An Introduction to Algebraic Topology

447 Pages·1988·11.847 MB·English

by  Joseph J. Rotman (auth.)

119
Graduate Texts in Mathematics
Editorial Board
s. Axler F.W. Gehring K.A. Ribet
Springer
New York
Berlin
Heidelberg
Barcelona
Budapest
Hong Kong
London
Milan
Paris
Singapore
Tokyo
Graduate Texts in Mathematics
TAKEUTJ!ZARING. Introduction to 33 HIRSCH. Differential Topology.
Axiomatic Set Theory. 2nd ed. 34 SPITZER. Principles of Random Walk.
2 OxrOBY. Measure and Category. 2nd ed. 2nd ed.
3 SCHAEFER. Topological Vector Spaces. 35 ALEXANDERIWERMER. Several Complex
4 HILTON/STAMMBACH. A Course in Variables and Banach Algebras. 3rd ed.
Homological Algebra. 2nd ed. 36 KELLEy/NAMIOKA et al. Linear
5 MAC LANE. Categories for the Working Topological Spaces.
Mathematician. 2nd ed. 37 MONK. Mathematical Logic.
6 HUGHESIPIPER. Projective Planes. 38 GRAUERTIFRITZSCHE. Several Complex
7 SERRE. A Course in Arithmetic. Variables.
8 TAKEUTIlZARING. Axiomatic Set Theory. 39 ARVESON. An Invitation to c*-Algebras.
9 HUMPHREYS. Introduction to Lie Algebras 40 KEMENy/SNELLIKNAPP. Denumerable
and Representation Theory. Markov Chains. 2nd ed.
10 COHEN. A Course in Simple Homotopy 41 ApOSTOL. Modular Functions and
Theory. Dirichlet Series in Number Theory.
11 CONWAY. Functions of One Complex 2nd ed.
Variable I. 2nd ed. 42 SERRE. Linear Representations of Finite
12 BEALS. Advanced Mathematical Analysis. Groups.
13 ANDERSONlFuLLER. Rings and Categories 43 GILLMAN/JERISON. Rings of Continuous
of Modules. 2nd ed. Functions.
14 GOLUBITSKy/GUILLEMIN. Stable Mappings 44 KENDIG. Elementary Algebraic Geometry.
and Their Singularities. 45 LoilVE. Probability Theory I. 4th ed.
IS BERBERIAN. Lectures in Functional 46 LOEVE. Probability Theory II. 4th ed.
Analysis and Operator Theory. 47 MOISE. Geometric Topology in
16 WINTER. The Structure of Fields. Dimensions 2 and 3.
17 ROSENBLATI. Random Processes. 2nd ed. 48 SACHS/WU. General Relativity for
18 HALMOS. Measure Theory. Mathematicians.
19 HALMOS. A Hilbert Space Problem Book. 49 GRUENBERG/WEIR. Linear Geometry.
2nd ed. 2nd ed.
20 HUSEMOLLER. Fibre Bundles. 3rd ed. 50 EDWARDS. Fermat's Last Theorem.
21 HUMPHREYs. Linear Algebraic Groups. 51 KLINGENBERG. A Course in Differential
22 BARNESiMACK. An Algebraic Introduction Geometry.
to Mathematical Logic. 52 HARTSHORNE. Algebraic Geometry.
23 GREUB. Linear Algebra. 4th ed. 53 MANIN. A Course in Mathematical Logic.
24 HOLMES. Geometric Functional Analysis 54 GRA VERIW ATKINS. Combinatorics with
and Its Applications. Emphasis on the Theory of Graphs.
25 HEwm/STROMBERG. Real and Abstract 55 BROWNIPEARCY. Introduction to Operator
Analysis. Theory I: Elements of Functional
26 MANES. Algebraic Theories. Analysis.
27 KELLEY. General Topology. 56 MASSEY. Algebraic Topology: An
28 ZARISKIISAMUEL. Commutative Algebra. Introduction.
Vol.I. 57 CROWELLIFox. Introduction to Knot
29 ZARISKIISAMUEL. Commutative Algebra. Theory.
Vol.II. 58 KOBLITZ. p-adic Numbers, p-adic
30 JACOBSON. Lectures in Abstract Algebra Analysis, and Zeta-Functions. 2nd ed.
I. Basic Concepts. 59 LANG. Cyclotomic Fields.
31 JACOBSON. Lectures in Abstract Algebra 60 ARNOLD. Mathematical Methods in
II. Linear Algebra. Classical Mechanics. 2nd ed.
32 JACOBSON. Lectures in Abstract Algebra
III. Theory of Fields and Galois Theory. continued after index
Joseph J. Rotman
An Introduction
to Algebraic Topology
With 92 Illustrations
, Springer
Joseph J. Rotman
Department of Mathematics
University of Illinois
Urbana, IL 61801
USA
Editorial Board
S. Axler F.W. Gehring K.A. Ribet
Mathematics Department Mathematics Department Department of Mathematics
San Francisco State East Hall University of California
University University of Michigan at Berkeley
San Francisco, CA 94132 Ann Arbor, MI 48109 Berkeley, CA 94720-3840
USA USA USA
Mathematics Subject Classification (1991): 55-01
Library of Congress Cataloging-in-Publication Data
Rotman, Joseph J.,
An introduction to algebraic topology.
(Graduate texts in mathematics; 119)
Bibliography: p.
Includes index.
1. Algebraic topology. I. Title. II. Series.
QA612.R69 1988 514'.2 87-37646
© 1988 by Springer-Verlag New York Inc.
Softcover reprint of the hardcover 15 t edition 1988
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, 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 of general descriptive names, trade names, trademarks, etc. in this publication, even if
the former are not especially identified, is not to be taken as a sign that such names, as understood
by the Trade Marks and Merchandise Marks Act, may accordingly be used freely by anyone.
Typeset by Asco Trade Typesetting Ltd., Hong Kong.
9 8 7 6 5 4 (Fourth corrected printing, 1998)
ISBN-13: 978-1-4612-8930-2 e-ISBN-13: 978-1-4612-4576-6
DOl: 10.1007/978-1-4612-4576-6
To my wife Marganit
and my children Ella Rose and Daniel Adam
without whom this book would have
been completed two years earlier
Preface
There is a canard that every textbook of algebraic topology either ends with
the definition of the Klein bottle or is a personal communication to J. H. C.
Whitehead. Of course, this is false, as a glance at the books of Hilton and
Wylie, Maunder, Munkres, and Schubert reveals. Still, the canard does reflect
some truth. Too often one finds too much generality and too little attention
to details.
There are two types of obstacle for the student learning algebraic topology.
The first is the formidable array of new techniques (e.g., most students know
very little homological algebra); the second obstacle is that the basic defini
tions have been so abstracted that their geometric or analytic origins have
been obscured. I have tried to overcome these barriers. In the first instance,
new definitions are introduced only when needed (e.g., homology with coeffi
cients and cohomology are deferred until after the Eilenberg-Steenrod axioms
have been verified for the three homology theories we treat-singular, sim
plicial, and cellular). Moreover, many exercises are given to help the reader
assimilate material. In the second instance, important definitions are often
accompanied by an informal discussion describing their origins (e.g., winding
numbers are discussed before computing 1tl (Sl), Green's theorem occurs
before defining homology, and differential forms appear before introducing
cohomology).
We assume that the reader has had a first course in point-set topology, but
we do discuss quotient spaces, path connectedness, and function spaces. We
assume that the reader is familiar with groups and rings, but we do discuss
free abelian groups, free groups, exact sequences, tensor products (always over
Z), categories, and functors.
I am an algebraist with an interest in topology. The basic outline of this
book corresponds to the syllabus of a first-year's course in algebraic topology
viii Preface
designed by geometers and topologists at the University of Illinois, Urbana;
other expert advice came (indirectly) from my teachers, E. H. Spanier and S.
Mac Lane, and from J. F. Adams's Algebraic Topology: A Student's Guide. This
latter book is strongly recommended to the reader who, having finished this
book, wants direction for further study.
I am indebted to the many authors of books on algebraic topology, with
a special bow to Spanier's now classic text. My colleagues in Urbana, es
pecially Ph. Tondeur, H. Osborn, and R. L. Bishop, listened and explained.
M.-E. Hamstrom took a particular interest in this book; she read almost the
entire manuscript and made many wise comments and suggestions that have
improved the text; my warmest thanks to her. Finally, I thank Mrs. Dee
Wrather for a superb job of typing and Springer-Verlag for its patience.
Joseph J. Rotman
Addendum to Second Corrected Printing
Though I did read the original galleys carefully, there were many errors that
eluded me. I thank all who apprised me of mistakes in the first printing,
especially David Carlton, Monica Nicolau, Howard Osborn, Rick Rarick,
and Lewis Stiller.
November 1992 Joseph J. Rotman
Addendum to Fourth Corrected Printing
Even though many errors in the first printing were corrected in the second
printing, some were unnoticed by me. I thank Bernhard J. Elsner and Martin
Meier for apprising me of errors that persisted into the the second and third
printings. I have corrected these errors, and the book is surely more readable
because of their kind efforts.
April,1998 Joseph Rotman
To the Reader
Doing exercises is an essential part of learning mathematics, and the serious
reader of this book should attempt to solve all the exercises as they arise. An
asterisk indicates only that an exercise is cited elsewhere in the text, sometimes
in a proof (those exercises used in proofs, however, are always routine).
I have never found references of the form 1.2.1.1 convenient (after all, one
decimal point suffices for the usual description of real numbers). Thus, Theorem
7.28 here means the 28th theorem in Chapter 7.
Contents
Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii
To the Reader. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix
CHAPTER 0
Introduction
Notation ................................................. 1
Brouwer Fixed Point Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
Categories and Functors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
CHAPTER 1
Some Basic Topological Notions 14
Homotopy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
Convexity, Contractibility, and Cones ........................... 18
Paths and Path Connectedness. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
CHAPTER 2
Simplexes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
Affine Spaces .............................................. 31
Affine Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
CHAPTER 3
The Fundamental Group. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
The Fundamental Groupoid. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
The Functor 'It 1 . • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • 44
'ltl(SI) ... " . .. . . .. . ... ... ... .. . .. . ..... ... . . . .. .... .. .... . 50