Spectral sequences. Abelian groups
The proof of the key result of the next section (Theorem 21.6) uses in an essential way the Roos spectral sequence and its consequences, which we describe in this section (see 20.3). In order to make the text as self-contained as possible, we develop gene
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Springer-Verlag Berlin Heidelberg GmbH
Sibe Mardesic
Strong Shape and Homology
,
Springer
Sibe Mardesic Department of Mathematics University of Zagreb Bijenicka cesta 30 10000 Zagreb, Croatia e-mail: [email protected]
Library of Congress Cataloging-in-Publication Data Mardesic, S. (Sibe),1927Strong shape and homology I Sibe Mardesic. p.cm. -- (Springer monographs in mathematics) Includes bibliographical references and index. ISBN 978-3-642-08546-8 ISBN 978-3-662-13064-3 (eBook) DOI 10.1007/978-3-662-13064-3 I. Shape theory (Topology) 2. Homology theory. I. Title. II Series QA612.7 .M353 1999 514'.24--dc21
99-047673
Mathematics Subject Classification (l991): 55NXX,55PXX, 18GXX
ISBN 978-3-642-08546-8
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Preface
It is well known that standard notions of homotopy theory are not adequate to study global properties of spaces with bad local behavior. For instance, all homotopy groups of the dyadic solenoid vanish in spite of the fact that the solenoid is globally a non-trivial object. Shape theory is designed to correct these shortcomings of homotopy theory. When restricted to spaces with good local behavior, like the ANR's, polyhedra or CW-complexes, shape theory coincides with homotopy theory, therefore, it can be viewed as the appropriate extension of homotopy theory to general spaces. Many constructions in topology lead naturally to spaces with bad local behavior even if one initially considers locally good spaces, e.g., manifolds. Standard examples include fibers of mappings, sets of fixed points, attractors of dynamical systems, spectra of operators, boundaries of certain groups. In all these areas shape theory has proved useful. It took some time to realize that beside ordinary shape, introduced in 1968 by K. Borsuk, there exists a finer theory, presently called strong shape theory, which has various advantages over ordinary shape. Its position is intermediate,
