L2-Invariants: Theory and Applications to Geometry and K-Theory
In algebraic topology some classical invariants - such as Betti numbers and Reidemeister torsion - are defined for compact spaces and finite group actions. They can be generalized using von Neumann algebras and their traces, and applied also to non-compac
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A Series of Modern Surveys in Mathematics
Editorial Board
S. Feferman, Stanford M. Gromov, Bures-sur-Yvette J. Jost, Leipzig J. Kollar, Princeton H.W. Lenstra, Jr., Berkeley P.-L. Lions, Paris M. Rapoport, K6in J.Tits, Paris D. B. Zagier, Bonn Managing Editor R. Remmert, Munster
Volume 44
Springer-Verlag Berlin Heidelberg GmbH
Wolfgang Luck
L2_ Invariants: Theory and Applications to Geometry and K-Theory
,
Springer
Wolfgang Li.ick Mathematisches Institut Universitat Mi.inster EinsteinstraBe 62 48149 Mi.inster, Germany e-mail: [email protected]
Library of Congress Cataloging-in-Publication Data applied for Die Deutsche Bibliothek - CIP-Einheitsaufnahme Liick, Wolfgang: [,-invariants: theory and applications to geometry and K-theory I Wolfgang Liick.Berlin; Heidelberg; New York; Barcelona; Hong Kong; London; Milan; Paris; Tokyo: Springer, 2002 (Ergebnisse der Mathematik und ihrer Grenzgebiete; Folge 3, Vol. 44)
Mathematics Subject Classification (2000): 57-99, 58J50, 46L99, 19A99
ISSN 0071-1136 ISBN 978-3-642-07810-1 ISBN 978-3-662-04687-6 (eBook) DOl 10.1007/978-3-662-04687-6 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 microfilms or in any other ways, 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.
http://www.springer.de © Springer-Verlag Berlin Heidelberg 2002 Originally published by Springer-Verlag Berlin Heidelberg in 2002. Sotkover reprint of the hardcover 1st edition 2002
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Preface
There is the general principle to consider a classical invariant of a closed Riemannian manifold M and to define its analog for the universal covering M taking the action of the fundamental group 7r = 7rl(M) on M into account. Prominent examples are the Euler characteristic and the signature of M, which lead to Wall's finiteness obstruction and to all kinds of surgery obstructions such as the symmetric signature or higher signatures. The pth L2-Betti number b~2) (M) arises from this principle applied to the p-th Betti number bp{M). Some effort is necessary to define L2-Betti numbe~ in the case where 7r is infinite. Typical problems for infinite 7r are that M is not compact and that the complex group ring C7r is a complicated ring, in general not Noetherian. Therefore some new technical input is needed from operator theory, namely, the group von Neumann algebra and its trace. Analytically Atiyah defined L2-Betti numbers in terms of the heat kernel on M. There also is an e~ivalent combinatorial approach based on the cellular C7
