Algebraic topology from a homotopical viewpoint PDF
Preview Algebraic topology from a homotopical viewpoint
Universitext Editorial Board (North America): S. Axler F.W. Gehring K.A. Ribet Springer New York Berlin Heidel berg Barcelona Hong Kong London Milan Paris Singapore Tokyo Universitext Editors (North America): S. Axler, F.W. Gehring, K.A. Rbet and Aguilar/Gitler/Prieto: Algebraic Topology from a Homotopical Viewpoint Aksoy/Khami: Nonstandard Methods in Fixed Point Theory Andersson: Topics in Complex Analysis Aupetit: A Primer on Spectral Theory BachmanNarici/Beckenstein:F ourier and Wavelet Analysis Badescu: Algebraic Surfaces A Textbook of Graph Theory Balakrishnan/Ranganathan: Baker: Formal Power Series and Linear Systems of Meromorphic Ordinary Differential Equations Bapat: Linear Algebra and Linear Models (2nd ed.) Berberian: Fundamentals of Real Analysis Boltyaaskii/Efremovich: Intuitive CombinatonalT opology. (Shenitzer, trans.) Booss/Bleecker: Topology and Analysis Borkar: Probability Theory: An Advanced Course Biittcher/Silbermann: Introduction to Large Truncated Toeplitz Matrices CarlesoxdGamelin:C omplex Dynamics Lie Sphere Geometry: With Applications to Submanifolds Cecil: Chae: Lebesgue Integration (2nd ed.) Charlap: Bieberbach Groups and Flat Manifolds Chern: Complex Manifolds Without Potential Theory Cob: A Classical Invitation to Algebraic Numbers and Class fields Curtis: Abstract Linear Algebra Curtis. Matrix Groups Debarre: Higher-DimensionalA lgebraic Geometry Deitmar: A First Course in Harmonic Analysis Wenedetto: Degenerate Parabolic Equations Dimca: Singularitis and Topology of Hypersurfaces Edwards: A Formal Background to Mathematics J ah Edwards: A Formal Background to Mathematics ll ah Farenick Algebras of Transformations Linear Foulds: Graph Applications Theory Friedman: Algebraic Surfaces and Holomophic Vector Bundles Fuhrmann: A Polynomal Approach to Linear Algebra Gardiner: A First Course in Group Theory GhdiqyT'ambour:A lgebra for Computer Science Goldblatt: Orthogonality and Spacetime Geometry GustafsonlRao: Numerical Range: The Field of Values of Linear Operaton and Mahices Hahn: Quadratic Algebras, Clifford Algebras, and Arithmetic Wid Groups Heinonen: Lectures on Analysis on Metric Spaces Holmgren: A Fust Course in Discrete Dynamical Systems Howe/Tan: Non-Abelian Harmonic Analysis: Applications of S42, R) Howes: Modem Analysis and Topology HsiehlSibuya: Basic Theory of Ordinary Differential Equaticus Humi/Miller: Second Course in Ordinary Differential Equations HurwitdKritikos: Lectures on Number Theory Jennings: Modem Geometry with Applications (continued aper index) Marcelo Aguilar Samuel Gitler Carlos Prieto Algebraic Topology from a Homotopical Viewpoint Springer Marcelo Aguilar Samuel Gitler Carlos Prieto Instituto de Matematicas Department a€ Mathematics Tnstituto de Matematicas Universidad Nacional University of Rochester Universidad Nacional Autonoma de Mexico Rochester, NY 14627-0001 Autonoma de Mexico 04510 Mexico, DF USA 04510 Mexico, DF Mexico [email protected] Mexico [email protected] cprieto @math.unam.mx Editorial Board (North America): S. Axler F.W. Gehring Mathematics Department Mathematics Department San Francisco State East Hall University University of Michigan San Francisco, CA 94132 AM Arbor, MI 48109-1109 USA USA K.A. Ribet Mathematics Department University of California Berkeley Berkeley, CA 94720-3840 USA Mathematics Subject Classification (2000): 55-01 Library of Congress Cataloging-in-Publication Data Aguilar, M.A. (Marcelo A.) Algebraic topology from a homotopical viewpoint / Marcelo Aguilar, Samuel Gitler, Carlos Prieto. p. cm. - (Universitext) Includes bibliographical references and index. (a. ISBN 0-387-95450-3 paper) 1. Algebraic topology. 2. Homotopy theory. I. Gitler, Samuel. 11. F’rieto, C. (Carlos) In. Title. QA612 2002 .A37 5 I 4’2-4~2 1 200201 9556 ISBN Printed on acid-free paper. 0-387-95450-3 0 2002 Springer-Verlag New York, Inc. All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer-Verlag New York, Inc., 175 Fifth Avenue, New York, NY 10010, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval. electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed in the United States of America. 987654321 SPIN 10867195 Typesetting: Pages created by authors using a Springer TEX macro package. wwwspringer-ny.com Springer-Verlag New York Berlin Heidelberg A member of BertelsmannSpringer Science+Business Media GmbH To my parents To Danny To Viola To Sebastian and Adrian This page intentionally left blank Preface Thisbookintroducesthebasicconceptsofalgebraictopologyusinghomoto- py-theoretical methods. We believe that this approach allows us to cover the material more efficiently than the more usual method using homological algebra. After an introduction to the basic concepts of homotopy theory, using homotopy groups, quasifibrations, and infinite symmetric products, we define homology groups. Furthermore, with the same tools, Eilenberg– Mac Lane spaces are constructed. These, in turn, are used to define the ordinary cohomology groups. In order to facilitate the computation, cellular homology and cohomology are defined. Inthesecondhalfofthebook,vectorbundlesarepresentedandthenused to define K-theory. We prove the classification theorems for vector bundles, whichprovideahomotopyapproachtoK-theory. Lateron,K-theoryisused to solve the Hopf invariant problem and to analyze the existence of multi- plicative structures in spheres. The relationship between cohomology and vector bundles is established introducing characteristic classes and related topics. To finish the book, we unify the presentation of cohomology and K-theory by proving the Brown representation theorem and giving a short account of spectra. In two appendices at the end of the book the proof of the Dold–Thom theoremonquasifibrationsandinfinitesymmetricproductsisgivenindetail, and a new proof of the complex Bott periodicity theorem, using quasifibra- tions, is presented. It is expected that the reader has a basic knowledge of general topology and algebra. In any case, the book is mainly aimed at advanced undergrad- uates and at graduate students and researchers for whose work algebraic- topological concepts are needed. ThistextoriginatedinapreliminaryversioninSpanish,whichwasajoint editionoftheMathematicsInstituteoftheNationalUniversityofMexicoand McGraw-Hill Interamericana Editores. To both institutions the authors are grateful. The translation of the main body of the text was the excellent vii viii Preface job of Stephen Bruce Sontz, to whom we express our deep thanks. Our gratitude goes also to Springer-Verlag, particularly to Ms. Ina Lindemann for her interest in our work, and to the referees for their valuable comments which certainly helped to improve the English version of the book. Its title is,ofcourse,atributetoJohnMilnor,fromwhosebooksandpaperswehave learnt many important concepts, which are included in this text. Last, but not least, we wish to acknowledge the support of Professor Albrecht Dold, who after reading the Spanish manuscript gave various im- portant comments to make some parts better. MexicoCity,Mexico Marcelo Aguilar Autumn2001 Samuel Gitler Carlos Prieto1 1ThisauthorwassupportedbyCONACYTgrants25406-Eand32223-E. Contents Preface vii Introduction xiii Basic Concepts and Notation xvii 1 Function Spaces 1 1.1 Admissible Topologies . . . . . . . . . . . . . . . . . . . . 1 1.2 Compact-Open Topology . . . . . . . . . . . . . . . . . . . 2 1.3 The Exponential Law . . . . . . . . . . . . . . . . . . . . . 3 2 Connectedness and Algebraic Invariants 9 2.1 Path Connectedness . . . . . . . . . . . . . . . . . . . . . 9 2.2 Homotopy Classes. . . . . . . . . . . . . . . . . . . . . . . 10 2.3 Topological Groups . . . . . . . . . . . . . . . . . . . . . . 13 2.4 Homotopy of Mappings of the Circle into Itself . . . . . . . 15 2.5 The Fundamental Group . . . . . . . . . . . . . . . . . . . 28 2.6 The fundamental Group of the Circle . . . . . . . . . . . . 41 2.7 H-Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 2.8 Loop Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 48 2.9 H-Cospaces . . . . . . . . . . . . . . . . . . . . . . . . . . 50 2.10 Suspensions . . . . . . . . . . . . . . . . . . . . . . . . . . 53 ix
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