Geometric Methods in Degree Theory for Equivariant Maps
The book introduces conceptually simple geometric ideas based on the existence of fundamental domains for metric G- spaces. A list of the problems discussed includes Borsuk-Ulam type theorems for degrees of equivariant maps in finite and infinite dimensio
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Springer Berlin Heidelberg New York Barcelona Budapest Hong Kong London Milan Paris Santa Clara Singapore Tokyo
Alexander Kushkuley Zalman Balanov
Geometric Methods in Degree Theory for Equivariant Maps
Springer
Authors Alexander Kushkuley 6 Carriage Drive Acton, MA 01720, USA Zalman Balanov Department of Mathematics and Computer Science Bar Han University 52900 Ramat-Gan, Israel Cataloging-in-Publication Data applied for
Die Deutsche Bibliothek - CIP-Einheitsaufnahme Kushkuley, Alexander: Geometric methods in degree theory for equivariant maps / Alexander Kushkuley ; Zalman Balanov. - Berlin; Heidelberg; New York; Barcelona; Budapest; Hong Kong; London; Milan; Paris; Santa Clara; Singapore; Tokyo: Springer, 1996 (Lecture notes in mathematics ; 1632) ISBN 3-540-61529-6 NE: Balanov, Zalman:; GT Mathematics Subject Classification (1991): Primary: 55M25, 55P91, 55S91, 58C30 Secondary: 55M35, 57Q91, 57R91, 57Sl5, 58E05, 58G45, 20C15 ISSN 0075-8434 ISBN 3-540-61529-6 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 1996 Printed in Germany
The use of general descriptive names, 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 protective laws and regulations and therefore free for general use. Typesetting: Camera-ready TEX output by the authors SPIN: 10479811 46/3142-543210 - Printed on acid-free paper
To: Vita and Gabriel; Larissa Tziporah, Sonya, Bronika and Jacob
Contents Introduction Chapter 1. Fundamental domains and extension of equivariant maps
1
13
Auxiliary information Existence theorem Equivariant Kuratowski-Dugundji Theorem Historical and bibliographical notes
13 20 23 30
Chapter 2. Degree theory for equivariant maps of finite-dimensional manifolds: topological actions
31
1.1. 1.2. 1.3. 1.4.
2.1. 2.2. 2.3. 2.4.
Comparison principle for a finite group Some special cases Some counterexamples Historical and bibliographical notes
Chapter 3. Degree theory for equivariant maps of finite-dimensional manifolds: smooth actions 3.1. 3.2. 3.3. 3.4. 3.5. 3.6. 3.7.
Statement of results Thorn class and cap-product Invariant foliations and equivariant transversality The case of a finite group The case of a compact Lie group Some special cases Historical and bibliographical notes
Chapter 4. A winding number of equivariant vector fields in infinite dimensional Banach spaces
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