Algebraic Topology PDF
Preview Algebraic Topology
Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix Standard Notations xii. Chapter 0. Some Underlying Geometric Notions . . . . . 1 Homotopy and Homotopy Type 1. Cell Complexes 5. Operations on Spaces 8. Two Criteria for Homotopy Equivalence 10. The Homotopy Extension Property 14. Chapter 1. The Fundamental Group . . . . . . . . . . . . . 21 1.1. Basic Constructions . . . . . . . . . . . . . . . . . . . . . 25 Paths and Homotopy 25. The Fundamental Group of the Circle 29. Induced Homomorphisms 34. 1.2. Van Kampen’s Theorem . . . . . . . . . . . . . . . . . . . 40 Free Products of Groups 41. The van Kampen Theorem 43. Applications to Cell Complexes 49. 1.3. Covering Spaces . . . . . . . . . . . . . . . . . . . . . . . . 56 Lifting Properties 60. The Classification of Covering Spaces 63. Deck Transformations and Group Actions 70. Additional Topics 1.A. Graphs and Free Groups 83. 1.B. K(G,1) Spaces and Graphs of Groups 87. Chapter 2. Homology . . . . . . . . . . . . . . . . . . . . . . . 97 2.1. Simplicial and Singular Homology . . . . . . . . . . . . . 102 Complexes 102. Simplicial Homology 104. Singular Homology 108. Homotopy Invariance 110. Exact Sequences and Excision 113. & The Equivalence of Simplicial and Singular Homology 128. 2.2. Computations and Applications . . . . . . . . . . . . . . 134 Degree 134. Cellular Homology 137. Mayer-Vietoris Sequences 149. Homology with Coefficients 153. 2.3. The Formal Viewpoint . . . . . . . . . . . . . . . . . . . . 160 Axioms for Homology 160. Categories and Functors 162. Additional Topics 2.A. Homology and Fundamental Group 166. 2.B. Classical Applications 169. 2.C. Simplicial Approximation 177. Chapter 3. Cohomology . . . . . . . . . . . . . . . . . . . . . 185 3.1. Cohomology Groups . . . . . . . . . . . . . . . . . . . . . 190 The Universal Coefficient Theorem 190. Cohomology of Spaces 197. 3.2. Cup Product . . . . . . . . . . . . . . . . . . . . . . . . . . 206 The Cohomology Ring 212. A Ku¨nneth Formula 214. Spaces with Polynomial Cohomology 220. 3.3. Poincar´e Duality . . . . . . . . . . . . . . . . . . . . . . . . 230 Orientations and Homology 233. The Duality Theorem 239. Connection with Cup Product 249. Other Forms of Duality 252. Additional Topics 3.A. Universal Coefficients for Homology 261. 3.B. The General Ku¨nneth Formula 268. 3.C. H–Spaces and Hopf Algebras 281. 3.D. The Cohomology of SO(n) 292. 3.E. Bockstein Homomorphisms 303. 3.F. Limits and Ext 311. 3.G. Transfer Homomorphisms 321. 3.H. Local Coefficients 327. Chapter 4. Homotopy Theory . . . . . . . . . . . . . . . . . 337 4.1. Homotopy Groups . . . . . . . . . . . . . . . . . . . . . . 339 Definitions and Basic Constructions 340. Whitehead’s Theorem 346. Cellular Approximation 348. CW Approximation 352. 4.2. Elementary Methods of Calculation . . . . . . . . . . . . 360 Excision for Homotopy Groups 360. The Hurewicz Theorem 366. Fiber Bundles 375. Stable Homotopy Groups 384. 4.3. Connections with Cohomology . . . . . . . . . . . . . . 393 The Homotopy Construction of Cohomology 393. Fibrations 405. Postnikov Towers 410. Obstruction Theory 415. Additional Topics 4.A. Basepoints and Homotopy 421. 4.B. The Hopf Invariant 427. 4.C. Minimal Cell Structures 429. 4.D. Cohomology of Fiber Bundles 431. 4.E. The Brown Representability Theorem 448. 4.F. Spectra and Homology Theories 452. 4.G. Gluing Constructions 456. 4.H. Eckmann-Hilton Duality 460. 4.I. Stable Splittings of Spaces 466. 4.J. The Loopspace of a Suspension 470. 4.K. The Dold-Thom Theorem 475. 4.L. Steenrod Squares and Powers 487. Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 519 Topology of Cell Complexes 519. The Compact-Open Topology 529. The Homotopy Extension Property 533. Simplicial CW Structures 534. Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 540 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 546 This book was written to be a readable introduction to algebraic topology with rather broad coverage of the subject. The viewpoint is quite classical in spirit, and stays well within the confines of pure algebraic topology. In a sense, the book could have been written thirty or forty years ago since virtually everything in it is at least that old. However, the passage of the intervening years has helped clarify what are the most important results and techniques. For example, CW complexes have proved over time to be the most natural class of spaces for algebraic topology, so they are emphasized here much more than in the books of an earlier generation. This empha- sis also illustrates the book’s general slant towards geometric, rather than algebraic, aspects of thesubject. Thegeometryofalgebraic topologyis sopretty, it wouldseem a pity to slight it and to miss all the intuition it provides. Attheelementarylevel,algebraictopologyseparatesnaturallyintothetwobroad channels of homology and homotopy. This material is here divided into four chap- ters, roughly according to increasing sophistication, with homotopy split between Chapters 1 and 4, and homology and its mirror variant cohomology in Chapters 2 and 3. These four chapters do not have to be read in this order, however. One could begin with homology and perhaps continue with cohomology before turning to ho- motopy. In the other direction, one could postpone homology and cohomology until after parts ofChapter 4. Ifthis latterstrategy ispushedto its natural limit,homology and cohomology can be developed just as branches of homotopy theory. Appealing asthisapproachisfromastrictlylogicalpointofview,itplacesmoredemandsonthe reader, and since readability is one of the first priorities of the book, this homotopic interpretationofhomologyandcohomologyisdescribedonlyafter thelattertheories have been developed independently of homotopy theory. Preceding the four main chapters there is a preliminary Chapter 0 introducing some of the basic geometric concepts and constructions that play a central role in both the homological and homotopical sides of the subject. This can either be read before the other chapters or skipped and referred back to later for specific topics as they become needed in the subsequent chapters. Eachofthefourmainchaptersconcludeswithaselectionofadditionaltopicsthat the reader can sample at will, independent of the basic core of the book contained in theearlierpartsofthechapters. Manyoftheseextratopicsareinfactratherimportant in the overall scheme of algebraic topology, though they might not fit into the time x Preface constraintsofafirstcourse. Altogether,theseadditionaltopicsamounttonearlyhalf thebook,and theyareincluded hereboth to makethebookmorecomprehensiveand to givethe reader who takes the time to delveinto them a moresubstantial sampleof the true richness and beauty of the subject. There is also an Appendix dealing mainly with a number of matters of a point- set topological nature that arise in algebraic topology. Since this is a textbook on algebraic topology, details involving point-set topology are often treated lightly or skipped entirely in the body of the text. Not included in this book is the important but somewhat more sophisticated topic of spectral sequences. It was very tempting to include something about this marvelous tool here, but spectral sequences are such a big topic that it seemed best tostartwiththemafreshinanewvolume. Thisistentativelytitled‘SpectralSequences inAlgebraicTopology’andisreferredtohereinas[SSAT].Thereisalsoathirdbookin progress,onvectorbundles,characteristicclasses,andK–theory,whichwillbelargely independent of [SSAT] and also of much of the present book. This is referred to as [VBKT], its provisional title being ‘Vector Bundles and K–Theory’. Intermsofprerequisites,thepresentbookassumesthereaderhassomefamiliar- ity with the content of the standard undergraduate courses in algebra and point-set topology. In particular, the reader should know about quotient spaces, or identifi- cation spaces as they are sometimes called, which are quite important for algebraic topology. Good sources for this concept are the textbooks [Armstrong 1983] and [J¨anich 1984] listed in the Bibliography. A book such as this one, whose aim is to present classical material from a rather classical viewpoint, is not the place to indulge in wild innovation. There is, however, onesmallnoveltyintheexpositionthatmaybeworthcommentingupon,eventhough in the book as a whole it plays a relatively minor role. This is the use of what we call complexes, which are a mild generalization of the classical notion of a simplicial complex. The idea is to decompose a space into simplices allowing different faces & of a simplex to coincide and dropping the requirement that simplices are uniquely determined by their vertices. For example, if one takes the standard picture of the torus as a square with opposite edges identified and divides the square into two tri- angles by cutting along a diagonal, then the result is a complex structure on the torus having 2 triangles, 3 edges, and 1 vertex. By contrast, a simplicial complex & structure on the torus must have at least 14 triangles, 21 edges, and 7 vertices. So complexesprovideasignificantimprovementinefficiency,whichisnicefromaped- agogical viewpoint since it simplifies calculations in examples. A more fundamental & reason forconsidering complexesis thattheyseemto beverynatural objects from the viewpoint of algebraic topology. They are the natural domain of definition for & simplicial homology, and a number of standard constructions produce complexes rather than simplicial complexes. Historically, complexes were first introduced by & & Preface xi Eilenberg and Zilber in 1950 under the name of semisimplicial complexes. Soon after this, additional structure in the form of certain ‘degeneracy maps’ was introduced, leading to a very useful class of objects that came to be called simplicial sets. The semisimplicial complexes of Eilenberg and Zilber then became ‘semisimplicial sets’, but in this book we have chosen to use the shorter term ‘ complex’. This book will remain availableonline in electronic form after ithas been printed & in the traditional fashion. The web address is http://www.math.cornell.edu/˜hatcher One can also find here the parts of the other two books in the sequence that are currently available. Although the present book has gone through countless revisions, including the correction of many small errors both typographical and mathematical foundbycarefulreadersofearlierversions,itisinevitablethatsomeerrorsremain,so the web page includes a list of corrections to the printed version. With the electronic version of the book it will be possible not only to incorporate corrections but also to make more substantial revisions and additions. Readers are encouraged to send comments and suggestions as well as corrections to the email address posted on the web page. Note on the 2015 reprinting. A large number of corrections are included in this reprinting. In addition there are two places in the book where the material was re- arranged to an extent requiring renumbering of theorems, etc. In §3.2 starting on page 210 the renumbering is the following: old 3.11 3.12 3.13 3.14 3.15 3.16 3.17 3.18 3.19 3.20 3.21 new 3.16 3.19 3.14 3.11 3.13 3.15 3.20 3.16 3.17 3.21 3.18 And in §4.1 the following renumbering occurs in pages 352–355: old 4.13 4.14 4.15 4.16 4.17 new 4.17 4.13 4.14 4.15 4.16 xii Standard Notations Z, Q, R, C, H, O : the integers, rationals, reals, complexes, quaternions, and octonions. Z : the integers modn. n Rn : n dimensional Euclidean space. Cn : complex n space. In particular, R0 = {0}= C0, zero-dimensional vector spaces. I = [0,1] : the unit interval. Sn : the unit sphere in Rn+1, all points of distance 1 from the origin. Dn : the unit disk or ball in Rn, all points of distance f 1 from the origin. "Dn = Sn21 : the boundary of the n disk. en : an n cell, homeomorphic to the open n disk Dn 2"Dn. In particular, D0 and e0 consist of a single point since R0 = {0}. But S0 consists of two points since it is "D1. 11 : the identity function from a set to itself. : disjoint union of sets or spaces. `×, : product of sets, groups, or spaces. j : Qisomorphism. A ¢ B or B £ A : set-theoretic containment, not necessarily proper. AB : the inclusion map A³B when A ¢ B. A2B : set-theoretic difference, all points in A that are not in B. iff : if and only if. There are also a few notations used in this book that are not completely standard. The union of a set X with a family of sets Y , with i ranging over some index set, i is usually written simply as X * Y rather than something more elaborate such as i i X * Y . Intersections and other similar operations are treated in the same way. i i (cid:0)S (cid:1) Definitions of mathematical terms are given within paragraphs of text, rather than displayed separately like theorems. These definitions are indicated by the use of boldfacetypeforthemoreimportantterms,withitalicsbeingusedforlessimportant or lessformal definitions as well as for simpleemphasisas in standardwritten prose. Terms defined using boldface appear in the Index, with the page number where the definition occurs listed first. The aim of this short preliminary chapter is to introduce a few of the most com- mon geometric concepts and constructions in algebraic topology. The exposition is somewhat informal, with no theorems or proofs until the last couple pages, and it should beread in this informal spirit, skippingbits here and there. In fact, this whole chapter could be skipped now, to be referred back to later for basic definitions. To avoid overusing the word ‘continuous’ we adopt the convention that maps be- tween spaces are always assumed to be continuous unless otherwise stated. Homotopy and Homotopy Type Oneofthemain ideas ofalgebraictopologyisto considertwo spaces tobeequiv- alent if they have ‘the same shape’ in a sense that is much broader than homeo- morphism. To take an everyday example, the letters of the alphabet can be writ- ten either as unions of finitely many straight and curved line segments, or in thickened forms that are compact regions in the plane bounded by one ormoresimpleclosedcurves. Ineach case the thin letter is a subspace of thethickletter,and wecan continuouslyshrinkthethicklettertothethinone. Anice way to do this is to decompose a thick letter, call it X, into line segments connecting each point on the outer boundary of X to a unique point of the thin subletter X, as indicated in the figure. Then we can shrink X to X by sliding each point of X2X into X along the line segment that contains it. Points that are already in X do not move. We can think of this shrinking process as taking place during a time interval ³ 0 f t f 1, and then it defines a family of functions f :X X parametrized by t * I = t [0,1], where f (x) is the point to which a given point x * X has moved at time t. t Naturally we would like f (x) to depend continuously on both t and x, and this will t
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