First Notions of K-Theory
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Editors
S. S. Chern J. L. Doob J. Douglas, jr. A. Grothendieck E. Heinz F. Hirzebruch E. Hopf S. Mac Lane W. Magnus M. M. Postnikov W. Schmidt D. S. Scott K. Stein J. Tits B. L. van der Waerden Managing Editors B. Eckmann
J. K. Moser
Max Karoubi
K-Theory An Introduction
With 26 Figures
Springer-Verlag Berlin Heidelberg GmbH 197 8
Max Karoubi Universite Paris VII, U.E.R. de Mathematiques, Tour 45-55, 5" Etage, 2 Place Jussieu F-75221 Paris Cedex 05
AMS Subject Classification (1970) Primary: 55B15, 18F25, 55F45, 55F10, 55020 Secondary: IOC05, 16A54, 46H25, 55E20, 55E50, 55El0, 55125, 55G25
ISBN 978-3-540-79889-7 ISBN 978-3-540-79890-3 (eBook) DOI 10.1007/978-3-540-79890-3 Library ofCongress Cataloging in Publication Data. Karoubi, Max. K·theory. (Grundlehren der mathematischen Wissenschaften; 226). Bibliography: p. Includes index. I. K·theory. I. Title. II. Series: Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen; 226. QA612.33.K373. 514'.23. 77-23162. This work is su bject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifi.cally those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and Storage in data banks. Under § 54 of the Gennan Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publisher. © by Springer-Verlag Berlin Heidelberg 1978. Originally published by Springer-Verlag Berlin Heidelberg New York in 1978 Softcoverreprint ofthe bardeover Istedition 1978 Typesetting: William Clowes & Sons Limited, London, Beccles and Colchester. Printing and 2141/314(}-543210
A Pierreet Thomas
Foreword
K-theory was introduced by A. Grothendieck in his formulation of the RiemannRoch theorem (cf. Borel and Serre [2]). For each projective algebraic variety, Grothendieck constructed a group from the category of coherent algebraic sheaves, and showed that it had many nice properties. AtiyaQ. and Hirzebruch [3] considered a topological analog defined for any compact space X, a group K(X) constructed from the category of vector bundles on X. It is this "topological K-theory" that this book will study. Topological K-theory has become an important tool in topology. Using Ktheory, Adams and Atiyah were able to give a simple proofthat the only spheres which can be provided with H-space structures are Si, S 3 and S 7 . Moreover, it is possible to derive a substantial part of stable homotopy theory from K-theory (cf. J. F. Adams [2]). Further applications to analysis and algebra are found in the work of Atiyah-Singer [2], Bass [1], Quillen [1], and others. A key factor in these applications is Bott periodicity (Bott [2]). The purpose of this book is to provide advanced students and mathematicians in other fields with the fundamental material in this subject. In addition, several applications of the type described above are included. In general we have tried to make this
