Table Of Content

Allen Hatcher Copyright c 2002 by Cambridge University Press Single paper or electronic copies for noncommercial use may be made freely without explicit permission from the author or publisher. All other rights reserved. Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix StandardNotations xii. Chapter 0. Some Underlying Geometric Notions . . . . . 1 HomotopyandHomotopyType1. CellComplexes5. OperationsonSpaces8. TwoCriteriaforHomotopyEquivalence10. TheHomotopyExtensionProperty14. Chapter 1. The Fundamental Group . . . . . . . . . . . . . 21 1.1. Basic Constructions . . . . . . . . . . . . . . . . . . . . . 25 PathsandHomotopy25. TheFundamentalGroupoftheCircle29. InducedHomomorphisms34. 1.2. Van Kampen’s Theorem . . . . . . . . . . . . . . . . . . . 40 FreeProductsofGroups41. ThevanKampenTheorem43. ApplicationstoCellComplexes50. 1.3. Covering Spaces . . . . . . . . . . . . . . . . . . . . . . . . 56 LiftingProperties60. TheClassificationofCoveringSpaces63. DeckTransformationsandGroupActions70. Additional Topics 1.A. GraphsandFreeGroups 83. 1.B. K(G,1)SpacesandGraphsofGroups 87. Chapter 2. Homology . . . . . . . . . . . . . . . . . . . . . . . 97 2.1. Simplicial and Singular Homology . . . . . . . . . . . . . 102 (cid:209) Complexes102. SimplicialHomology104. SingularHomology108. HomotopyInvariance110. ExactSequencesandExcision113. TheEquivalenceofSimplicialandSingularHomology128. 2.2. Computations and Applications . . . . . . . . . . . . . . 134 Degree134. CellularHomology137. Mayer-VietorisSequences149. HomologywithCoefficients153. 2.3. The Formal Viewpoint . . . . . . . . . . . . . . . . . . . . 160 AxiomsforHomology160. CategoriesandFunctors162. Additional Topics 2.A. HomologyandFundamentalGroup 166. 2.B. ClassicalApplications 169. 2.C. SimplicialApproximation 177. Chapter 3. Cohomology . . . . . . . . . . . . . . . . . . . . . 185 3.1. Cohomology Groups . . . . . . . . . . . . . . . . . . . . . 190 TheUniversalCoefficientTheorem190. CohomologyofSpaces197. 3.2. Cup Product . . . . . . . . . . . . . . . . . . . . . . . . . . 206 TheCohomologyRing211. AKu¨nnethFormula218. SpaceswithPolynomialCohomology224. 3.3. Poincaré Duality . . . . . . . . . . . . . . . . . . . . . . . . 230 OrientationsandHomology233. TheDualityTheorem239. ConnectionwithCupProduct249. OtherFormsofDuality252. Additional Topics 3.A. UniversalCoefficientsforHomology 261. 3.B. TheGeneralKu¨nnethFormula 268. 3.C. H–SpacesandHopfAlgebras 281. 3.D. TheCohomologyofSO(n) 292. 3.E. BocksteinHomomorphisms 303. 3.F. LimitsandExt 311. 3.G. TransferHomomorphisms 321. 3.H. LocalCoefficients 327. Chapter 4. Homotopy Theory . . . . . . . . . . . . . . . . . 337 4.1. Homotopy Groups . . . . . . . . . . . . . . . . . . . . . . 339 DefinitionsandBasicConstructions340. Whitehead’sTheorem346. CellularApproximation348. CWApproximation352. 4.2. Elementary Methods of Calculation . . . . . . . . . . . . 360 ExcisionforHomotopyGroups360. TheHurewiczTheorem366. FiberBundles375. StableHomotopyGroups384. 4.3. Connections with Cohomology . . . . . . . . . . . . . . 393 TheHomotopyConstructionofCohomology393. Fibrations405. PostnikovTowers410. ObstructionTheory415. Additional Topics 4.A. BasepointsandHomotopy 421. 4.B. TheHopfInvariant 427. 4.C. MinimalCellStructures 429. 4.D. CohomologyofFiberBundles 431. 4.E. TheBrownRepresentabilityTheorem 448. 4.F. SpectraandHomologyTheories 452. 4.G. GluingConstructions 456. 4.H. Eckmann-HiltonDuality 460. 4.I. StableSplittingsofSpaces 466. 4.J. TheLoopspaceofaSuspension 470. 4.K. TheDold-ThomTheorem 475. 4.L. SteenrodSquaresandPowers 487. Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 519 TopologyofCellComplexes519. TheCompact-OpenTopology529. Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 533 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 539 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 themostimportantresultsandtechniques. Forexample,CWcomplexeshaveproved over time to be the most natural class of spaces for algebraic topology, so they are emphasizedheremuchmorethaninthebooksofanearliergeneration. Thisempha- sisalsoillustratesthebook’sgeneralslanttowardsgeometric, ratherthanalgebraic, aspectsofthesubject. Thegeometryofalgebraictopologyissopretty,itwouldseem apitytoslightitandtomissalltheintuitionitprovides. Attheelementarylevel,algebraictopologyseparatesnaturallyintothetwobroad channels of homology and homotopy. This material is here divided into four chapters, roughly according to increasing sophistication, with homotopy split between Chapters 1 and 4, and homology and its mirror variant cohomology in Chapters 2 and3. Thesefourchaptersdonothavetobereadinthisorder,however. Onecould begin with homology and perhaps continue with cohomology before turning to homotopy. Intheotherdirection,onecouldpostponehomologyandcohomologyuntil afterpartsofChapter4. Ifthislatterstrategyispushedtoitsnaturallimit,homology and cohomology can be developed just as branches of homotopy theory. Appealing asthisapproachisfromastrictlylogicalpointofview,itplacesmoredemandsonthe reader,andsincereadabilityisoneofthefirstprioritiesofthebook,thishomotopic interpretationofhomologyandcohomologyisdescribedonlyafterthelattertheories havebeendevelopedindependentlyofhomotopytheory. 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 beforetheotherchaptersorskippedandreferredbacktolaterforspecifictopicsas theybecomeneededinthesubsequentchapters. Eachofthefourmainchaptersconcludeswithaselectionofadditionaltopicsthat thereadercansampleatwill,independentofthebasiccoreofthebookcontainedin theearlierpartsofthechapters. Manyoftheseextratopicsareinfactratherimportant in the overall scheme of algebraic topology, though they might not fit into the time constraintsofafirstcourse. Altogether,theseadditionaltopicsamounttonearlyhalf thebook,andtheyareincludedherebothtomakethebookmorecomprehensiveand togivethereaderwhotakesthetimetodelveintothemamoresubstantialsampleof thetruerichnessandbeautyofthesubject. 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],itsprovisionaltitlebeing‘VectorBundlesandK–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 [Jänich1984]listedintheBibliography. Abooksuchasthisone,whoseaimistopresentclassicalmaterialfromarather classicalviewpoint,isnottheplacetoindulgeinwildinnovation. Neverthelessthereis onenewfeatureoftheexpositionthatmaybeworthcommentingupon,eventhough in the book as a whole it plays a relatively minor role. This is a modest extension of theclassicalnotionofasimplicialcomplexthatgoesunderthenameofa (cid:209) complex inthisbook. Theideaistoallowdifferentfacesofasimplextocoincide,soonlythe interiorsofsimplicesareembeddedandsimplicesarenolongeruniquelydetermined by their vertices. (As a technical point, an ordering of the vertices of each simplex is also part of the structure of a (cid:209) complex.) For example, if one takes the standard picture of the torus as a square with opposite edges identified and divides the squareintotwotrianglesbycuttingalongadiagonal,thentheresultisa (cid:209) complex structure on the torus having 2 triangles, 3 edges, and 1 vertex. By contrast, it is known that a simplicial complex structure on the torus must have at least 14 triangles,21edges,and7vertices. So (cid:209) complexesprovideasignificantimprovementin efficiency, which is nice from a pedagogical viewpoint since it cuts down on tedious calculations in examples. A more fundamental reason for considering (cid:209) complexes isthattheyseemtobeverynaturalobjectsfromtheviewpointofalgebraictopology. They are the natural domain of definition for simplicial homology, and a number of standard constructions produce (cid:209) complexes rather than simplicial complexes, for instancethesingularcomplexofaspace,ortheclassifyingspaceofadiscretegroup or category. In spite of this naturality, (cid:209) complexes have appeared explicitly in the literatureonlyrarely,andnostandardnameforthenotionhasemerged. Thisbookwillremainavailableonlineinelectronicformafterithasbeenprinted inthetraditionalfashion. Thewebaddressis http://www.math.cornell.edu/˜hatcher One can also find here the parts of the other two books in the sequence that are currentlyavailable. Althoughthepresentbookhasgonethroughcountlessrevisions, including the correction of many small errors both typographical and mathematical found by careful readers of earlier versions, it is inevitable that some errors remain, so the web page will include 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 tosendcommentsandsuggestionsaswellascorrectionstotheemailaddressposted onthewebpage. Standard Notations Z, Q, R, C, H, O: theintegers, rationals, reals, complexes, quaternions, andCayleyoctonions Z : theintegersmodn n Rn: n dimensionalEuclideanspace Cn: complex n space I (cid:131)(cid:134)0;1(cid:135): theunitinterval Sn: theunitspherein Rn(cid:130)1,allvectorsoflength1 Dn: theunitdiskorballin Rn,allvectorsoflength (cid:20)1 @Dn (cid:131)Sn−1: theboundaryofthe n disk 11: theidentityfunctionfromasettoitself q: disjointunionofsetsorspaces Q (cid:2), : productofsets,groups,orspaces (cid:25): isomorphism A(cid:26)B or B (cid:27)A: set-theoreticcontainment,notnecessarilyproper iff: ifandonlyif Theaimofthisshortpreliminarychapteristointroduceafewofthemostcom- 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 shouldbereadinthisinformalspirit,skippingbitshereandthere. Infact,thiswhole chaptercouldbeskippednow,tobereferredbacktolaterforbasicdefinitions. To avoid overusing the word ‘continuous’ we adopt the convention that maps be- tweenspacesarealwaysassumedtobecontinuousunlessotherwisestated. Homotopy and Homotopy Type Oneofthemainideasofalgebraictopologyistoconsidertwospacestobeequiv- 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 written either as unions of finitely many straightandcurvedlinesegments,or in thickened forms that are compact subsurfaces of the plane bounded by simpleclosedcurves. Ineachcasethe thin letter is a subspace of the thick letter, andwecancontinuouslyshrinkthethicklettertothethinone. Anicewayto 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 indi- cated inthe figure. Thenwecan shrink X to X by sliding eachpoint of X−X into X alongthelinesegmentthatcontainsit. Pointsthatarealreadyin X donotmove. We can think of this shrinking process as taking place during a time interval ! 0(cid:20)t(cid:20)1,andthenitdefinesafamilyoffunctions f :X X parametrizedby t2I (cid:131) t (cid:134)0;1(cid:135), where f (cid:132)x(cid:133) is the point to which a given point x 2 X has moved at time t. t