Algebraic topology, william fulton

Graduate Texts in Mathematics 153 Editorial Board S Axler F.w Gehring K.A Ribet Graduate Texts in Mathematics 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 TAKEUTI/ZARING Introduction to Axiomatic Set Theory 2nd ed OXTOBY Measure and Category 2nd ed SCHAEFER Topological Vector Spaces 2nd ed HILTON/STAMMBACH A Cour~e in Homological Algebra 2nd ed MAC LANE Categories for the Working Mathematician 2nd ed HUGHES/PIPER Projective Planes SERRE A Course in Arithmetic TAKEUTI/ZARING Axiomatic Set Theory HUMPHREYS Introduction to Lie Algebras and Representation Theory COHEN A Course in Simple Homotopy Theory CONWAY Functions of One Complex Variable I 2nd ed BEALS Advanced Mathematical Analysis ANDERSON/FuLLER Rings and Categories of Modules 2nd ed GOLUBITSKY/GUILLEMIN Stable Mappings and Their Singularities BERBERIAN Lectures in Functional Analysis and Operator Theory WINTER The Structure of Fields ROSENBLAn Random Processes 2nd ed HALMOS Measure Theory HALMOS A Hilbert Space Problem Book 2nd ed HUSEMOLLER Fibre Bundles 3rd ed HUMPHREYS Linear Algebraic Groups BARNES/MACK An Algebraic Introduction to Mathematical Logic GREUB Linear Algebra 4th ed HOLMES Geometric Functional Analysis and Its Applications HEWITT/STROMBERG Real and Abstract Analysis MANES Algebraic Theories KELLEY General Topology ZARISKI/SAMUEL Commutative Algebra Vol.1 ZARISKI/SAMUEL Commutative Algebra VoU! JACOBSON Lectures in Abstract Algebra I Basic Concepts JACOBSON Lectures in Abstract Algebra 11 Linear Algebra JACOBSON Lectures in Abstract Algebra 111 Theory of Fields and Galois Theory 33 HIRSCH Differential Topology 34 SPITZER Principles of Random Walk 2nd ed 35 ALEXANDER/WERMER Several Complex Variables and Banach Algebras 3rd ed 36 KELLEy/NAMIOKA et al Linear Topological Spaces 37 MONK Mathematical Logic 38 GRAUERT/FRITZSCHE Several Complex Variables 39 ARVESON An Invitation to CO-Algebras 40 KEMENy/SNELLiKNAPP Denumerable Markov Chains 2nd ed 41 ApOSTOL Modular Functions and Dirichlet Series in Number Theory 2nd ed 42 SERRE Linear Representations of Finite Groups 43 GILLMAN/JERISON Rings of Continuous Functions 44 KENDIG Elementary Algebraic Geometry 45 LOEVE Probability Theory I 4th ed 46 LOEVE Probability Theory II 4th ed 47 MOISE Geometric Topology in Dimensions and 48 SACHS/WU General Relativity for Mathematicians 49 GRUENBERG/WEIR Linear Geometry 2nd ed 50 EDWARDS Fermat's Last Theorem 51 KLINGENBERG A Course in Differential Geometry 52 HARTSHORNE Algebraic Geometry 53 MANIN A Course in Mathematical Logic 54 GRAVER/WATKINS Combinatorics with Emphasis on the Theory of Graphs 55 BROWN/PEARCY Introduction to Operator Theory I: Elements of Functional Analysis 56 MASSEY Algebraic Topology: An Introduction 57 CROWELL/Fox Introduction to Knot Theory 58 KOBLITZ p-adic Numbers, p-adic Analysis, and Zeta-Functions 2nd ed 59 LANG Cyclotomic Fields 60 ARNOLD Mathematical Methods in Classical Mechanics 2nd ed 61 WHITEHEAD Elements of Homotopy (continued after index) William Fulton Algebraic Topology A First Course With 137 Illustrations ~ Springer William Fulton Mathematics Department University of Chicago Chicago, IL 60637 USA Editorial Board S Axler Mathematics Department San Francisco State University San Francisco, CA 94132 USA F.w Gehring Mathematics Department East Hall University of Michigan Ann Arbor, MI 48109 USA K.A Ribet Mathematics Department University of California at Berkeley Berkeley, CA 94720-3840 USA Mathematics Subject Classifications (1991): 55-0 I Library of Congress Cataloging-in-Publication Data Fulton, William, 1939Algebraic topologylWilliam Fulton p cm - (Graduate texts in mathematics) Includes bibliographical references and index ISBN-13: 978-0-387-94327-5 I Algebraic topology QA612.F85 1995 I Title II Series 514'.2-dc20 ISBN-13: 978-0-387-94327-5 94-21786 e-ISI3N: 978-1-4612-4180-5 DOl: 10.1007/978-1-4612-4180-5 © 1995 Springer Science+Business Media, 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 Science+Business Media, Inc., 233 Spring Street, New York, NY 10013, 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 know or hereafter developed is forbidden The use in this publication of trade names, trademarks, service marks and similar terms, even if the 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 (EB) springeronline.com To the memory of my parents Preface To the Teacher This book is designed to introduce a student to some of the important ideas of algebraic topology by emphasizing the relations of these ideas with other areas of mathematics Rather than choosing one point of view of modem topology (homotopy theory, simplicial complexes, singular theory, axiomatic homology, differential topology, etc.), we concentrate our attention on concrete problems in low dimensions, introducing only as much algebraic machinery as necessary for the problems we meet This makes it possible to see a wider variety of important features of the subject than is usual in a beginning text The book is designed for students of mathematics or science who are not aiming to become practicing algebraic topologists-without, we hope, discouraging budding topologists We also feel that this approach is in better harmony with the historical development of the subject What would we like a student to know after a first course in topology (assuming we reject the answer: half of what one would like the student to know after a second course in topology)? Our answers to this have guided the choice of material, which includes: understanding the relation between homology and integration, first on plane domains, later on Riemann surfaces and in higher dimensions; winding numbers and degrees of mappings, fixed-point theorems; applications such as the Jordan curve theorem, invariance of domain; indices of vector fields and Euler characteristics; fundamental groups and covering spaces; the topology of surfaces, including intersection numbers; relations with complex analysis, especially on Riemann survii viii Preface faces; ideas of homology, De Rham cohomology, Cech cohomology, and the relations between them and with fundamental groups; methods of calculation such as the Mayer-Vietoris and Van Kampen theorems; and a taste of the way algebra and "functorial" ideas are used in the subject To achieve this variety at an elementary level, we have looked at the first nontrivial instances of most of these notions: the first homology group, the first De Rham group, the first Cech group, etc In the case of the fundamental group and covering spaces, however, we have bowed to tradition and included the whole story; here the novelty is on the emphasis on coverings arising from group actions, since these are what one is most likely to meet elsewhere in mathematics We have tried to this without assuming a graduate-level knowledge or sophistication The notes grew from undergraduate courses taught at Brown University and the University of Chicago, where about half the material was covered in one-semester and one-quarter courses By choosing what parts of the book to cover-and how many of the challenging problems to assign-it should be possible to fashion courses lasting from a quarter to a year, for students with many backgrounds Although we stress relations with analysis, the analysis we require or develop is certainly not "hard analysis." We start by studying questions on open sets in the plane that are probably familiar from calculus: When are path integrals independent of path? When are I-forms exact? (When vector fields have potential functions?) This leads to the notion of winding number, which we introduce first for differentiable paths, and then for continuous paths We give a wide variety of applications of winding numbers, both for their own interest and as a sampling of what can be done with a little topology This can be regarded as a glimpse of the general principle that algebra can be used to distinguish topological features, although the algebra (an integer!) is fairly meager We introduce the first De Rham cohomology group of a plane domain, which measures the failure of closed forms to be exact We use these groups, with the ideas of earlier chapters, to prove the Jordan curve theorem We also use winding numbers to study the singularities of vector fields Then I-chains are introduced as convenient objects to integrate over, and these are used to construct the first homology group We show that for plane open sets homology, winding numbers, and integrals all measure the same thing; the proof follows ideas of Brouwer, Artin, and Ahlfors, by approximating with grids As a first excursion outside the plane, we apply these ideas to sur- Preface ix faces, seeing how the global topology of a surface relates to local behavior of vector fields We also include applications to complex analysis The ideas used in the proof of the Jordan curve theorem are developed more fully into the Mayer-Vietoris story, which becomes our main tool for calculations of homology and cohomology groups Standard facts about covering spaces and fundamental groups are discussed, with emphasis on group actions We emphasize the construction of coverings by patching together trivial coverings, since these ideas are widely used elsewhere in mathematics (vector bundles, sheaf theory, etc.), and Cech cocycles and cohomology, which are widely used in geometry and algebra; they also allow, following Grothendieck, a very short proof of the Van Kampen theorem We prove the relation among the fundamental group, the first homology group, the first De Rham cohomology group, and the first Cech cohomology group, and the relation between cohomology classes, differential forms, and the coverings arising from multivalued functions We then turn to the study of surfaces, especially compact oriented surfaces We include the standard classification theorem, and work out the homology and cohomology, including the intersection pairing and duality theorems in this context This is used to give a brief introduction to Riemann surfaces, emphasizing features that are accessible with little background and have a topological flavor In particular, we use our knowledge of coverings to construct the Riemann surface of an algebraic curve; this construction is simple enough to be better known than it is The Riemann-Roch theorem is included, since it epitomizes the way topology can influence analysis Finally the last part of the book contains a hint of the directions the subject can go in higher dimensions Here we include the construction and basic properties of general singular (cubical) homology theory, and use it for some basic applications For those familiar with differential forms on manifolds, we include the generalization of De Rham theory and the duality theorems The variety of topics treated allows a similar variety of ways to use this book in a course, since many chapters or sections can be skipped without making others inaccessible The first few chapters could be used to follow or complement a course in point set topology A course with more algebraic topology could include the chapters on fundamental groups and covering spaces, and some of the chapters on surfaces It is hoped that, even if a course does not get near the last third of the book, students will be tempted to look there for some idea of where the subject can lead There is some progression in the level of difficulty, both in the text and the problems The last few chapters x Preface may be best suited for a graduate course or a year-long undergraduate course for mathematics majors We should also point out some of the many topics that are omitted or slighted in this treatment: relative theory, homotopy theory, fibrations, simplicial complexes or simplicial approximation, cell complexes, homology or cohomology with coefficients, and serious homological algebra To the Student Algebraic topology can be thought of as the study of the shapes of geometric objects It is sometimes referred to in popular accounts as "rubber-sheet geometry." In practice this means we are looking for properties of spaces that are unchanged when one space is deformed into another "Doughnuts and teacups are topologically the same." One problem of this type goes back to Euler: What relations are there among the numbers of vertices, edges, and faces in a convex polytope, such as a regular solid, in space? Another early manifestation of a topological idea came also from Euler, in the Konigsberg bridge problem: When can one trace out a graph without traveling over any edge twice? Both these problems have a feature that characterizes one of the main attractions, as well as the power, of modern algebraic topology-that a global question, depending on the overall shape of a geometric object, can be answered by data that are collected locally Since these are so appealing-and perhaps to capture your interest while we turn to other topics-they are included as problems with hints at the end of this Preface In fact, modern topology grew primarily out of its relation with other subjects, particularly analysis From this point of view, we are interested in how the shape of a geometric object relates to, or controls, the answers to problems in analysis Some typical and historically important problems here are: (i) whether differential forms w on a region that are closed (dw = 0) must be exact (w = djL) depends on the topology of the region; (ii) the behavior of vector fields on a surface depends on the topology of the surface; and (iii) the behavior of integrals f dx/YR(x) depends on the topology of the surface l = R(x), here with x and y complex variables In this book we will begin with the first of these problems, working primarily in open sets in the plane There is one disadvantage that must be admitted right away: this geometry is certainly flat, and lacks some of the appeal of doughnuts and teacups Later in the book we 418 Hints and Answers = to in Hom(A, IR), f vanishes on the image of a Since B /Image(a) C' , there is a homomorphism g' from C' to IR such that g' W= f By the lemma, there is a homomorphism g from C to IR that restricts to g' on C' Then go f3 = f, which means that g in Hom( C, IR) maps to f C.16 Let B =A2/Al' and show that there are exact sequences O-AI-A2-B-O and O-B-A3- -An-O References Included in this list are books and articles that were referred to in the text They can be consulted for more on topics covered in this book L.V Ahlfors, Complex Analysis, 3rd edn, McGraw-Hill, 1979 M.A Armstrong, Basic Topology, Springer-Verlag, 1983 R Bott and L.W Tu, Differential Forms in Algebraic Topology, SpringerVerlag, 1980 G.E Bredon, Topology and Geometry, Springer-Verlag, 1993 S.S Cairns, Introductory Topology, The Ronald Press, 1961 C Chevalley, Introduction to the Theory of Algebraic Functions of One Variable, American Mathematical Society, 1951 W.G Chinn and N.E Steenrod, First Concepts of Topology, Random House, 1966 H.S.M Coxeter, Introduction to Geometry, 2nd edn, Wiley, 1961, 1980; Wiley Classics, 1989 R.H Crowell and R.H Fox, Introduction to Knot Theory, Springer-Verlag, 1963 Dieudonne, A History of Algebraic and Differential Topology, 1990-1960, Birkhauser, 1989 C Godbillon, Elements de Topologie Algebrique, Hermann, 1971 M.J Greenberg and J.R Harper, Algebraic Topology, A First Course, Benjamin/Cummings, 1981 P.A Griffiths, Introduction to Algebraic Curves, American Mathematical Society, 1989 V Guillemin and A Pollack, Differential Topology, Prentice-Hall, 1974 419 420 References D Hilbert and S Cohn-Vossen, Geometry and the Imagination, Chelsea, 1952 P.I Hilton, An Introduction to Homotopy Theory, Cambridge University Press, 1961 J.G Hocking and G.S Young, Topology, Addison-Wesley, 1961, Dover, 1988 H Hopf, Differential Geometry in the Large, Springer Lecture Notes 1000, 1983 F Klein, On Riemann's Theory of Algebraic Functions and their Integrals, MacMillan and Bowes, 1893; Dover, 1963 S Lang, Introduction to Algebraic and Abelian Functions, 2nd edn, SpringerVerlag, 1982 W.S Massey, A Basic Course in Algebraic Topology, Springer-Verlag, 1991 J Milnor, Morse Theory, Annals of Mathematics Studies, Princeton University Press, 1963 J Milnor, Topology from the Differentiable Viewpoint, University Press of Virginia, 1965 R Narasimhan, Compact Riemann Surfaces, Birkhauser, 1992 M.H.A Newman, Elements of the Topology of Plane Sets of Points, Cambridge University Press, 1939 V.V Nikulin and I.R Shafarevich, Geometries and Groups, Springer-Verlag Universitext, 1987 H Rademacher and O Toeplitz, The Enjoyment of Mathematics, Princeton University Press, 1957 E.G Rees, Notes on Geometry, Springer-Verlag, 1983 G Springer, Introduction to Riemann Surfaces, Addison-Wesley, 1957 A Wallace, Differential Topology, First Steps, W.A Benjamin, 1968 Index of Symbols For some general notation, see page 365 smooth, I-form, differential, J~w, integral of I-form along smooth path, along segmented path, along a continuous path or I-chain, 127, 132 d, df, differential of function, form, 6,247,318,326,329,391 w{}, I-form for angle, aR, boundary of rectangle, 11, 80 ",p,p path around circle, 15 Wp,{}, I-form for angle around P, 16, 22 {}(t), angle function along path, 18 W(""O), winding number around 0, 19 :y, lifting of path, 21, 156 W("" P), winding number of", around P, 23, 36, 84 ",-t, inverse of path, 23, 165 grad (f), gradient, 28 Supp(",), support of path or chain, 42 W(F,P), winding number, 44, 328 deg(F), degree of mapping of circles,45 CQ,OO, degp(F), local degree, 46 D, disk, 50 C = aD, boundary circle of disk, 50, 80 P*, antipode of P, 53 HOU, Oth De Rham cohomology group, 63 HI U, 1st De Rham cohomology group, 63 [w], cohomology class of form w, 64 Wp = (]I27T)wp,{), 64 0, coboundarymap, 65-67, 224, 326 I-chain, 78-79 O-chain, 80-81 ZoU, group of O-chains on U, 81, 91 BoU, group of O-boundaries on U, 81, 91 HoU = ZoU /BoU, Oth homology group on U, 81, 91 C]U, group of I-chains on U, 82, 91 ZIU, group of I-chains on U, 82, 91 B 1U, group of I-boundaries on U, 83, 91 H]U=ZIU/B1U, 1st homology group on U, 83, 91 F *, map on chains or homology induced by F, 89, 92 IndexpV, index of vector field at point, 97, 104, 107 vic, restriction of V to C, 97 421 Index of Symbols 422 TpS, tangent space, 102 g, genus of surface, 108, 112 IRp2, projective plane, 115 W('Y,A), winding number around set A, 123 A"" infinite part of complement, 125 (n + I)-connected plane domain, 125 lJj(w) = lJ(w,A j ) , period, 130 Resif), residue off at a, 133 ordif), order off at a, 134 a, boundary map, 137-140, 334 S, subdivision operator, 139, 334335 MV(i) to MV(vi), Mayer-Vietoris properties, 140-142, 148, 258 +, -, maps on homology and co_ homology, 144, 148-149 HoX, reduced homology group, 145 flOx, reduced cohomology group, 149 n-sheeted covering, 155 y * 'Y, endpoint of lift of 'Y starting at y, 156-157 IRpn, real projective space, 159 Aut(YIX), group of deck transformations, 163 0"' T, product of paths, 165 En constant path at x, 165 '!TI(X,X), fundamental group of X at x, 168 ['Y], class of loop 'Y in '!TI(X,X), 168 e = [Ex], identity in '!TI(X, x), 168 T#, map induced by path T, 169 '!TI(X, X)abeh 173 y * [0"], endpoint of lift of 0" starting at y, 180 [0"] z, left action of fundamental group on covering, 182-184 PH: YH~X' covering from He '!TI(X, x), 189 Kabel, universal abelian covering, 192 Pp: Yp ~ X, covering from p: '!TI(X,X)~G, 193-194 (z x g), element of Yp from z E Y, gEG,194 HI(fJU;G), first Cech cohomology set, 209 E[(fJU,x;G), with base point, 210 M~M, orientation covering, 219 p",: X"'~X, covering from I-form, 221 HO(X;G), HI(X;G), cohomology, 222-225 covering from G-set and Gcovering, 225 Y(IjI)~X, covering from 1jI: G~G' and G-covering, 227 aj and bj, basic loops on a surface, 244 (0", T), intersection number, 245-246, 255-256, 357-358 (J.j and I3j, basic I-forms on a surface, 248-251 ffxv, integral of 2-form on surface, 251 1'1, wedge of forms, 252, 325, 355, 392 (w,I1), intersection number for 1forms, 252, 289 H 2X, second De Rham group, 257 e(P) = e/P) , ramification index, 265 ordp(f) , 267 CIA, Riemann surface of genus 1, 264, 275-276, 291-293 gx, genus of Riemann surface X, 273 Fw(Z, W), partial derivative, 277-278 M = M(X) , field of meromorphic functions, 281 C(z, w), field of rational functions in z and w, 282 n = nl,o = nl,o(X), space of holomorphic I-forms, 284 nO,1 = nO.I(X), antiholomorphic 1forms, 285 ordp(w), order of meromorphic 1form, 287 Resp(w), residue of meromorphic 1form, 288, 299 WI, • • • , w g , basis of holomorphic I-forms, 289 YT~X, Index of Symbols Z = (Tj.k), period matrix, 290 A, Abel-Jacobi mapping, 291 Div(!), divisor off, 291, 295 ordp(D), order of divisor at point, 295 deg(D), degree of divisor, 295 E 2: D, E - D is effective, 295 Div(w), divisor of meromorphic 1fonn,295 L(D), functions with poles allowed at D, 296 a(D), meromorphic I-fonns with zeros at D, 296 E, divisor of poles, 297 f = (jp), adele, 299 R, space of adeles, 299 R(D), adeles with poles allowed at D,300 SeD) = R/(R(D) + M), 300 a'(D), dual space to SeD), 300 ai, union of all a'(D), 302 HkU, De Rham cohomology group, 319,325 'lTiX, x), higher homotopy group, 324 H;X, De Rham cohomology with compact supports, 328 af, boundary slices, 332 CkX, k-chains on X, 332 a, af, boundary of cube or chain, 333 HkX = ZkX/BkX, homology group, 333 R, R(f), operator on chains, 333 423 A, A(r), operator on chains, 335 Rp, operator on chains, 335 Hk(X)CiJl, homology with small cubes, 338 a, boundary homomorphism, 348 H;X, homology using