, and Dehn's approach to the theorem became forgotten when Nielsen (1927) buried it in a l80-page paper. In fact, Dehn's approach can be made to work quite simply, as the appendix demonstrates.
In surface topology and the associated theory of fuchsian groups,
Dehn's work fills a gap between the works of Poincare and Nielsen.
The three together are the main sources of contemporary work in this field, by Thurston in particular. with the recent publication of my translations of poincare's Papers on Fuchsian Functions (Springer-Verlag
1985) and Nielsen's Collected Mathematical Papers (Birkhauser 1986), these sources are now available in what I hope is a convenient form.
The present volume would not have been possible without the generosity and encouragement of Wilhelm Magnus. He provided me with copies of important unpublished manuscripts of Dehn and put me in contact with
Dehn's widow, Mrs. Toni Dehn, who very kindly gave permission for trans- lations of these manuscripts to be published. Thanks are also due to
B.G. Teubner and Co. for permission to publish the papers which originally appeared in Mathematische Annalen, to the Institut Mittag-Leffler for permission to publish the paper from Acta Mathematica, and to the Mathe• matisches Seminar of the University of Hamburg for permission to publish the
Otto Schreier paper whose translation is included as an appendix to
Paper 6.
The various papers in this book were typed by Anne-Marie Vandenberg and Joan Williams, at a time when publication was not foreseen. It is a tribute to the quality of their work that the book could be photographed directly from their typescripts. Wilhelm Magnus and Dave Johnson read large portions of the book and saved me from many errors. To all these friends and colleagues I offer my sincere thanks. CONTENTS
Translator's Introduction 1 1
Paper 1 : Lectures on group theory 5
Translator's Introduction 2 47
Paper 2 : Lectures on surface topology 52
Translator's Introduction 3 86
Paper 3 : On the topology of three-dimensional space
(Uber die Topologie des dreidimensionalen Raumes. Math. Ann. 69, (1910), 137-168) 92
Translator's Introduction 4 127
Paper 4 : On infinite discontinuous groups
(Uber unendliche diskontinuierliche Gruppen. Math. Ann. 71 (1912), 116-144) 133
Translator's Introduction 5 179
Paper 5 : Transformation of curves on two-sided surfaces
(Transformation der Kurven auf zweiseitigen Flachen. Math. Ann. 72 (1912), 413-421) 183
Translator's Introduction 6 200
Paper 6 : The two trefoil knots
(Die beiden Kleeblattschlingen. Math. Ann. 75 (1914), 402-413) 203
Appendix to Paper 6
(Uber die Gruppen AaBb = 1, by Otto Schreier, Abh. Math. Sem. Univ. Hamburg 3 (1924), 167-169) 224
Translator's Introduction 7 229
Paper 7 : On curve systems on two-sided surfaces, with application to the mapping problem.
(Uber Kurvensysteme auf zweiseitigen Flachen mit Anwendung auf das Abbildungsproblem. Vortrag (erganzt) im math. Kolloquium, Breslau 11/2/1922) 234 Translator's Introduction 8 253
Paper 8 : The group of mapping classes
(Die Gruppe der Abbildungsklassen. Acta Math. 69 (1938), 135-206) 256
Appendix The Dehn-Nielsen theorem 363