Algebraic Topology from a Homotopical Viewpoint

511 Pages·2010·3.02 MB·English

Universitext
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(North America):
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F.W. Gehring
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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,
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Printed in the United States of America.
987654321 SPIN 10867195
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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