Table Of ContentGraduate 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
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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.
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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)
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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 homo-
topical 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
