An Invitation to Morse Theory

370 Pages·2011·2.24 MB·English

by  Liviu Nicolaescu (auth.)

Universitext
Universitext
SeriesEditors:
SheldonAxler
SanFranciscoStateUniversity
VincenzoCapasso
Universita` degliStudidiMilano
CarlesCasacuberta
UniversitatdeBarcelona
AngusJ.MacIntyre
QueenMary,UniversityofLondon
KennethRibet
UniversityofCalifornia,Berkeley
ClaudeSabbah
CNRS,E´colePolytechnique
EndreSu¨li
UniversityofOxford
WojborA.Woyczynski
CaseWesternReserveUniversity
Universitextisaseriesoftextbooksthatpresentsmaterialfromawidevarietyofmathematical
disciplinesatmaster’slevelandbeyond.Thebooks,oftenwellclass-testedbytheirauthor,
mayhaveaninformal,personalevenexperimentalapproachtotheirsubjectmatter.Someof
themostsuccessfulandestablishedbooksintheserieshaveevolvedthroughseveraleditions,
alwaysfollowingtheevolutionofteachingcurricula,toverypolishedtexts.
Thusasresearchtopicstrickledownintograduate-levelteaching,firsttextbookswrittenfor
new,cutting-edgecoursesmaymaketheirwayintoUniversitext.
Forfurthervolumes:
http://www.springer.com/series/223
Liviu Nicolaescu
An Invitation to Morse
Theory
Second Edition
123
LiviuNicolaescu
DepartmentofMathematics
UniversityofNotreDame
NotreDame
USA
[email protected]
ISSN0172-5939 e-ISSN2191-6675
ISBN978-1-4614-1104-8 e-ISBN978-1-4614-1105-5
DOI10.1007/978-1-4614-1105-5
SpringerNewYorkDordrechtHeidelbergLondon
LibraryofCongressControlNumber:2011939083
Mathematics Subject Classification (2010): 53A04, 53A05, 53D20, 57R17, 57R58, 57R65, 57R70,
57R91,58A35,58K05,58K10
©SpringerScience+BusinessMedia,LLC2011
Allrightsreserved.Thisworkmaynotbetranslatedorcopiedinwholeorinpartwithoutthewritten
permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York,
NY10013, USA),except forbrief excerpts inconnection with reviews orscholarly analysis. Usein
connectionwithanyformofinformationstorageandretrieval,electronicadaptation,computersoftware,
orbysimilarordissimilarmethodologynowknownorhereafterdevelopedisforbidden.
Theuseinthispublicationoftradenames,trademarks,servicemarks,andsimilarterms,eveniftheyare
notidentifiedassuch,isnottobetakenasanexpressionofopinionastowhetherornottheyaresubject
toproprietaryrights.
Printedonacid-freepaper
SpringerispartofSpringerScience+BusinessMedia(www.springer.com)
To mymother,withdeepest gratitude

Preface
As the title suggests, the goal of this book is to give the reader a taste of the
“unreasonableeffectiveness”ofMorsetheory.Themainideabehindthistechnique
canbeeasilyvisualized.
SupposeM is a smooth,compactmanifold,whichfor simplicitywe assume is
embeddedin a Euclideanspace E.We wouldlike to understandbasic topological
invariantsofM suchasitshomology,andweattempta“slicing”technique.
We fix a unit vector u in E and we start slicing M with the family of
hyperplanesperpendiculartou.SuchahyperplanewillingeneralintersectM along
asubmanifold(slice).Themanifoldcanberecoveredbycontinuouslystackingthe
slicesontopofeachotherinthesameorderastheywerecutoutofM.
Think of the collectionof slices as a deck of cardsof variousshapes. If we let
these slices continuously pile up in the order they were produced, we notice an
increasingstackofslices. As thisstack grows,we observethattherearemoments
oftimewhenitsshapesuffersaqualitativechange.Morsetheoryisaboutextracting
quantifiableinformationbystudyingtheevolutionoftheshapeofthisgrowingstack
ofslices.
Fromamathematicalpointofview,wehaveasmoothfunction
hWM !R; h.x/Dhu;xi:
Theaboveslicesarethelevelsetsofh,
˚ (cid:2)
x 2MI h.x/Dconst ;
andthegrowingstackisthetime-dependentsublevelset
fx 2MI h.x/(cid:2)tg; t 2R:
Themomentsoftimewhenthepilechangesitsshapearecalledthecriticalvalues
ofhandcorrespondtothemomentsoftimet whenthecorrespondinghyperplane
fhu;xi D tgintersectsM tangentially.Morsetheoryexplainshowtodescribethe
shapechangeintermsoflocalinvariantsofh.
vii
viii Preface
A related slicing technique was employed in the study of the topology of
algebraic manifolds called the Picard–Lefschetz theory. This theory is back in
fashion due mainly to Donaldson’s pioneering work on symplectic Lefschetz
pencils.
Thepresentbookisdividedintothreeconceptuallydistinctparts.Inthefirstpart,
welaythefoundationsofMorsetheory(overthereals).Thesecondpartconsistsof
applicationsofMorsetheoryoverthereals,whilethelastpartdescribesthebasics
and some applications of complex Morse theory, a.k.a. Picard–Lefschetz theory.
Hereisamoredetailedpresentationofthecontents.
InChap.1,weintroducethebasicnotionsofthetheoryandwedescribethemain
propertiesof Morse functions:their rigid local structure (Morse lemma) and their
abundance.(Morsefunctionsaregeneric.)Toaidthereader,wehavesprinkledthe
presentation with many examples and figures. One recurring simple example that
weuseasatestinggroundisthatofanaturalMorsefunctionarisinginthedesign
of robot arms. We conclude this chapter with a simple but famous application of
Morsetheory.Weshowthattheexpectednumberofcriticalpointsoftherestriction
of a randomlinear map ` W R3 ! R to a knotK ,! R3 is describedby the total
curvatureoftheknot.Asaconsequence,weobtainMilnor’scelebratedresult[M0]
statingthatifaclosedcurveinR3is“nottoocurved,”thenitisnotknotted.
Chapter 2 is the technical core of the book. Here we prove the fundamental
facts of Morse theory: crossing a critical level corresponds to attaching a handle
and Morse inequalities. Inescapably, our approach was greatly influenced by the
classicalsourcesonthissubject,morepreciselyMilnor’sbeautifulbooksonMorse
theoryandh-cobordism[M3,M4].
The operationof handleaddition is muchmore subtle than it first appears, and
sinceitisthefundamentaldeviceformanifold(re)construction,wedevotedanentire
sectiontothisoperationanditsrelationshiptocobordismandsurgery.Inparticular,
wediscussinsomedetailthetopologicaleffectsoftheoperationofsurgeryonknots
inS3andillustratethisinthecaseofthetrefoilknot.
In Chap.2, we also discuss in some detail dynamical aspects of Morse theory.
More precisely, we present the techniques of Smale about modifying a Morse
function so that it is self-indexing and its stable/unstable manifolds intersect
transversally. This allows us to give a very simple description of an isomorphism
between the singular homology of a compact smooth manifold and the (finite
dimensional)Morse–Floerhomologydeterminedby a Morse function,thatis, the
homology of a complex whose chains are formal linear combinations of critical
points and whose boundary is described by the connecting trajectories of the
gradient flow. We have also included a brief section on Morse–Bott theory, since
itcomesinhandyinmanyconcretesituations.
We conclude this chapter with a section of a slightly different flavor. Whereas
Morsetheorytriestoextracttopologicalinformationfrominformationaboutcritical
points of a function, min–max theory tries to achieve the opposite goal, namely,
to transform topologicalknowledgeinto informationaboutthe critical points of a
function.Inparticular,wediscusstheLusternik–Schnirelmanncategoryofaspace,
whichisahomotopyinvariantparticularlyadeptatdetectingcriticalpoints.
Preface ix
Chapter 3 is devoted entirely to applications of Morse theory. We present
relativelyfew examples,butwe use them as pretextsfor wanderingin manyparts
ofmathematicsthatarestillactiveareasofresearch.Westartbypresentingarecent
result of Farber and Schu¨tz, [FaSch], on the Betti numbersof the space of planar
polygons,orequivalently,thespaceofconfigurationsofplanarrobotarmssuchthat
theend-pointofthearmcoincideswiththeinitialjoint.Besidesitsintrinsicinterest,
this application has an added academic bonus: it gives the reader the chance to
witnessMorsetheoryinaction,inallitssplendor.Additionallyitexposesthereader
to the conceptof Bott–Samelsoncycle whichis usefulin manyotherapplications
ofMorsetheory.
We next discuss two classical applications: the computation of the Poincare´
polynomialsofcomplexGrassmanniansandanoldresultofLefschetzconcerning
thetopologyofSteinmanifolds.
The complex Grassmannians give us a pretext to discuss at length the Morse
theoryofmomentmapsofHamiltoniantorusactions.Weprovethatthesemoment
maps are Morse–Bott functions.We then proceed to give a complete presentation
of the equivariantlocalization theorem of Atiyah, Borel, and Bott (for S1-actions
only),andweusethistheoremtoprovearesultofConner[Co]:thesumoftheBetti
numbersofacompact,orientedsmoothmanifoldisgreaterthanthesumoftheBetti
numbersofthefixedpointsetofanysmoothS1-action.Conner’stheoremimplies
amongotherthingsthatthemomentmapsofHamiltoniantorusactionsareperfect
Morse–Bottfunction.The(complex)Grassmanniansarecoadjointorbitsofunitary
groups,andassuchtheyareequippedwithmanyHamiltoniantorusactionsleading
tomanychoicesofperfectMorsefunctionsonGrassmannians.Weconcludewitha
sectiononthecelebratedDuistermaat–Heckmanformula.
Chapter 4 is more theoretical in nature but it opens the door to an active area
of research, namely Floer homology.While still in the finite dimensionalcontext,
we take a closer look at the topological structure of a Morse–Smale flow. The
main resultsare inspiredby ourrecentinvestigations[Ni2] and,to the bestof our
knowledge,theyseemtohaveneverappearedintheMorsetheoreticliterature.
Thekeyresultsofthischapter(Theorem4.32and4.33)statethataMorseflow
on a compact manifold satisfies the Smale transversality condition if and only if
the stratification given by the unstable manifolds satisfies the Whitney regularity
conditions. Because the theory of Whitney stratifications is not part a standard
graduatecurriculum,we devoteda largepartof thischaptersurveyingthistheory.
Since the proofs of the main results in this area are notoriously complex, we
decidedto skip mostof them opting instead for copiousreferencesand numerous
illuminatingexamples.
TheseresultsprovidearigorousfoundationtoThom’soriginalinsight[Th].One
immediate consequence of Theorem 4.33 is a result of Laudenbach [Lau] on the
natureofthesingularitiesoftheclosureofanunstablemanifoldofaMorse–Smale
flow.
InSect.4.4weinvestigatethespacesoftunnelingsbetweentwocriticalpointsof
aMorse–Smaleflow.Usinga recentideaofKronheimerandMrowka[KrMr],we
show thatthese spaces admitnaturalcompactificationsas manifoldswith corners.