Invariants for Homology 3-Spheres
Homology 3-sphere is a closed 3-dimensional manifold whose homology equals that of the 3-sphere. These objects may look rather special but they have played an outstanding role in geometric topology for the past fifty years. The book gives a systematic exp
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Low-Dimensional Topology Subseries Editors: R. V. Gamkrelidze V.A. Vassiliev
Springer-Verlag Berlin Heidelberg GmbH
Nikolai Saveliev
Invariants for HOßlology 3-Spheres
Springer
Nikolai Saveliev Department of Mathematics University of Miami Coral Gables, FL 33124 USA e-mail: [email protected]
Founding editor of the Encyclopedia of Mathematical Sciences: R. V. Gamkrelidze
Mathematics Subject Classification (2000): 57-02, 57M, 57N, 57Q, 57R, 58E, 58J, 58K
The figure on the cover of the book was created with KnotPlot (http://www.pims.math.ca/knotplotl) with kind permission and assistance of Robert Scharein, University of British Columbia, Canada
ISSN 0938-0396 ISBN 978-3-642-07849-1 ISBN 978-3-662-04705-7 (eBook) DOI 10.1007/978-3-662-04705-7 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. reuse of illustrations, recitation. broadcasting. reproduction on microfilm 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.
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Originally published by Springer-Verlag Berlin Heidelberg New York in 2002. Softcover reprint ofthe hardcover 1st edition 2002 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 eJrempt from the relevant protective laws and regulations and therefore free for general use. Typeset by the author using a Springer TEX macro package Cover Design: E. Kirchner. Heidelberg. Germany Printed onacid-free paper SPIN:46/3111db 54321
To rny rnorn, with love and gratitude
Preface
Homology 3-spheres are closed oriented 3-manifolds having the homology of the 3-sphere. These objects may look rather special but they have played a prominent role in manifold topology for a long time. This book is a survey of various ideas and constructions used in their study, from the classical Rokhlin invariant through Casson's theory and its numerous generalizations up to the most recent gauge theoretical invariants. It is hardly possible to account for all the ramifications of this theory in one book. While trying to give as complete an account as possible, I was largely guided by my personal research interests in deciding which topics to cover in more detail. Some results in the book have never appeared in monograph form before while others are rather well documented in the literature. The state of affairs in low dimensional topology by the early 1980s is described in Mandelbaum's survey [203]. The book [275] can be recommended as a gentle introduction to the Casson invariant aimed at graduate students. Akbulut and McCarthy's
