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PublishedonlineJune4,2022JournalofFixedPointTheory
cJonathanDavidEvans2022andApplications
ALagrangianKleinbottleyoucan’tsqueeze
JonathanDavidEvans
DedicatedtoClaudeViterbo,onhisfifty-tenthbirthday.
asymplectic4-
beforethesmoothisotopyclassofLcontainsnoLagrangians?Isolve
somerelatedconjectures.
,53D35,53D42.
,symplecticgeometry,Kleinbottle,
genus,pseudoholomorphiccurve.
Herearetwooverlappingquestions:
.(Minimalnonorientablegenus)Givenasymplectic4-manifold
(X,ω)andaZ/2-homologyclassβ∈H2(X;Z/2),whatistheminimal
nonorientablegenusofanonorientableLagrangiansurfaceL⊂Xwith
[L]=β?
.(Nonsqueezing)Givenasymplectic4-manifold(X,ω)anda
nonorientableLagrangiansurfaceL⊂X,howfarcanyoudeformωinco-
homologybeforethereisnoLagrangiansmoothlyisotopictoL?
IfLisorientablethenthesequestionsarelessinteresting:thegenusis
2
determinedby[L]=−χ(L)and,inQuestion,itisnecessarytodeform
[]=0
,ifLis
nonorientable,wehaveH2(L;R)=0,whichmeansthatitispossibleto
deformω,keepingLLagrangian,insuchawaythat[ω]rangesoveranopen
setinH2(X;R).
Thisarticleispartofthetopicalcollection“Symplecticgeometry-AFestschriftinhonour
ofClaudeViterbo’s60thbirthday”editedbyHelmutHofer,AlbertoAbbondandolo,Urs
Frauenfelder,andFelixSchlenk.
Iwillgivesomegeneraldiscussionofthesequestionsinturn,thengive
beansweredcompletely().
Onerunningthemethroughoutthediscussionistheuseofvisibleand
tropicalLagrangiansinalmosttoric4-manifolds:theseprovidearichsource
ofLagrangiansubmanifoldscomingrespectivelyfromstraightlinesandtrop-
aboutsomeofthephenomenaunderdiscussion,andforformulatingconjec-
’swork[16];tropical
LagrangianswereintroducedindependentlybyMikhalkin[9]andMatessi[8].
2
#[4]showsthatanyZ/2-homologyclassin
asymplectic4-manifoldcanberepresentedbysomeembeddednonorientable
Lagrangian,-definedanswer,whichIwilldenote1
byN(X,ω,β).
[1]showedthat
P2(β)=χ(L)=2−kmod4,
whereP2denotesthePontryaginsquareoperationandχistheEulerchar-
performaHamiltonianfingermovelocallytointroducepairsofintersec-
tionswithindexdifference1andthenperformPolterovichsurgery[13]on
theseself-intersectionstogetanembeddedLagrangianwithnonorientable
genusk+
{N(X,ω,β),N(X,ω,β)+4,...}.
(X,ω,β)isknowninasmallrangeofcases,the
lowerboundbeingtheprincipaldifficulty.
[ω]·c1(X)>0,weknowthatN(X,ω,0)=
2
followsfromGivental’sconstruction[5]ofaLagrangian#6RPinthe
4-ballandfromthefact,provedbyShevchishin[14]thatXcontainsno
nullhomologousLagrangianKleinbottles(seealsothebeautifulpapers
byNemirovski[11,12]).
,b,cbetheblow-upofthe4-ballinthreesubballssothatthesym-
plecticareasoftheexceptionalspheresE1,E2,E3area,b,
2
andSmirnov[15]showthatE1+E2+E3containsaLagrangianRP
ifandonlyifthefollowinginequalitiesallhold
a<b+c,b<c+a,c<a+b.
boundN(Xa,b,c,ω,E1+E2+E3)≥5whena,b,cviolatethetriangle
inequalities.
1“ng”istheInternationalPhoneticAlphabetsymbolforthe”ng”sound.
(2022)ALagrangianKleinbottleyoucan’tsqueezePage3of1047
aa
××
×
×
c⊕b
bc
cc
,b,cwitha
:Thesymplectictriangleinequali-
tiesandtheassociatedtropicalLagrangianisdiffeomorphic
toRP2(withthecorecircleofacross-caplivingoverthe
pointmarkedbythecross-hairsymbol).Right:Thesym-
plectictriangleinequalitiesareviolatedandtheassociated
tropicalLagrangianisdiffeomorphictoadisc
,weseethatthereisatropicaloralmosttoric
motivationfortheShevchishin-
-upXa,b,c;theaffinelengthsa,b,c
indicatedcorrespondtothesizesoftheexceptionalspheresE1,E2,
youcanseeatropicalcurve;usingtheideasofMikhalkin[9]andMatessi[8],
wecanconstructaLagrangiansubmanifoldLlivingovera(smallthickening
ofa)
andonlyiftheinequalitiesallhold:thepreimageofthepointmarkedwith
cross-hairsisacircleinLwhoseneighbourhoodisaMöbiusstrip.
×S2
LetX=S2×,anysymplecticformonX
∗∗2
isdiffeomorphictoonefromthefamilyλp1σ+p2σ,wherep1,p2:X→Sare
(X,ω,0)=
6,whichleavestwointerestingZ/2-homologyclassesuptodiffeomorphism:
β=[×S2]
P2(β)=0andP2(Δ)=2,sothereisachancetorepresentβbyLagrangian
Kleinbottles.
<2thenβisrepresentedbyaLagrangianKleinbottle.
22
HamiltoniantorusactiononS×
LagrangianKleinbottlelivingovertheline(slope1/2)inthediagram.
23
Toseethis,considerthetwoSfactorssittinginsideRandlet(pj,θj)be
cylindricalcoordinatesonthejthfactor(j=1,2).Theseareaction-angle
coordinates,soω=dpj∧isgivenby2p2=p1andthe
LagrangianKleinbottleiscutoutbythisequationtogetherwithθ2=−2θ1.
Kleinbottle,noticethattheregularlevelsetsofp1restrictedtoLarecircles
θ2=−2θ1inthe(θ1,θ2)-torus,whichcollapse2-to-1ontothecirclesof
maximaandminimaatp1=±λ(asthetoruscollapsestothecirclewith
coordinateθ2).Theprojectionsofthesecirclesaredenotedwithcross-hairs
.
[16]as
wellasbeingatropicalLagrangianinthesenseofMatessi[8]andMikhalkin
[9].ThisKleinbottleiswell-known:itappearsin[3]asaHamiltonianmin-
imalLagrangian,in[6]asaHamiltoniansuspension,andin[4]asafibre
connect-
representativeinitsLagrangianisotopyclassifλ=1.
Ifλ≥2thentheline
conjectureseemsnatural;whileIcannotproveit,
below.
≥2.
notoolstoprovelowerboundswhentheLagrangiansareofhighgenusand
maybeFloer-
thatLagrangianswithhighgenusbecomeflexibleenoughthat:
→∞N(X,ωλ,β)<∞.
ThefollowinglemmagivesanupperboundonN(X,ωλ,β),butitgoes
toinfinitywithλ.
(X,ωλ,β)≤20+2whenλ<10+2.
<10+1thenthereisatropicalLagrangianintheclassβwith
nonorientablegenus20+=
below;forgeneralwesimplyrepeatthepatternbetweentheverticalblue
barsasoftenasrequiredtogetfromtheleft-handsidetotheright-handside
oftherectangle.
Theedgesofthistropicalcurveare:
•internaledgesparalleltoeither(3,1)or(2,−1),
⊗
⊗
λ
22
(S×S,ωλ)
forλ<-capsareindicatedwith
cross-hairs
(2022)ALagrangianKleinbottleyoucan’tsqueezePage5of1047
20+2inthecase=2
•externaledgesparallelto(2,−1)or(1,2).
ThecorrespondingtropicalLagrangianintersectsthehorizontalsphereswith
evenmultiplicityandtheverticalsphereswithoddmultiplicity,soitinhabits
:eachhasself-
[9,],thistropicalcurvetherefore
yieldsanimmersedLagrangianwith8doublepointsand2+4cross-caps
whereithitstheboundary(markedwithcross-).Whenwe
performPolterovichsurgeryatthedoublepoints,weobtainaLagrangian
whichistopologicallyasurfaceofgenus8with4+2cross-
Eulercharacteristic2−16−4−2=−20,sothenonorientablegenusis
2+20.
-
ingtropicalLagrangians,butthereisnoreasontobelievethattropicalLa-
grangiansshouldgiveasharpupperboundforN.
ForeachconnectedopenintervalI⊂R(length|I|),letCIdenotethecylinder
1
I×(R/2πZ)withcoordinates(p,θ),equippedwiththesymplecticform2πdp∧
2
dθ;thishastotalarea|I|.LetSdenotethe2-sphereequippedwithitsarea
=2
formσsatisfyingS2σ.
222
LetUI=S×(S×S,ω|I|)by
excisingthespheresS2×{n,s},wheren,sdenotethepolesofthesecond
,weseethatif|I|>1,theonlynontrivial
classβ∈H2(UI;Z/2)isrepresentedbyaLagrangianKleinbottle(seeFig.
4).
|I|≤:K→UIisaLagrangianembedding
oftheKleinbottleintheclassβthenι∗:Q=H1(K;Q)→H1(UI;Q)=Q
isthezeromap.
usesSFTandneck-stretching.
2
Ateachvertexofatropicalcurve,theoutgoingedgesv1,v2,v3mustsumtozero;ifwe
writemforthedeterminant|v1∧v2|=|v2∧v3|=|v3∧v1|thentheself-intersectionof
m−1
-intersectionzero.
|I|>1thenH1(L;Q)→H1(UI;Q)isanisomor-
,
takeeitheroneofthecircleslivingoverthepointsmarkedwithcross-hairs
;thisisageneratorforbothH1(L;Q)andH1(UI;Q).Wededuce:
(0,1+)-
notbesqueezedintoU(0,1).
,youwouldneedtopro-
duceapairofsymplecticspheresintheclass[S2×]which“link”yourLa-
-minimal
symplecticarea,itisdifficulttocontroltheSFTlimitofsuchspheres.
.
’salmostcomplexstructure
(thecanonical
1-form)ontheunitcotangentbundleM⊂T∗KwhoseclosedReeborbits
betweengeodesicsandthecorrespondingReeborbitsandwewillwrite−γfor
γ0,γ1whicharethecorecirclesfortwodisjointembeddedMöbiusstripsin
geodesicsoccurinone-
andtheothergeodesicseven.
.(Mohnke[10,])Thereexistsanalmostcomplex
structureJ−onthecotangentbundleT∗Kwiththefollowingproperties:
−iscylindricalatinfinityandsuitableforneck-stretching.
−
-energyJ-holomorphiccylinderfγ
inT∗Kasymptotictoγand−γ.
3.[10,Lemma7(2)]AnyJ−-holomorphiccylinderinT∗Kwhichintersects
thezero-sectionisoneofthesefγforsomeclosedgeodesicγ.
:=T∗Kdenotethecompactificationofthecotan-
gentbundleobtainedbygluingonitsidealcontactboundaryMthenthere
isawell-definedintersectionpairingH2(W,M;Z/2)⊗H2(W;Z/2)→Z/2.
ThecylindersfγdefineelementsofH2(W,M;Z/2)andwehave[10,Lemma
⊗
|I|
⊗
2
|I|>1
(2022)ALagrangianKleinbottleyoucan’tsqueezePage7of1047
7(3)]
1ifγisodd
fγ·K=
0ifγiseven.
.[10,Lemma7(1)]Notethattherearealsonofiniteenergyplanes
inT∗K,norinthesymplectisationR×M,foranycylindricalalmostcom-
arenocontractibleReeborbits,andafiniteenergyplanewouldprovidea
nullhomotopyofitsasymptote.
-stretching
LetI=(0,1)andI¯=[0,1].SupposethereisaLagrangianKleinbottle
K⊂UIsuchthatQ=H1(K;Q)→H1(UI;Q)=Qisnonzero(inparticular,
itisinjective).ThinkofKsittingin