Fibrewise Homotopy Theory
Topology occupies a central position in the mathematics of today. One of the most useful ideas to be introduced in the past sixty years is the concept of fibre bundle, which provides an appropriate framework for studying differential geometry and much els
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Springer London Berlin Heidelberg New York Barcelona Budapest Hong Kong Milan Paris Santa Clara Singapore Tokyo
Michael Crabb
loan James
Fibrewise Homotopy Theory
,
Springer
Michael Charles Crabb Department of Mathematics University of Aberdeen Aberdeen AB243UE
UK
loan Mackenzie James Mathematical Institute Oxford University 24-29 St Giles Oxford OX13LB
UK
British Ubrary Cataloguing in Publication Data Crabb,M.C. Fibrewise homotopy theory. - (Springer monographs in mathematics I. Homotopy theory 2. Fibre bundles (Mathematics) I. Title II. James, loan Mackenzie 514.2'4
Library of Congress Cataloging-in-Publication Data Crabb, M.C. (Michael Charles) Fibrewise homotopy theory / Michael Crabb and loan James p. cm. -- (Springer monographs in mathematics) Includes bibliographical references (p. - ) and index. [SBN-13: 978-1-4471-1267-9 e-[SBN-13: 978-1-4471-1265-5 00[: 10.1007/978-1-4471-1265-5 1. Homotopy theory. 2. Fibre bundles (Mathematics) I. James, I.M. (loan Mackenzie), 1928- . II. Title. III. Series. QA612.7.C685 1998 98-7022 514'.24--dc21 CIP Mathematics Subject Classification (1991): 55-02, 55M20, 55M30, 55P25, 55P42, 55P99, 55Q15, 55R65, 55R99, 57R22, 19L20 Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms of licences issued by the Copyright Licensing Agency. Enquiries concerning reproduction outside those terms should be sent to the publishers. © Springer-Verlag London Limited 1998
Sof'tcover reprint of the hardcover 1st edition 1998 The use of registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant laws and regulations and therefore free for general use. The publisher makes no representation, express or implied, with regard to the accuracy of the information contained in this book and cannot accept any legal responsibility or liability for any errors or omissions that may be made. Typesetting: Camera-ready by the authors and Michael Mackey 12/3830-543210 Printed on acid-free paper
Preface
Topology occupies a central position in the mathematics of today. One of the most useful ideas to be introduced in the past sixty years is the concept of fibre bundle, which provides an appropriate framework for studying differential geometry and much else. Fibre bundles are examples of the kind of structures studied in fibrewise topology. Just as homotopy theory arises from topology, so fibrewise homotopy theory arises from fibrewise topology. In this monograph we provide an overview of fibrewise homotopy theory as it stands at present. It is hoped that this may stimulate further research. The literature on the subject is already quite extensive but clearly there is a great
