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Graduate Texts in Mathematics Anatoly Fomenko Dmitry Fuchs Homotopical Topology Second Edition Graduate Texts in Mathematics 273 Graduate Texts in Mathematics SeriesEditors: SheldonAxler SanFranciscoStateUniversity,SanFrancisco,CA,USA KennethRibet UniversityofCalifornia,Berkeley,CA,USA AdvisoryBoard: AlejandroAdem,UniversityofBritishColumbia DavidEisenbud,UniversityofCalifornia,Berkeley&MSRI IreneM.Gamba,TheUniversityofTexasatAustin J.F.Jardine,UniversityofWesternOntario JeffreyC.Lagarias,UniversityofMichigan KenOno,EmoryUniversity JeremyQuastel,UniversityofToronto FadilSantosa,UniversityofMinnesota BarrySimon,CaliforniaInstituteofTechnology Graduate Texts in Mathematics bridge the gap between passive study and creative understanding, offering graduate-level introductions to advanced topics in mathematics. The volumes are carefully written as teaching aids and highlight characteristic features of the theory. Although these books are frequently used as textbooksingraduatecourses,theyarealsosuitableforindividualstudy. Moreinformationaboutthisseriesathttp://www.springer.com/series/136 Anatoly Fomenko • Dmitry Fuchs Homotopical Topology Second Edition 123 AnatolyFomenko DmitryFuchs DepartmentofMathematics DepartmentofMathematics andMechanics UniversityofCalifornia MoscowStateUniversity Davis,CA,USA Moscow,Russia ISSN0072-5285 ISSN2197-5612 (electronic) GraduateTextsinMathematics ISBN978-3-319-23487-8 ISBN978-3-319-23488-5 (eBook) DOI10.1007/978-3-319-23488-5 LibraryofCongressControlNumber:2015958884 ©MoscowUniversityPress1969(FirsteditioninRussian) ©AkadémiaiKiadó1989(FirsteditioninEnglish) ©SpringerInternationalPublishingSwitzerland2016 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation, broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology nowknownorhereafterdeveloped. Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublication doesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevant protectivelawsandregulationsandthereforefreeforgeneraluse. Thepublisher,theauthorsandtheeditorsaresafetoassumethattheadviceandinformationinthisbook arebelievedtobetrueandaccurateatthedateofpublication.Neitherthepublishernortheauthorsor theeditorsgiveawarranty,expressorimplied,withrespecttothematerialcontainedhereinorforany errorsoromissionsthatmayhavebeenmade. Printedonacid-freepaper ThisSpringerimprintispublishedbySpringerNature TheregisteredcompanyisSpringerInternationalPublishingAGSwitzerland Preface The book we offer to the reader was conceived as a comprehensive course of homotopical topology, starting with the most elementary notions, such as paths, homotopies, and products of spaces, and ending with the most advanced topics, suchastheAdamsspectralsequenceandK-theory.Thehistoryofhomotopical,or algebraic,topologyisshortbutfullofsharpturnsandbreathtakingevents,andthis bookseekstofollowthishistoryasitunfolded. It is fair to say that homotopicaltopologybegan withAnalysisSitusby Henri Poincaré (1895). Poincaré showed that global analytic properties of functions, vectorfields,anddifferentialformsaregreatlyinfluencedbyhomotopicproperties of the relevant domains of definition. Poincaré’s methods were developed, in the half-century that followed AnalysisSitus, by a constellation of great topologists whichincludedsuchfiguresasJamesAlexander,HeinzHopf,AndreyKolmogorov, Hassler Whitney, and Lev Pontryagin. Gradually, it became clear that the homotopical properties of domains which were singled out by Poincaré could be best understoodintheformofgroupsandringsassociated,inahomotopyinvariantway, withatopologicalspacewhichhastobe“good”fromthegeometricpointofview, like smooth manifolds or triangulations. Thus, AnalysisSitus became algebraic topology. Stillthealgebrausedbythealgebraictopologyofthatepochwasveryelemen- tary:ItdidnotgofarbeyondtheclassificationoffinitelygeneratedAbeliangroups (withsomeexceptions,however,suchasVanKampen’stheoremaboutfundamental groups).Moreadvancedalgebra(which,actually,wasnotdevelopedbyalgebraists of that time and had to be started from scratch by topologists under the name “homological algebra,” aka “abstract nonsense”) invaded topology, which made algebraictopologytrulyalgebraic.Thishappenedinthelate1940sandearly1950s. Theleadingitemofthisnewalgebraappearedintheformofa“spectralsequence.” The true role of a spectral sequence in topology was discovered mainly by Jean- Pierre Serre (who was greatly influenced by older representatives of the French school of mathematics, mostly by Henri Cartan, Armand Borel, and JeanLeray). v vi Preface The impact of spectral sequences on algebraic topology was tremendous: Many major problems of topology, both solved and unsolved, became exercises for students. Theprogressofthenewalgebraictopologywasveryimpressivebutshort-lived: As early as in the late 1950s, the results became less and less interesting, and the proofsbecamemoreandmoreinvolved.Thelastbigachievementofthealgebraic topologywhichwasstartedbySerrewastheAdamsspectralsequence,which,ina sense,absorbedallmajornotionsandmethodsofcontemporaryalgebraictopology. Usinghisspectralsequence,J.FrankAdamswasabletoprovethefamousFrobenius conjecture(thedimensionofarealdivisionalgebramustbe1,2,4,or8);itwasalso usedbyRenéThominhisseminalwork,becomingthestartingpointoftheso-called cobordismtheory. Revivingailingalgebraictopologyrequiredstrongmeans,andsuchmeanswere found in the newly developed K-theory. Created by J. Frank Adams, Michael Atiyah, Raoul Bott, and Friedrich Hirzebruch, K-theory (which may be regarded as a branch of the broader “algebraic K-theory”) had applications which were unthinkable from the viewpoint of “classical” algebraic topology. It is sufficient to say that the Frobenius conjecture was reduced, via K-theory, to the following question: For which positive integers n is 3n (cid:2) 1 divisible by 2n? (Answer: for nD1;2;and4.) Developing K-theory was more or less completed in the mid-1960s.Certainly, it was not the end of algebraic topology. Very important results were obtained later; some of them, belonging to Sergei Novikov, Victor Buchstaber, Alexander Mishchenko,JamesBecker,andDanielGottlieb,arediscussedinthelastchapterof thisbook.Manyexcellentmathematicianscontinuetoworkinalgebraictopology. Still,onecansaythat,fromthestudents’pointofview,algebraictopologycannow beseenasacompleteddomain,anditispossibletostudyitfromthebeginningtothe end.(Wecanaddthatthisisnotonlypossible,butalsohighlyadvisable:Algebraic topology provides a necessary background for geometry, analysis, mathematical physics,etc.)Thisbookisintendedtohelpthereaderachievethisgoal. The book consists of an introduction and six chapters. The introduction intro- ducesthemostoftenusedtopologicalspaces(fromspherestotheCayleyprojective plane) and major operations over topological spaces (products, bouquets, sus- pensions, etc.). The chapter titles are as follows: “Homotopy”; “Homology”; “SpectralSequencesofFibrations”;“CohomologyOperations”;“TheAdamsSpec- tral Sequence”; and “K-theory and Other Extraordinary Theories.” Chapters are dividedintopartscalled“Lectures,”whicharenumeratedthroughoutthebookfrom Lecture1toLecture44.LecturesaredividedintosectionsnumeratedwithArabic numbers,andsomesectionsaredividedintosubsectionslabeledwithcapitalletters oftheRomanalphabet.Forexample,Lecture13consistsofSects.13.1,13.2,13.3, :::,13.11,andSect.13.8consistsofthesubsectionsA;B;:::;E(whicharereferred to,infurtherpartsofthebook,asSects.13.8.A,13.8.B,andsoon). Topresentthishugematerialinonevolumeofmoderatesize,wehadtobevery selective in presenting details of proofs. Many proofs in this book are algebraic, and they often involve routine verifications of independence of the result of a Preface vii constructionofsomearbitrarychoiceswithinthisconstruction,ofexactnessofthis orthatsequence,ofgrouporringaxiomsforthisorthatadditionormultiplication. Theseverificationsarenecessary,buttheyoftenrepeateachother,andifincludedin thebook,theywillonlyleadtoexcessivelyincreasingthevolumeandirritatingthe reader,whowillprobablyskipthem.Ontheotherhand,wedidnotwanttofollow some authorsof bookswhoskip detailsof proofswhichare inconvenient,for this orthatreason,foranhonestpresentation.Wedidourbesttoavoidthispattern:Ifa partofaproofislefttothereaderasanexercise,thenwearesurethatthisexercise shouldnotbe difficultfor somebodywho hasconsciouslyabsorbedthe preceding material. As should be clear from the preceding sentences, the book contains many exercises;actually,thereareapproximately500ofthem.Theyarenumeratedwithin each lecture. They may be serve the usual purposes of exercises: Instructors can use them for homework and tests, while readers can solve them to check their understanding or to get some additional information. But at least some of them must be regardedas a necessary part of the course; in some (not very numerous) caseswewillmakereferencestoexercisesfromprecedingsectionsorlectures.We hopethatthereaderwillappreciatethisstyle.Themostvisibleconsequenceofthis approachtoexercisesisthattheyarenotconcentratedinonespecialsection(which iscommonformanytextbooks)butratherscatteredthroughouteverylecture. Being a part of geometry, homotopic topology requires, for its understanding, a lot of graphic material. Our book contains more than 100 drawings (“figures”), which are supposed to clarify definitions, theorems, or proofs. But the book also containsachainofdrawingsthatarepiecesofartratherthanrigorousmathematical figures.ThesepicturesweredrawnbyA. Fomenko;someofthemweredisplayed at various exhibitions. All of them are supposed to present not the rigorous mathematicalmeaning,butratherthespiritandemotionalcontentsofnotionsand results of homotopicaltopology.They are located in the appropriateplaces in the book.Ashortexplanationforthesepicturescanbefoundattheendofthebook. We owe our gratitudeto manypeople.The first, mimeographed,versionof the beginningofthisbook(whichroughlycorrespondedtoChap.1andaconsiderable part of Chap. 2) was written in collaboration with Victor Gutenmacher; we are deeplygratefultohimforhishelp.Theideaofformallypublishingthisbookwas suggestedtousbySergeiNovikov.Wearegratefultohimforthissuggestion.Some improvements to the book were suggested by several students of the University of California; we are grateful to all of them, especially to Colin Hagemeyer. The wholeideaofpublishingthisbookundertheauspicesofSpringerbelongedtoBoris KhesinandAntonZorich;wethankthemheartily.Andthelastbut,maybe,themost importantthanksgotothebrilliantteamofeditorsatSpringer,especiallytoEugene HaandJayPopham.Itistheresultoftheirworkthatthebooklooksasattractiveas itdoes. Moscow,Russia AnatolyFomenko Davis,CA,USA DmitryFuchs viii Preface Contents Introduction:TheMostImportantTopologicalSpaces...................... 1 Lecture1. ClassicalSpaces .................................................. 1 Lecture2. BasicOperationsoverTopologicalSpaces...................... 16 Chapter1. Homotopy ......................................................... 25 Lecture3. HomotopyandHomotopyEquivalence ......................... 25 Lecture4. NaturalGroupStructuresintheSets(cid:2).X;Y/................... 33 Lecture5. CWComplexes................................................... 38 Lecture6. TheFundamentalGroupandCoverings......................... 60 Lecture7. VanKampen’sTheoremandFundamentalGroups ofCWComplexes................................................ 79 Lecture8. HomotopyGroups................................................ 93 Lecture9. Fibrations ......................................................... 104 Lecture10. TheSuspensionTheoremandHomotopyGroups ofSpheres......................................................... 121 Lecture11. HomotopyGroupsandCWComplexes......................... 129 Chapter2. Homology.......................................................... 143 Lecture12. MainDefinitionsandConstructions............................. 143 Lecture13. HomologyofCWComplexes.................................... 158 Lecture14. HomologyandHomotopyGroups............................... 178 Lecture15. HomologywithCoefficientsandCohomology ................. 183 Lecture16. Multiplications.................................................... 203 Lecture17. HomologyandManifolds........................................ 215 Lecture18. TheObstructionTheory.......................................... 253 Lecture19. VectorBundlesandTheirCharacteristicClasses............... 270 Chapter3. SpectralSequencesofFibrations ............................... 305 Lecture20. AnAlgebraicIntroduction....................................... 305 Lecture21. SpectralSequencesofaFilteredTopologicalSpace............ 321 Lecture22. SpectralSequencesof Fibrations:Definitions andBasicProperties.............................................. 324 Lecture23. AdditionalPropertiesofSpectralSequencesofFibrations..... 336 ix