Spaces of Homotopy Self-Equivalences A Survey
This survey covers groups of homotopy self-equivalence classes of topological spaces, and the homotopy type of spaces of homotopy self-equivalences. For manifolds, the full group of equivalences and the mapping class group are compared, as are the corresp
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1662
Springer Berlin Heidelberg New York Barcelona Budapest Hong Kong London Milan Paris Santa Clara Singapore Tokyo
John W. Rutter
Spaces of Homotopy Self-Equivalences A Survey
Springer
Author John W. Rutter Division of Pure Mathematics University of Liverpool Liverpool L69 3BX, England e-mail: [email protected]
Cataloging-in-Publication Data applied for
Die Deutsche Bibliothek - CIP-Einheitsaufnahme Rutter, John W.: Spaces of homotopy self-equivalences: a survey / John W. Rutter. Berlin ; Heidelberg ; New York ; Barcelona ; Budapest ; Hong Kong ; London ; Milan ; Paris ; Santa Clara ; Singapore ; Tokyo : Springer,
1997
(Lecture notes in mathematics; 1662) ISBN 3-540-63103-8
Mathematics Subject Classification (1991): 55-02, 55Q05, 55PlO, 55P45, 55P62, 55P91, 57-02, 57M07, 57QlO ISSN 0075-8434 ISBN 3-540-63103-8 Springer-Verlag Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer- Verlag. Violations are liable for prosecution under the German Copyright Law. © Springer-Verlag Berlin Heidelberg 1997 Printed in Germany
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Preface The group of homotopy self-equivalence classes of a topological space X probably first appeared as the mapping-class group of the torus in the book by Clebsch and Gordan [1866], implicit there in the theory of elliptic functions. We now know from the work of Baer and others that the group of free homotopy self-equivalence classes £(X) is isomorphic to the homeotopy group H(X) for almost all compact surfaces, and can be faithfully represented into the automorphism group of the fundamental group (see Chapter 4). Also a simple presentation using a minimum set of twist-homeomorphisms as generators was given by Wajnryb [1983] in the case of orientable closed surfaces and also of compact orientable surfaces with one boundary component: presentations of surface braid groups are also known (see Chapter 5). For higher dimensional manifolds we note in Chapter 6 cases where H(X) -+ £(X) is isomorphic or epimorphic and cases where it is not, the finite generation and presentation problems for £(X), and the relation between the homotopy types of H(X) and £(X). It was shown, by Dror, Dwyer and Kan [1981]' that £(X) is of finite type, and is
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