Cellular Spaces, Null Spaces and Homotopy Localization
In this monograph we give an exposition of some recent development in homotopy theory. It relates to advances in periodicity in homotopy localization and in cellular spaces. The notion of homotopy localization is treated quite generally and encompasses al
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1622
Lecture Notes in Mathematics Editors: A. Dold, Heidelberg F. Takens, Groningen
1622
Springer
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Emmanuel Dror Farjoun
Cellular Spaces, Null Spaces and Homotopy Localization
Springer
Author Emmanuel Dror Farjoun Mathematics Department Hebrew University of Jerusalem Jerusalem, Israel E-Mail: [email protected]
Cataloging-in-Publication Data applied for
Die Deutsche Bibliothek - CIP-Einheitsaufnahme Farjoun , Emmanuel Dror: Cellular spaces, null spaces and homotopy localization I Emmanuel Dror Farjoun. - Berlin; Heidelberg; New York; Barcelona ; Budapest; Hong Kong; London; Milan ; Paris; Tokyo: Springer, 1995 (Lecture notes in mathematics ; 1622) ISBN 3-540-60604-1 NE:GT
Mathematics Subject Classification (1991): 55 ISBN 3-540-60604-1 Springer-Verlag Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part ofthe 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 Typesetting: Camera-ready TEX output by the author SPIN: 10479706 46/3142-543210 - Printed on acid-free paper
CONTENTS
Introduction
Vll
1. Coaugmented homotopy idempotent localization functors
Introduction " .. _ '" 1 A. Local spaces, null spaces, localization functors, elementary facts. . . .. 1 B. Construction of L f . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 C. Universality and continuity of L f " 17 D. L f and homotopy colimits, f -local equivalence " 23 E. Examples: Localization according to Quillen-Sullivan, Bousfield-Kan, homological localizations, and vi-periodic localization 26 F. Fibrewise localization " 28 G. Proof of elementary facts " 35 H. The fibre of the localization map. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 37 2. Augmented homotopy idempotent functors
A. Introduction, A-equivalence " B. Construction of CW A, the universal A-equivalence C. A common generalization of L f and CW A and model category structures " D. Closed classes and A-cellular spaces '" E. A-Homotopy theory and universal properties
39 40 42 45 53
3. Commutation rules for n, L f and CW A, preservation of fibrations and cofibrations
Introduction " A. Commutation with the loop functor " B. Relations between CW A and P A ..........................•...•.. C. Examples of cellular spaces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. D. Localization Lj and cofibrations, fibrations
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