User's guide to spectral sequences

575 Pages·2000·3.481 MB·English

by  John McCleary

Tomyfamily:
tomyDadandtothememoryofmyMom,
toCarlie,
andtomyboys,JohnandAnthony.

Preface
tothesecondedition
“ForIknowmytransgressions,::: ”
Psalm51
Thefirsteditionofthisbookservedasmyintroductiontothemysteriesof
spectralsequences. Sincewritingit,Ihavelearnedalittlemoreintryingtodo
somealgebraictopologyusingthesetools. ThesensethatIhadmisrepresented
sometopics,misledthereader,evenwrittendownmistakennotions,grewover
theyears. Whenthefirsteditioncametotheendofitsrun,andwasgoingout
ofprint,Iwasencouragedbysomegeneroussoulstoconsiderasecondedition
withthegoalofeliminatingmanyoftheerrorsthathadbeenfoundandbringing
itsomewhatuptodate.
The most conspicuous change to the first edition is the addition of new
chapters—Chapter 8bis on nontrivial fundamental groups and Chapter 10 on
the Bockstein spectral sequence. In Chapter 8bis (an address, added on after
Chapters 5 through 8, but certainly belonging in that neighborhood), I have
foundanaturalplacetodiscusstheCartan-LerayandtheLyndon-Hochschild-
Serre spectral sequences, as well as the important class of nilpotent spaces.
Thischapterisanoddmixtureoftopics,butIbelievetheyhangtogetherwell
andadddetailstoearlierdiscussionsthatdependedonthefundamentalgroup.
Chapter10acknowledgesthefundamentalrolethattheBocksteinspectralse-
quenceplaysinhomotopytheory,especiallyinthestudyofH-spaces. Itisas
muchabasictoolastheotherspectralsequencesofPartII.
Lessconspicuously,IhavechangedtheorderoftopicsinChapters2and
3 in order to focus better on convergence in Chapter 3, which includes an
expositionoftheimportantpaperof[Boardman99]. Ihavereorderedthetopics
inChapter8tomakeitmoreparalleltoChapter6. Theproofoftheexistence
andstructureoftheLeray-Serrespectralsequenceisalsosignificantlychanged.
I have followed the nice paper of [Brown, E94]. With this change, I have
added a proof of the multiplicative structure that was not in the first edition.
Chapter9nowsportsadiscussionoftheroleoftheAdamsspectralsequence
inthecomputationofcobordismrings.
Manyoftheintendedimprovementsinthiseditionaresmalldetailsthat
arementionedintheacknowledgments. Detailsthatarenoticeablethroughout
includeachangeintheconventionforcitation. (WhatwasIthinkinginthefirst
viii Preface
edition?) Inthiseditionthereaderisinvitedtoreadthecitationsasanintegral
partofthetext. Inthecaseofmultiplepapersinagivenyear,Ihaveaddeda
primetotheyeartodistinguishpapers. Theotherlittleglobalchangeisanend
ofproofmarkerut(suggestedbyMicheleIntermont).
Ihadoncethoughtthatwritingabookwouldbeeasyandthefirstedition
curedmeofthatmisconception. Ihavediscoveredthatwritingasecondedition
isn’teasyeither. IwillthankothersattheendoftheIntroduction,butIwishto
thankcertainfolkswhoseencouragement,kindwords,andsteadfastnessmade
thecompletionofthesecondeditionpossible. Firstaremyteachersintheuse
ofspectralsequences, andinthewritingofbooks, JimStasheff, BobBruner,
andLarrySmith. Theyhaveallgivenmorethanonecouldexpectofafriend.
To you, I owe many thanks for so many kindnesses. In an effort to avoid a
second edition full of little errors that frustrate even the most diligent reader,
Hal Sadofsky organized an army of folks who read the penultimate version
of most chapters. This act of organization was most welcome, helpful and
generous. ThoughImayhaveaddednewtyposinanefforttofixfounderrors,I
amsurethatthebookismuchbetterforHal’sefforts. AtVassar,DianeWinkler
gavesomeofhervaluabletimetohelpinthepreparationofthebibliography
andindex. BenLottosolvedallmycomputerproblems,andFloraGrabowska
hunteddownreferencematerialIalwaysseemedtoneedyesterday. Muchof
theworkonthiseditionwasdoneduringuneanne´esabbatiquea` Strasbourg.
MythankstoChristianKasselandJean-LouisLodayfortheirhospitalityduring
that stay. In the department of steadfastness, many thanks go to my editor at
CambridgeUniversityPress,LaurenCowles,whosepatienceisextraordinary.
Finally, my thanks to my family—Carlie, John, and Anthony—for tolerating
myprojectsandfortheirlovethroughwhatseemedlikeaneverendingstory.
JohnMcCleary July17,2000
Poughkeepsie,NY
Introduction
“Itisnowabundantlyclearthatthespectralsequenceis
oneofthefundamentalalgebraicstructuresneededfor
dealingwithtopologicalproblems.”
W.S.Massey
Topologistsarefondoftheirmachinery. Asthetitleofthisbookindicates,
myintentionistoprovideauser’smanualfortheclassofcomplicatedalgebraic
gadgetsknownasspectralsequences.
A‘good’user’smanualforanyapparatusshouldsatisfycertainexpecta-
tions. It should provide the beginner with sufficient details in exposition and
examples to feel comfortable in starting to apply the new apparatus to his or
herproblems. Themanualshouldalsoincludeenoughdetailsabouttheinner
workings of the apparatus to allow a user to determine what is going on if it
failswhileinoperation. Finally,auser’smanualshouldincludeplentyofinfor-
mationfortheexpertwhoislookingfornewwaystousethedevice. Inwriting
thisbook,Ihavekeptthesegoalsinmind.
Thereareseveralclassesofreadersforwhomthisbookiswritten. There
isthestudentofalgebraictopologywhoisinterestedinlearninghowtoapply
spectralsequencestoquestionsintopology. Thisreaderisexpectedtohaveseen
abasiccourseintopologyatthelevelofthetextsby[Massey91]and[May99]
onsingularhomologytheoryandincludingthedefinitionofhomotopygroups
andtheirbasicproperties. Thisbeginneralsoneedsanacquaintancewiththe
basictopicsofthehomologicalalgebraofringsandmodules,atthelevelofthe
firstthreechaptersofthebookof[Weibel94].
Thenextclassofreaderisprincipallyinterestedinalgebraandheorshe
wants an exposition of the basic notions about spectral sequences, hopefully
withouttoomuchtopologyasprerequisite. PartIandChapter12areintended
forthesereaders,alongwithx7.1,x8bis.2,andx9.2.
Somesectionsofthebookareintendedfortheexperienceduserandwould
offeranunenlighteningdetourforthenovice. Ihavemarkedthesesectionswith
thesymbol
fi
N
for ‘not for the novice.’ As with other users’ manuals, these sections will
becomeusefulwhenthereaderbecomesfamiliarwithspectralsequencesand
hasaneedforparticularresults.
x Introduction
The material in the book is organized into three parts. Part I is called
Algebra and consists of Chapters 1, 2, and 3. The intention in Part I is to
lay the algebraic foundations on which the construction and manipulation of
all subsequent examples will stand. Chapter 1 is a gentle introduction to the
manipulationoffirstquadrantspectralsequences;theproblemofhowtocon-
struct a spectral sequence is set aside and some of the formal aspects of the
algebraof these objectsare developed. In Chapter 2, thealgebraic origins of
spectralsequencesaretreatedinthreeclassiccases—filtereddifferentialgraded
modules,exactcouples,anddoublecomplexes—-alongwithexamplesofthese
ideasinhomologicalalgebra. Thesubtlenotionofconvergenceisthefocusof
Chapter3. Comparisontheoremsareintroducedhereandtheunderlyingtheory
oflimitsandcolimitsispresented.
PartIIiscalledTopology;itistheheartofthebookandconsistsofChap-
ters4through10. PartIItreatsthefourclassicalexamplesofspectralsequences
that are found in homotopy theory. The introduction to each chapter gives a
detailedsummaryofitscontents. Wedescribethechaptersbrieflyhere. Chap-
ter4isathumbnailsketchofthetopicsinbasichomotopytheorythatwillbe
encounteredinthedevelopmentoftheclassicalspectralsequences. Chapters5
and 6 treat the Leray-Serre spectral sequence, and Chapters 7 and 8 treat the
Eilenberg-Moorespectralsequence. Chapters5and7, labeledasI,containa
constructionofeachspectralsequenceanddeveloptheirbasicpropertiesand
applications. Chapters 6 and 8, labeled as II, go into the deeper structures
of the spectral sequences and apply these structures to less elementary prob-
lems. AlternateconstructionsofeachspectralsequenceappearinChapters6
and 8. Chapter 8bis gives an account of the effect of a nontrivial group on
theLeray-SerrespectralsequenceandtheEilenberg-Moorespectralsequence.
Importanttopics,includingnilpotentspaces,thehomologyofgroups,andthe
Cartan-LerayandLyndon-Hochschild-Serrespectralsequences,aredeveloped.
Chapter 9 treats the classical Adams spectral sequence (as constructed in the
daysbeforespectra). Chapter10treatstheBocksteinspectralsequence,espe-
ciallyasatoolinthestudyofH-spaces. Throughoutthebook,Ihavefollowed
anhistoricaldevelopmentofthetopicsinordertomaintainasenseofthemoti-
vationforeachdevelopment. Insomeoftheproofsfoundinthebook,however,
Ihavestrayedfromtheoriginalpapersandfoundother(hopefullymoredirect)
proofs,especiallybasedontheresultsofPartI.
Part III is called Sins of Omission and consists of Chapters 11 and 12.
My first intention was to provide a catalogue of everyone’s favorite spectral
sequence,ifitdoesn’thappentobeinChapters4through9. Thishasbecome
too large an assignment as spectral sequences have become almost common-
place in many branches of mathematics. I have chosen some of the major
examplesandafewexoticatodemonstratethebreadthofapplicationsofspec-
tral sequences. Chapter 11 consists of spectral sequences of use in topology.
Chapter12includesexamplesfromcommutativealgebra,algebraicgeometry,
algebraicK-theory,andanalysis,evenmathematicalphysics.
Introduction xi
ThereareexercisesattheendofallofthechaptersinPartsIandII.They
offer further applications, missing details, and alternate points of view. The
noviceshouldfindtheseexerciseshelpful.
Thebibliographyconsistsofpapersandbookscitedinthetext. Attheend
ofeachbibliographicentryisalistofthepageswherethepaperhasbeencited.
The idea of a comprehensive bibliography on spectral sequences is unneces-
sarywithaccesstoMathSciNetortheZentralblattMATHDatabase. These
databasesalloweasysearchesoftitlesandreviewsofmostofthepublications
writtenaftertheintroductionofspectralsequences.
Howtousethisbook
Theseinstructionsareintendedforthenovicewhoisseekingtheshortest
pathtosomeofthesignificantapplicationsofspectralsequencesinhomotopy
theory. The following program should take the least amount of time, incur
the least amount of pain, and provide a good working knowledge of spectral
sequences.
(1) AllofChapter1.
(2) x2.1,x2.2(butskiptheproofofTheorem2.6),x2.3.
(3) x3.1andx3.3.
(4) Chapter4,asneeded.
(5) x5.1andx5.2.
(6) x6.1,x6.2,andx6.3.
From this grounding, the Bockstein (Chapter 10), the Cartan-Leray, and the
Lyndon-Hochschild-Serre spectral sequences (x8bis.2) are accessible. The
novicewhoisinterestedintheEilenberg-Moorespectralsequenceshouldin-
cludex2.4withtheaboveandthengoontoChapter7asdesired. Thenovice
whoisinterestedintheAdamsspectralsequenceshouldalsoreadx2.4aswell
asx7.1fortherelevanthomologicalalgebrabeforeembarkingonChapter9.
DetailsandAcknowledgments
Inthewritingofbotheditionsofthisbook,manypeoplehaveofferedtheir
time,expertise,andsupporttowhomIacknowledgeagreatdebt. Alongwith
athanksgiving,Iwillsayalittleaboutthesourcesofeachchapter.
ThisprojectbeganinPhiladelphia,inthecarwithBruceConrad,between
Germantown and Temple University. It was going to be a handy pamphlet,
listing E -terms and convergence results, but it has since run amok. At the
2
beginning, chats with Jim Stasheff, Lee Riddle, and Alan Coppola were en-
couraging. Chapter 1 is modeled on the second graduate course in algebraic
topologyItookfromJimandonunpublishednotesforsuchacoursewritten
byDavidKraines. DavidLyonsspottedacrucialmisstatementinthischapter
in the first edition. Michele Intermont gave it a good close reading for the
xii Introduction
second edition. Chapter 2 is classical in outline and owes much to my read-
ing of the classics by [Cartan-Eilenberg56], [Eckmann-Hilton66], and [Mac
Lane63]. Chapter3ismyaccountofthefoundationalpaperof[Boardman99],
the 1981 version of which I count among the classics in algebraic topology.
Alan Hatcher’s close reading of the first edition version of the Zeeman com-
parisontheoremresultedinsomesignificantchanges. BrookeShipleyreadmy
firstversionofthenewChapter3andgavemanyhelpfulremarks. Chapter4
was a suggestion of Lee Riddle whose notes on beginning homotopy theory
werehelpful.
Chapter5isbasedonthethesisof[Serre51]—itisaremarkablepaperand
Ihaveaddedmoreofittothesecondedition. Thepaperof[Brown,E94]isthe
backboneofx5.3. SpecialthankstoEdBrownandDonDavisforclosereadings
ofpartofthischapter. Chapter6isbasedonfurtherworkof[Serre51,53],the
thesisof[Borel53],thelovelypaperof[Dress67],andsubsequentdevelopments.
Chapter7isbasedontheYalethesisof[Smith,L70]. Ihavelearnedalotfrom
the papers of and conversations with Larry Smith. His remarks on an early
version of Chapters 7, 8, and 8bis made a big impact on the final version of
thesechapters. JimStashefffirsttaughtmemanyoftheideasinChapters7and
8, and deeply influenced their formulation. I wrote these chapters first in the
firsteditionandIbelievetheybearthestampofJim’steaching.
Chapter 8bis was an idea based on a remark of Serre in a list of errata
forthefirsteditionhekindlysentmein1985. Hesuggestedthattheclassof
nilpotentspacesshouldbepresentedinadiscussionofspectralsequences. This
chapteristheresult. IhavelearnedalotfromEmmanuelDror-FarjounandBill
Dwyerinthewritingofthischapter. CoffeehourswithRichardGoldstonehave
been very helpful as well. Chapter 9 is a result of my reading of the papers
ofAdamsandLiuleviciusandthebeautifulbookof[Mosher-Tangora68]. Bob
Bruner and Norihiko Minami added much to the chapter in their readings of
earlyversionsofit. Chapter10isbasedonthepapersofBillBrowder,JimLin,
and Richard Kane. An outline of Chapter 10 appeared as the introduction to
mypaper,[McCleary87].
Manypeoplehaveofferedsuggestionsthatenhancedmypresentationand
myunderstanding;theyincludeClaudeSchochet,BillMassey,NathanHabeg-
ger,DanGrayson,BobThomason,JohnMoore,AndrewRanicki,FrankAdams,
IanLeary,AlanDurfee,Carl-FriedrichBo¨digheimer,JohannesHuebschmann,
LarsHasselholt,JerryLodder,GuidoMislin,andJasonCantarella.
ThearmyofreadersorganizedbyHalSadofskyareowedamightythanks
forthedegreeofcareandcommitmenttheygaveinmakingthebookabetter
effort. Theyare: ZoranPetrovic(Chapter2),DonDavisandMartinCrossley
(Chapter 4), Martin Cadek and Dan Christensen (Chapter 5), Chris French
(Chapter6),JimStasheff(Chapters7,8,and12),TomHunterandKathrynHess
(Chapter8),RichardGoldstone(Chapter8bis),ChristianNassau(Chapter9),
EthanBerkoveandKathrynLesh(Chapter10),andFrankNeumann(Chapters
11and12).