Cellular Chain Algebra Models
Our aim in this chapter is to present cellular chain algebra models for \(C_* D_N (\Omega (\mathbb{F}_k (M)))\) , the normalized chain algebra of singular cubes, where M is either ℝ n+1 or S n+1. The models are based on the RPT-models for \(\Omega (\mathb
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Springer-Verlag Berlin Heidelberg GmbH
Edward R. Fadell • Sufian Y. Husseini
Geometry and Topology of Configuration Spaces
Springer
Edward R. Fadell University of Wisconsin-Madison Department of Mathematics Madison, WI 53706 USA e-mail: [email protected] Sußan Y. Husseini University of Wisconsin-Madison Department of Mathematics Madison, WI 53706 USA e-mail: husseini@[email protected]
Library of Congress Cataloging-in-Publication Data Fadell, Edward R., 1926Geometry and topology of configuration spaces / Edward R. Fadell, Sufian Husseini. p. cm. ~ (Springer monographs in mathematics, ISSN 1439-7382) Includes bibliosraDhical references and index. ISBN 978-3-642-63077-4 ISBN 978-3-642-56446-8 (eBook) DOI 10.1007/978-3-642-56446-8 1. Configuration space. 2. Algebraic topology. I. Husseini, S. Y. II. Title. III. Series. QA607 .F34 2000 516.3'5~dc21 00-049221
Mathematics Subject Classification (2000): 20F36,55M30,55R10,55R20,55T10, 55T20,58B05,58E05,70G25 ISSN 1439-7382 ISBN 978-3-642-63077-4 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.
© Springer-Verlag Berlin Heidelberg 2001 Originally published by Springer-Verlag Berlin Heidelberg New York in 2001 Softcover reprint of the hardcover 1st edition 2001 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. Cover design: Erich Kirchner, Heidelberg Typesetting by the authors using a Springer I*lgX macro package Printed on acid-free paper SPIN 10724127
41/3142ck-5 43210
To Patricia and Barbara
Preface
The configuration space of k particles in the smooth manifold M is the space
These spaces and the associated free and based loop spaces, nlFk (M) and AlFk(M), respectively, play an important role in topology and geometry and related areas. For example, the space IFk(M) provides additional topological invariants for the mainfold M and, more generally, for an imbedding f : M ~ M' of one manifold into another (see, for example [11, Bott], [13, Bott-Taubes], [69, Kohno], [108, Vasiliev], [51, Haefiiger], [52, Haefiiger], [99, Shapiro], [111, Wu], and [112, Wu]). Also, IFk(an +!) is the space of k noncolliding particles Xl>· .. , Xk in an +1 , and its free loop space AlFk(an +1 ) is the space of its k closed curves (orbits) in an+!. The existence of periodic solutions to a Hamiltonian system of the k-body type is deduced f
