Free Loop Spaces in Geometry and Topology: Including the Monograph Symplectic Cohomology and Viterbo's Theorem

500 Pages·2015·2.527 MB·English

by  Abouzaid, Mohammed, Latschev, Janko, Oancea, Alexandru (eds.)

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Free Loop Spaces in
Geometry and Topology
Including the monograph
Symplectic cohomology and
Viterbo’s theorem
by Mohammed Abouzaid
Janko Latschev
Alexandru Oancea
Editors
Editors:
Janko Latschev Alexandru Oancea
Fachbereich Mathematik Sorbonne Universités, UPMC Univ Paris 06
Universität Hamburg UMR 7586, Institut de Mathématiques de Jussieu-Paris Rive Gauche
Bundesstrasse 55 Case 247, 4 place Jussieu
20146 Hamburg, Germany F-75005, Paris, France
[e-mail] [e-mail]
Mohammed Abouzaid
Department of Mathematics
Columbia University
Room 509, MC 4406
2990 Broadway
New York, NY 10027, USA
[e-mail]
2010 Mathematics Subject Classification: Primary: 53D40, 53D12, 53D25, 53D35, 53D37, 55P35, 55P50,
55P62, 55P92, 13D03, 13D07, 13D09, 13D10, 16E40, 16E45; Secondary: 57R15, 57R19, 57R56, 57R70, 57R91
Key words: Loop space, symplectic geometry, symplectic topology, string topology, Morse theory, Hochschild
and cyclic homology, operations on Hochschild and cyclic homology, rational homotopy theory, minimal
models, Lagrangian embeddings, pseudo-holomorphic curves, moduli spaces of pseudo-holomorphic curves,
Cauchy–Riemann operator, Floer theory, symplectic cohomology
ISBN 978-3-03719-153-8
The Swiss National Library lists this publication in The Swiss Book, the Swiss national bibliography, and the
detailed bibliographic data are available on the Internet at http://www.helveticat.ch.
This work is subject to copyright. All rights are reserved, whether the whole or part of the material is
concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting,
reproduction on microfilms or in other ways, and storage in data banks. For any kind of use permission of the
copyright owner must be obtained.
© 2015 European Mathematical Society
Contact address:
European Mathematical Society Publishing House
Seminar for Applied Mathematics
ETH-Zentrum SEW A27
CH-8092 Zürich
Switzerland
Phone: [phone]
Email: [e-mail]
Homepage: www.ems-ph.org
Typeset using the authors’ TEX files: le-tex publishing services GmbH, Leipzig, Germany
Printing and binding: Beltz Bad Langensalza GmbH, Bad Langensalza, Germany
∞ Printed on acid free paper
9 8 7 6 5 4 3 2 1
Contents
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
I A panorama oftopology,geometry and algebra . . . . . . . . . . 19
1 Basicsonfreeloopspaces . . . . . . . . . . . . . . . . . . . . . . . . . . 21
DavidChataurandAlexandruOancea
2 Morsetheory,closedgeodesics,andthehomologyoffreeloopspaces . . 67
AlexandruOancea
3 Rationalhomotopy–Sullivanmodels . . . . . . . . . . . . . . . . . . . 111
LucMenichi
4 Freeloopspaceandhomology . . . . . . . . . . . . . . . . . . . . . . . 137
Jean-LouisLoday
5 AppendixtothechapterbyJ.-L.Loday . . . . . . . . . . . . . . . . . . 157
JankoLatschev
6 OnalgebraicstructuresoftheHochschildcomplex . . . . . . . . . . . . 165
HosseinAbbaspour
7 Basicrationalstringtopology . . . . . . . . . . . . . . . . . . . . . . . . 223
YvesFe´lix
8 Fukaya’sworkonLagrangianembeddings . . . . . . . . . . . . . . . . 243
JankoLatschev
II Symplectic cohomologyand Viterbo’s theorem . . . . . . . . . . 271
MohammedAbouzaid
9 Symplecticcohomologyofcotangentbundles . . . . . . . . . . . . . . . 279
10 Operationsinsymplecticcohomology . . . . . . . . . . . . . . . . . . . 323
11 Stringtopologyusingpiecewisegeodesics . . . . . . . . . . . . . . . . . 355
vi Contents
12 Fromsymplecticcohomologytoloophomology . . . . . . . . . . . . . . 377
13 Viterbo’stheorem: surjectivity . . . . . . . . . . . . . . . . . . . . . . . 405
14 Viterbo’stheorem: isomorphism . . . . . . . . . . . . . . . . . . . . . . 455
BibliographytoPartII . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 481
Listofcontributors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 487
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 489
Introduction
Thereisnouniversalagreementonwhatconstitutesa“good”mathematicalproblem.
One possible measurement could be the amount of interesting mathematics that it
leadsto. Fromthatpointofview,thesearchfor(closed)geodesics,originatingwith
the works of Hadamard [38] and Poincare´ [53] and with substantial early contribu-
tions by Birkhoff [8], Morse [52], and Lyusternik and Schnirel’man [48] certainly
qualifies. The general problem is to find geodesics connecting two given points of
aRiemannianmanifoldorto find periodicgeodesics,andto givea meaningto their
count. The most important offspring of this problem is the development of topo-
logicalmethodsin variationalcalculus,generallyreferredtoasMorsetheory(or,as
Bottputsit, “Morsetheoryindomitable”[9]). Oneofits mostrecentincarnationsis
Floer theory, a central tool in modern symplectic topology. The geodesic problem
also led to the development of computational tools in algebraic topology (spectral
sequences),andisconnectedtothetheoryofminimalmodelsandtoHochschildand
cyclichomology.
Inattemptstosolvethegeodesicproblemoneisquicklyledtothestudyofspaces
ofpathsandloopsonmanifolds. Thesespaceshavebeentheobjectofmuchinterest
in recent years, both for topologists and symplectic geometers. The main purpose
of this book is to facilitate communication between these two communities by de-
veloping a common basis. From a topological point of view, a lot is known about
pathandloopspaces,buttheresultsare oftenscatteredthroughouttheliterature. In
particular, it can be difficult for a newcomerto the subject to extractthe main lines
ofthought. Hopefully,thisbookwillserveasaguidetothesetopologicaltechniques
andresults. At the same time, the symplectic pointofview emphasizescertain fea-
turesandalgebraicstructuresthathavebeenlittleornotstudiedatall. Thisrelatesin
particulartothemoderndevelopmentknownasstringtopology. Itseemsreasonable
toexpectthatquestionsfromsymplectictopologywillmotivatenewdevelopmentsin
thetopologicalstudyoffreeloopspaces,andconversely.
Genesis of this book
This book grew out of a learning seminar on “Free loop spaces” held at Strasbourg
Universityin2008–2009.Theseminarattractedamuchbiggeraudiencethaninitially
expected,andmanyofthespeakersagreedtoexpandtheirtalksintochaptersforthis
book. The guiding rule that we tried to follow was to keep the level of exposition
accessible to a graduate student. Our goal in this book is not to present the latest
developments, but rather to build from the basics up to some level from which the
interestedreadercouldcontinueonherorhisown. Someofthechaptersalsocontain
newresearchmaterial,mostnotablytheonebyHosseinAbbaspour.
2 Introduction
The contribution that stands out particularly is Mohammed Abouzaid’s “Sym-
plectic cohomology and Viterbo’s theorem”, which constitutes Part II of this book.
Thoughinitiallyintendedasoneofthechapters,itgrewintoafullyfledgedresearch
monograph.Itdoesstartgentlybydiscussingsomefoundationalfactsfromsymplec-
ticgeometry,andalsofromtheMorsetheoryoffinite-dimensionalapproximationsof
free loopspaces. But ratherthan sketchingone ofthepublishedproofsofViterbo’s
result,itthenproceedstogiveanewproof,buildingonideasofpreviousapproaches
butdevelopinganoriginalpointofview.
Unfolding the story
Let us now proceed with a more detailed description of the story that is told in this
book. Ouraimhereis notsomuchto offera strictly historicalperspective–though
wedoprovidesomehistoricalbackground–butrathertointroducethemathematical
subjectsandobjectsdiscussedinthebook.
WedenoteS1 R=Zand,foramanifoldM,wedenoteitsfreeloopspaceby
D
LM
S1 M
continuous :
WDf W ! W g
InthesubsequentdiscussionsonesometimeshastoconsidersubspacesofLM con-
sisting of loops that satisfy additional regularity properties, but in this introduction
wewillusetheuniformnotationLM.
Riemanniangeometry. GivenaRiemannianmanifold.M;g/theclosedgeode-
sicsparametrizedbyS1 arethecriticalpointsoftheenergyfunctional
1
E LM R; E.
/
.t/ 2dt:
W ! WD 2ZS1kP k
HerethemostconvenientsetupisthatofloopsofSobolevclassH1. Withthischoice,
the energyfunctionalis well-behavedin severalrespects: (i) it is boundedfrom be-
low, (ii) it satisfies the Palais–Smale condition(any sequence
such that E.
/ is
k k
boundedand
0hasaconvergentsubsequence),(iii)foragenericmetricthe
k
kr k!
criticalsetisadisjointunionofsubmanifolds(onecopyofM thatcorrespondstothe
constantloopsandacountableunionofdisjointcirclesgivenbynontrivialgeodesics
and their shift reparametrizations), and (iv) the Hessian d2E is non-degenerateand
hasfiniteindexinthenormaldirectiontoanycriticalsubmanifold(wesaythatE is
Morse–Bott). Morse theory is designedto handleprecisely this kind offunctionals.
The outcome is a description of the loop space LM by successive attachments of
bundles over the critical submanifolds with rank given by the index of d2E. This
allowsagriponthetopologyofLM providedonehasenoughinformationonthese
indices and on the attaching maps. Conversely, knowledge of the topology of LM
impliesexistenceresultsforcriticalpointsofE.
Introduction 3
One significant difficulty in converting existence results for critical points of E
intoexistenceresultsforgeometricallydistinctclosedgeodesicsisthateverynoncon-
stantclosedgeodesiccanbeiteratedandhencegivesrisetocountablymanydistinct
criticalsubmanifolds. Still, inmanycasesthis problemcanbeovercome. Themost
powerfulresultinthisdirectionisthefollowingtheoremduetoGromoll–Meyer.
Theorem(Gromoll–Meyer[35]). LetM beasimplyconnectedclosedmanifoldsuch
thatthesequence b .LM/ ,k 0ofBettinumbersofLM withcoefficientsinsome
k
f g 
fieldisunbounded.ThenforanyRiemannianmetriconM thereexistinfinitelymany
geometricallydistinctclosedgeodesics.
Thedifficultcontentofthetheoremisthattheconclusionholdsforanymetric,not
onlyforagenericone. ThusthecriticalsetofE isnotassumedtobewell-behaved,
andso theproofneedsideas beyondthe Morse theorypicturesketchedabove. This
theoremandrelatedideasfromthecalculusofvariationsarediscussedinChapter2of
thisbook.Tomakeeffectiveuseofthisresult,oneneedstoknowwhenitstopological
assumptionshold,andthisbringsustothediscussioninthenextsection.
Minimal models. The starting point for the topological study of the free loop
spaceLM istheloop-loopfibration(seeChapter1)
M //LM
ev
(cid:15)(cid:15)
M
Here ev is the evaluation at the origin of a loop, and M is the based loop space,
consistingofloopsstartingandendingatafixedbasepointinM. Thisfibrationcan
be used to determine the homotopy groups of LM, namely  .LM/  .M/
k k
' ˚
 .M/: indeed, the section given by the inclusion of constant loops determines
k
asplittingofthehomotopylongexactsequence(Chapter1). However,thesituation
isverydifferentasfarashomologygroupsareconcerned.ItturnsoutthattheLeray–
Serre spectral sequence is effective in simple cases (spheres [50, 21]) but of very
limited use in general, unless one has additional geometric information about the
differentials. Indeed, the path-loop fibration with the same fiber M and the same
base M has contractible total space, so any successful reasoning must use specific
featuresoftheloop-loopfibration.
AgeneralsolutionforthecomputationofH
.LM Q/(andhenceofH.LM Q/)


I I
forsimplyconnectedspacesM wasmadeavailablebySullivan’stheoryofminimal
models([57],seealso[56]). Itturnsoutthatthistheoryispowerfulenoughtoclarify
exactlywhentheassumptionsoftheGromoll–Meyertheoremhold.
Theorem(Sullivan–Vigue´-Poirrier[58]). LetM beasimplyconnectedclosedman-
ifold. The sequence of Betti numbers of LM with coefficients in Q is unboundedif
andonlyifH
.M Q/requiresatleasttwogeneratorsasaring.
I
4 Introduction
This theorem and the theory of minimal models are explainedin Luc Menichi’s
Chapter3.
A minimal model for a commutative differential graded algebra (cdga) A
over
a field k is a cdga .M
;d/ with a quasi-isomorphism M
A
such that M
is
free (i.e. M
ƒV
with V
a graded vector space) and!its differential satisfies
d.M
/ MCDMC, with MC Mk. We callM
a “model”becauseitis
k1
quasi-isomorphi
ctoA
anditisfreDea˚sanalgebra(thus,thecomplexityofthealgebra
structureofA
hasbeenmovedintothedifferentialforM
). WecallM
“minimal”
because it contains no unnecessary generators, which is expressed by the condition
d.M
/ MC MC. TherealwaysexistsaminimalmodelprovidedH0 k;when

D
aminimalmodelexists,itisuniqueuptoisomorphism[23,Thm.14.12].
The construction of a (real) minimal model M for a simply connected mani-
foldM proceedsinductively,startingfromthedeRhamalgebraofdifferentialforms
.
.M/; ;d/andbuildingasequence
^
Q M //M // ::: //M //M //:::
D 1 ❲❲❲❲❲❲❲❲❲2❲❲❘f❘❲1❘❲❘❲❘❲❘❲❘❲❘f❲❘2❲❘❲❘❲❘❲❘❲❘❲❘++((
.M(cid:15)(cid:15)frr/zztttttftrtCt1t rC1
whereM isbuiltfromM byadjoininggeneratorsofdegreer insuchawaythat
r r(cid:0)1
the map f inducesan isomorphismoncohomologyin degrees r andis injective
r

in degree r 1, and the horizontalmaps are inclusions. One of Sullivan’s insights
was that oveCr Q one can still build a cdga .A
;d/ of rational differential forms, so
thatthe sameconstructionappliesandbuilds a rationalminimalmodelforM. The
existenceofSteenrodoperationsshowsthatthereisnohopeofextendingthisdiscus-
siontointegercoefficients,sincetheirconstructionisdirectlybasedonthefailureof
commutativityofthecupproductonthechainleveloverZ(seee.g. [55,÷5.9]or[40,
÷4.L]fortheconstructionofSteenrodsquares).
Theconstructionofminimalmodelsis analgebraicanalogue(in awaywhichis
discribed e.g. by Fe´lix, Oprea and Tanre´ in [24, ÷2.5.4], see also Bott–Tu [10] or
Sullivan [56] for intuitive discussions) of the Postnikov tower of the manifold M,
whichisasequence
Y1 oojj❯❯❯❯❯❯Y❯❯2❯oohhP❯P❯iP1❯P❯P❯P❯P:❯P:i❯P2:❯Poo❯P❯P❯P❯P❯PYOOriroo②②②②②i②r②C②Y1<< rC1 oo :::
M
where the maps i M Y are inclusions that induce isomorphisms of homotopy
r r
W !
groups up to degree r, and all homotopy groups of Y in dimensions larger than r
r
vanish. Moreover,each map Y Y is a fibration whosefiber is an Eilenberg–
r r(cid:0)1
!
MacLanespaceK. .M/;r/.
r