The cylindrical contact homology of universally tight sutured contact solid tori

0.31 MB·English

by  Roman Golovko

THE CYLINDRICAL CONTACT HOMOLOGY OF UNIVERSALLY TIGHT
SUTURED CONTACT SOLID TORI
2
1
0 ROMANGOLOVKO
2

n
ABSTRACT. Wecalculatethesuturedversionofcylindricalcontacthomologyofasuturedcontact
a solidtorus(S1×D2,Γ,ξ),whereΓisarbitraryandξisauniversallytightcontactstructure.
J

4
2

]
G 1. INTRODUCTION
S
The cylindrical contact homology of a (closed) contact manifold was introduced by Eliashberg
.
h and Hofer and is the simplest version of the symplectic field theory of Eliashberg, Givental and
t
a Hofer[6]. It isthehomologyofadifferentialgraded modulewhosedifferentialcountsgenus zero
m
holomorphiccurves inthesymplectizationwithonepositivepunctureand onenegativepuncture.
[
A natural condition to impose on a compact, oriented contact (2m + 1)-manifold (M,ξ) with

5 boundary is to require that ∂M be convex, i.e., there is a contact vector field X transverse to ∂M.
v
ToatransversecontactvectorfieldX wecanassociatethedividingsetΓ = Γ ⊂ ∂M,namelythe
3 X
7 setofpointsx ∈ ∂M suchthatX(x) ∈ ξ(x). Bythecontactcondition,(Γ,ξ∩TΓ)isa(2m−1)-
0 dimensionalcontact submanifoldof(M,ξ); theisotopyclass of(Γ,ξ ∩TΓ) is independentofthe
4
. choice of X. We will denote by (M,Γ,ξ) the contact manifold (M,ξ) with convex boundary and
6
dividing set Γ = Γ ⊂ ∂M with respect to some transverse contact vector field X. Note that the
0 X
0 actualboundaryconditionweneedisslightlydifferentandiscalledasuturedboundarycondition.
1
(Intheearly1980’s,Gabaidevelopedthetheoryofsuturedmanifolds[7],whichbecameapowerful
:
v toolinstudying3-manifoldswithboundary.) Forthemomentwewrite(M,Γ,ξ)toindicateeither
i
X theconvexboundaryconditionorthesutured boundarycondition.
r It turns out that there is a way to generalize cylindrical contact homology to sutured manifolds.
a
This is possibleby imposinga certain convexity condition on the contact form. This construction
is described in the paper of Colin, Ghiggini, Honda and Hutchings [3] and will be summarized in
Section 2.
In this paper, we construct a sutured contact solid torus with 2n parallel sutures of slope k
l
using the gluing method of Colin, Ghiggini, Honda and Hutchings [3], and calculate the sutured
cylindrical contact homology of it. Here n ∈ N, (k,l) = 1 and |k| > l > 0. In order to define
the slope, we choose an oriented identification ∂(S1 × D2) ≃ T2 = (R/Z)2 as follows: map
{pt}×∂D2 (themeridian)to(1,0)(slopeis 0)andS1 ×{pt} (alongitude)to(0,1).
Thiscalculation,togetherwiththecalculationofthesuturedcylindricalcontacthomologyofthe
sutured contact solid torus with 2n parallel longitudinal sutures, where n ≥ 2, that has been done
in [8], finishes the calculation of the cylindrical contact homology of (S1 ×D2,Γ,ξ), where Γ is
arbitrary and ξ isa universallytightcontactstructure.
2000MathematicsSubjectClassification. Primary53D42;Secondary57M50,53D10.
Keywordsandphrases. suturedmanifolds,contacthomology.
1
2 ROMANGOLOVKO
Ourgoal isto provethefollowingtheorem:
Theorem1.1. Let(S1×D2,Γ)beasuturedmanifold,whereΓisasetof2nparallelclosedcurves
of slope k, where (k,l) = 1, |k| > l > 0 and n ∈ N. Then there is a contact form α which makes
l
(S1 ×D2,Γ,α) a sutured contact manifold with a universally tight contact structrure ξ = kerα,
HCcyl(S1×D2,Γ,α)isdefined,isindependentofthecontactformαforξ = kerαandthealmost
complexstructureJ and
Q, fork ∤ h > 0;
HCcyl,h(S1 ×D2,Γ,ξ) ≃ Qn 1, fork | h > 0;
−

0, otherwise.

Herehcorrespondsto thehomologicalgrading.

ACKNOWLEDGEMENTS
The author is deeply grateful to Ko Honda for his guidance, help and support. He also thanks
DmytroChebotarov,OliverFabert,PaoloGhiggini,JianHe,MichaelHutchingsandMarkMcLean
for helpful suggestions and interest in his work. In addition, the author is extremely grateful to
AndrewCotton-Clay forhiscritical commentson thefirst versionofthepaper. Finally,theauthor
thanks the Mathematical Sciences Research Institute and the organizers of the “Symplectic and
Contact Geometryand Topology”programfortheirhospitality.
2. BACKGROUND
The goal of this section is to review definitions of sutured contact manifold and the relative
versionofcylindricalcontact homology. Thissectioncan beconsidered as asummaryof[3].
2.1. Reviewofsuturedcontactmanifolds. Inthissectionwerecallsomedefinitionsanddescribe
someconstructionsfrom [3]. Wefirst start withthenotionofaLiouvillemanifold.
Definition 2.1. A Liouville manifold (often also called a Liouville domain) is a pair (W,β) con-
sisting of a compact, oriented 2n-dimensional manifold W with boundary and a 1-form β on W,
where ω = dβ is a positive symplectic form on W and the Liouville vector field Y given by
i (ω) = β is positively transverse to ∂W. It follows that the 1-form β = β| (this notation
Y 0 ∂W
meansβ pulledback to∂W) isapositivecontact formwithkernel ζ.
Wenowrecall thedefinitionofa suturedcontactmanifold.
Definition 2.2. A compact oriented 2n+ 1-dimensional manifold M with boundary and corners
is a sutured contact manifold if it comes with an oriented, not necessarily connected submanifold
Γ ⊂ ∂M ofdimension2n−1(calledthesuture),togetherwithaneighborhoodU(Γ) = [−1,0]×
[−1,1]×Γ ofΓ = {0}×{0}×Γ inM, withcoordinates (τ,t) ∈ [−1,0]×[−1,1], such thatthe
followingholds:
(1) U ∩∂M = ({0}×[−1,1]×Γ)∪([−1,0]×{−1}×Γ)∪([−1,0]×{1}×Γ);
(2) ∂M\({0}×(−1,1)×Γ) = R (Γ)⊔R (Γ),wheretheorientationof∂M agreeswiththat
+
−
of R (Γ) and is oppositethat of R (Γ) and the orientation of Γ agrees with the boundary
+
−
orientationofR (Γ);
+
(3) thecorners ofM are precisely {0}×{±1}×Γ.
THECYLINDRICALCONTACTHOMOLOGYOFUNIVERSALLYTIGHTSUTUREDCONTACTSOLIDTORI 3
In addition, M is equipped with a contact structure ξ, which is given by the kernel of a positive
contact 1-formα suchthat:
(i) (R (Γ),β = α| ) isaLiouvillemanifold;
R±(Γ)
± ±
(ii) α = Cdt + β inside U(Γ), where C > 0 and β is independent of t and does not have a
dt-term;
(iii) ∂ = Y ,where Y is aLiouvillevectorfield forβ .
τ
± ± ±
Such acontact form α issaid tobeadaptedto (M,Γ,U(Γ)).
Herewebrieflydescribethewaytogluesuturedcontactmanifolds. Thisprocedurewasfirstde-
scribedby ColinandHondain[4]and thengeneralized byColin,Ghiggini,HondaandHutchings
in[3].
Let (M ,Γ,U(Γ),ξ ) be a sutured contact 3-manifold with an adapted contact form α. We
′ ′ ′ ′ ′
denotebyπ theprojectionalong ∂ defined onU(Γ).
t ′
Take 2-dimensional submanifolds P ⊂ R (Γ) such that ∂P is the union of (∂P ) ⊂
′ ∂
± ± ± ±
∂R (Γ), (∂P ) ⊂ int(R (Γ)) and ∂P is positively transversal to the Liouville vector field
′ int ′
± ± ± ±
Y on R (Γ). Whenever we refer to (∂P ) and (∂P ) , we assume that closures are taken as
′ ′ int ∂
ap±propria±te. Moreoverwemaketheassum±ptionthat π((±∂P ) )∩π(∂P ) ) = ∅.
∂ + ∂
−
Let ϕ be a diffeomorphism which sends (P ,β | ) to (P ,β | ) and takes (∂P ) to
+ +′ P+ ′ P− + int
(∂P ) and(∂P ) to(∂P ) . Notethat,sincedimM = 3, we−onl−yneedβ | and ϕ (β | )
∂ + ∂ int +′ P+ ∗ ′ P−
tom−atchupon∂P ,sincew−ecanlinearlyinterpolatebetweenprimitivesofpositiveareaform−son
+
asurface.
Topologically, we construct the sutured manifold (M,Γ) from (M ,Γ) and the gluing data
′ ′
(P ,P ,ϕ) as follows: LetM = M / ∼, where
+ ′
−
• x ∼ ϕ(x) for allx ∈ P ;
+
• x ∼ x ifx,x ∈ π 1(Γ) andπ(x) = π(x) ∈ Γ.
′ ′ − ′ ′ ′
Then
R (Γ)\P
R (Γ) = ± ′ ± ∼ π ((∂P )∂)
± (∂P )int ± ∓
±
and
Γ \π(∂P ⊔∂P )
Γ = ′ + − ∼ π((∂P )int ∩(∂P )∂).
π((∂P+)int ∩(∂P+)∂) − −
Forthedetaileddescriptionofthegluingprocedurewerefer to [3].
Finally, we describe the way to complete sutured contact manifold (M,α) to a noncompact
contact manifold(M ,α ). Thisconstructionwas first described in[3].
∗ ∗
Let (M,Γ,U(Γ),ξ) be a sutured contact manifold with an adapted contact form α. The form α
isthengivenbyCdt+β on[1−ε,1]×R (Γ)and[−1,−1+ε]×R (Γ)ofR (Γ) = {1}×R (Γ)
+ + +
± −
and R (Γ) = {−1}×R (Γ), where t ∈ [−1,−1+ε]∪[1−ε,1]extendsthet-coordinate onU.
− −
On U, α = Cdt + β, β = β = β and ∂ is a Liouville vector field Y for β. We first extend
+ τ
−
α to [1,∞)×R (Γ) and (−∞,−1]×R (Γ) by taking Cdt+β as appropriate. The boundary
+
− ±
of this new manifold is {0} × R × Γ. Notice that since ∂ = Y, the form dβ| is the
τ [ 1,0] t Γ
symplectization of β| in the positiveτ-direction. We glue [0,∞)×R×−Γ w×i{th}×the form
0 t Γ
Cdt+eτβ , whereβ{i}s×t{he}×pullbackofβ to{0}×{t}×Γ.
0 0
4 ROMANGOLOVKO
We denote by M the noncompact extension of M described above and by α the extension of
∗ ∗
α toM .
∗
2.2. Reviewofcylindricalcontacthomology. Let(M,Γ,U(Γ),ξ)beasuturedcontactmanifold
withan adapted contactform α and(M ,α ) beitscompletion.
∗ ∗
The Reeb vector field R that is associated to a contact form α is given by dα (R ,·) = 0
α∗ ∗ ∗ α∗
and α (R ) = 1.
∗ α∗
We assume that R is nondegenerate, i.e., the first return map along each (not necessarily
α∗
simple) periodic orbit does not have 1 as an eigenvalue. Observe that nondegeneracy can always
beachievedbya smallperturbation.
Remark2.3. NotethateveryperiodicorbitofR liesinM. Hence,thesetofperiodicReeborbits
α∗
ofR coincideswiththeset ofperiodicReeb orbitsofR .
α∗ α
A Reeb orbit γ is called ellipticor positive(respectivelynegative)hyperboliciftheeigenvalues
ofP are ontheunitcircleorthepositive(resp. negative)real linerespectively.
γ
Ifτ isatrivializationofξoverγ,wecanthendefinetheConley-Zehnderindex. In3-dimensional
situation, we can explicitly describe the Conley-Zehnder index and its behavior under multiple
coversas follows:
Proposition 2.4 ([10]). If γ is elliptic, then there is an irrational number φ ∈ R such that P is
γ
conjugateinSL (R) toa rotationbyangle2πφ and
2
µ (γk) = 2⌊kφ⌋+1,
τ
where 2πφisthetotalrotationanglewith respect toτ ofthelinearizedflowaroundtheorbit.
Ifγ ispositive(respectivelynegative)hyperbolic,thenthereisaneven(respectivelyodd)integer
r such that the linearized flow around the orbit rotates the eigenspaces of P by angle πr with
γ
respect toτ and
µ (γk) = kr.
τ
A closed orbit of R is said to be good if it does not cover a simple orbit γ an even number
α∗
of times, where the first return map ξ → ξ has an odd number of eigenvalues in the interval
γ0 γT
(−1,0). Here T istheperiodoftheorbitγ. An orbitthat isnotgoodis calledbad.
WenowrecallthenotionofanalmostcomplexstructureonR×M thatistailoredto(M ,α ).
∗ ∗ ∗
Let(W,β)beaLiouvillemanifoldandζ bethecontactstructuregivenon∂W byker(β ),where
0
β = β| . In addition, let (W,β) be the completion of (W,β), i.e., W = W ∪([0,∞)× ∂W)
0 ∂W
andβ| = eτβ ,whereτ isthe[0,∞)-coordinate. AnalmostcomplexstructureJ onW is
[0, ) ∂W 0 0
β-adapt∞ed×ifJ isβ -adaptedcon[b0,∞)×∂W;anddβ(v,J v) > 0forcallnonzerotangentvectors
0 0 0
v onbW. c
b
Definition 2.5. Let (M,Γ,U(Γ),ξ) be a sutured contact manifold, α be an adapted contact form
and(M ,α )beitscompletion. Wesaythatan almostcomplexstructureJ onR×M istailored
∗ ∗ ∗
to(M ,α ) ifthefollowinghold:
∗ ∗
(1) J is α -adapted, i.e, J is R-invariant, J(ξ) = ξ, dα(v,Jv) > 0 for nonzero v ∈ ξ and
∗
J(∂ ) = R , wheres denotes theR-coordinate;
s α∗
(2) J is ∂ -invariantin aneighborhoodofM \int(M);
t ∗
THECYLINDRICALCONTACTHOMOLOGYOFUNIVERSALLYTIGHTSUTUREDCONTACTSOLIDTORI 5
(3) The projection of J to TR\(Γ) is a β -adapted almost complex structure J on the com-
0
± ±
pletion(R\(Γ),β ) (R\(Γ),β )oftheLiouvillemanifold(R (Γ),β ) (R (Γ),β ).
+ + + +
Moreover,theflow of∂ i−dentifie−sJ b| and J | . − −
F t 0 R\+(Γ) R+(Γ) 0 R\−(Γ) R−(Γ) F
b b \ \
Given a sutured contact manifold (M,Γ,U(Γ),α) and an α -adapted almost complex structure
∗
J,wedefinethesuturedcylindricalcontacthomologygroupHCcyl(M,Γ,α,J)tobethecylindri-
calcontacthomologyof(M ,α ,J). ThecylindricalcontacthomologychaincomplexC(α,J)is
∗ ∗
aQ-modulefreelygeneratedbyallgoodReeborbits,wherethegrading|·|andtheboundarymap
∂ are defined as in [1] with respect to the α -adapted almost complex structure J. The homology
∗
ofC(α,J)is thesuturedcylindricalcontacthomologygroup HCcyl(M,Γ,α,J).
For our calculations we need the following construction of a “global” symplectic trivialization
describedin[1]. AssumethatalltheReeborbitsofR aregood. Letusnowchoosetrivializations
α
τ(γ)consistentlyforallReeborbitsγ. AssumethatH (M;Z)isafreemodule. Wepickrepresen-
1
tatives C ,...,C in H (M;Z) for a basis of H (M;Z), together with a trivialization of ξ along
1 s 1 1
each representativeC , i = 1,...,s. Now foraReeb orbitγ, wedistinguishthefollowingcases:
i
(1) [γ] = 0 ∈ H (M;Z). Chooseaspanningsurface S and useitto trivializeξ along γ.
1 γ
(2) 0 6= [γ] ∈ H (M;Z). WechooseasurfaceS realizingahomologybetweenγ andalinear
1 γ
combination of the representatives C , i = 1,...,s. We then use S to extend the chosen
i γ
trivializationsofξ alongtheC ,i = 1,...,s toγ.
i
Wedenotetheobtainedtrivializationbyτ.
ToaJ-holomorphiccurveinMJ(γ;γ ),wecan gluethechosensurfacesS andS andobtain
′ γ γ′
a closed surface in M. Let A ∈ H (M;Z) be its homology class; we can use it to decorate the
2
correspondingconnected componentMJ(γ;γ ) ofthemodulispace. Usingτ wecan write
A ′
(2.2.1) ind(u) = |γ|−|γ |+2hc (ξ),Ai
′ 1
foru ∈ MJ(γ;γ ), where|γ|is theConley-Zehnder gradingofγ defined by
A ′
(2.2.2) |γ| := µ (γ)−1.
τ
WewilluseFormulas2.2.1and 2.2.2forourcalculations.
In addition,wewillneed thefollowingfact, whichisaconsequenceofLemma5.4 in[2]:
Fact2.6. Let (M,α)bea closed,orientedcontactmanifoldwithnondegenerateReeb orbitsand
u = (a,f) : (S˙,j) → (R×M,J)
be a J-holomorphiccurve in MJ(γ;γ ), where γ and γ are good Reeb orbits, J is an α-adapted
′ ′
almost complex structure on R × M and MJ(γ;γ ) is a moduli space of J-holomorphic curves
′
thatwe considerincylindricalcontacthomology. Then thefollowinginequalityholds:
def def
A(γ) = α ≥ α = A(γ )
′
Z Z
γ γ′
with equality if and only if γ = γ and in this case the moduli space consists of a single element
′
R×γ.
Nowwerecall thefollowingtheorem:
6 ROMANGOLOVKO
Theorem 2.7 ([1]). Let (M,α) be a closed, oriented contact manifold with nondegenerate Reeb
orbits. LetCh(M,α)bethecylindricalcontacthomologycomplex,wherehisahomotopyclassof
m
ReeborbitsandmcorrespondstotheConley-Zehndergrading. IfC0(M,α) = 0fork = −1,0,1,
k
thenfor everyfreehomotopyclassh
(1) ∂2 = 0;
(2) H(Ch(M,α),∂) is independent of the contact form α for ξ, the almost complex structure
J and∗ thechoiceofperturbationforthemodulispaces.
When M is closed and R × M is 4-dimensional, the following transversality result has been
provenbyMomin,seeProposition2.10in[11]:
Theorem 2.8 ([11]). Let u ∈ MJ(γ;γ ) be such that ind(u) = 1. Then the linearization of the
′
Cauchy-Riemannoperatoris surjectiveatu.
Remark 2.9. ObservethatTheorem 2.8does notrequireJ tobegeneric.
Remark2.10. NotethatTheorem2.8canbeconsideredasaconsequenceoftheautomatictransver-
salityresultofWendl,see Theorem0.1 in[12].
Finally,werecall thefollowingresultofColin,Ghiggini,Hondaand Hutchingsfrom [3]:
Theorem 2.11([3]). Let(M,Γ,U(Γ),ξ)beasuturedcontact3-manifoldwithanadaptedcontact
form α, (M ,α ) be its completion and J be an almost complex structure on R × M which is
∗ ∗ ∗
tailoredto(M ,α ). ThenthecontacthomologyalgebraHC(M,Γ,ξ)isdefinedandindependent
∗ ∗
of the choice of contact 1-form α with ker(α) = ξ, adapted almost complex structure J, and
abstractperturbation.
Remark 2.12. Fact 2.6, Theorems 2.7 and 2.8, Formulas 2.2.1 and 2.2.2 hold for J-holomorphic
curves in the symplectization of the completion of a sutured contact manifold, provided that we
choosethealmostcomplexstructureJ onR×M to betailoredto(M ,α ).
∗ ∗ ∗
Remark 2.13. Observe that Theorem 2.11 and Remark 2.12 rely on the assumption that the ma-
chinery, needed to prove the analogous properties for contact homology and cylindrical contact
homologyintheclosed case, works.
3. CONSTRUCTION
3.1. Gluing map. First we construct H ∈ C (R2). The time-1 flow of the Hamiltonian vector
∞
field associated to H composed with an appropriate rotation will play a role of the gluing map
when we will apply the gluing construction described in Section 2.1 to the sutured contact solid
cylinderconstructedinSection 3.2.
We fix p ∈ R2 and consider H : R2 → R given by H = µr2cos(n|k|θ) in polar coordi-
sing sing
nates about p, where µ > 0, n ≥ 1 and k ∈ Z\{−1,0,1}. Note that H is singular only at p.
sing
We obtainH ∈ C (R2) from H by perturbingH on a diskD(r ) about p in such away
∞ sing sing sing
that H has n|k| equally spaced saddle points, critical point at p and interpolates with no critical
points with H on D(r ). In other words, H = H on R2 \D(r ). For the level sets of
sing sing sing sing
H and H inthecasen = 1, |k| = 3werefer to Figure1.
sing
TheconstructionofH is amodificationoftheconstructiondescribed in[5].
Weproceed infoursteps.
THECYLINDRICALCONTACTHOMOLOGYOFUNIVERSALLYTIGHTSUTUREDCONTACTSOLIDTORI 7
FIGURE 1. The level sets of H (left) and the level sets of H (right) in the case
sing
n = 1, |k| = 3
(1) Weconsider
H = H +f(r,θ) = H +f (r,θ)+g(r,θ)
1 sing sing exp
= µr2cos(n|k|θ)−Ae mr2 +g(r,θ),
−
where A and m are positiveconstants, and g(r,θ) is a smooth function to be chosen later.
Weare interestedin thecritical pointsofH away from theorigin.
1
Wecalculate
∂H ∂g
1 = 2µrcos(n|k|θ)+2mrAe mr2 + ,
−
∂r ∂r
∂H
1 = −n|k|µr2sin(n|k|θ).
∂θ
Thus, at the critical points of H we must have sin(n|k|θ) = 0. In this case, cos(n|k|θ) =
1
±1. Ifcos(n|k|θ) = 1,then ∂H1 − ∂g cannotbezero. Whencos(n|k|θ) = −1, ∂H1 − ∂g =
∂r ∂r ∂r ∂r
−2µr + 2mrAe mr2. For r > 0, ∂H1 − ∂g = 0 when emr2 = mA, i.e., when r = r :=
− ∂r ∂r µ c
1 ln(mA). We impose the restriction that mA > µ. Note that by making m large, we
m µ
cqan make r arbitrarily small. When cos(n|k|θ) = −1, H −g(r,θ) = −µ(ln(mA)+1).
c 1 m µ
Let g(r) be equal to µ(ln(mA) + 1) on the annular neighborhood of r = r . For such g,
m µ c
H is 0at thecriticalpoints,i.e.,at thepoints(r ,θ), wherecos(n|k|θ) = −1.
1 c
Insummary,wegetcriticalpointsatonevalueofr atthevaluesofθ whencos(n|k|θ) =
−1, that is, for n|k| values of θ. These are our n|k| saddle points (it’s not hard to see they
are saddlepoints;alternatively,wecan deducethattheymustbeforindexreasons).
8 ROMANGOLOVKO
(2) Keepingf solelyafunctionofrandkeepinggconstant,wecutofff smoothlystarting
exp exp
atsomepointpastr togiveaHamiltonianH whichagrees withH +g outsideaball.
c 2 sing
As longas ∂fexp < 2µr,thereare nonew criticalpoints.
∂r
Notethatf (r ) = −µ. Keeping ∂fexp nearµr (which,usinge.g. A = eµ,is 1 ),we
exp c m ∂r c m √m
can bring f to zero in a radial distance of a constant times 1 ; i.e. for m large we can
exp √m
makeH agree withH +g outsidean arbitrarilysmallball.
2 sing
ForA = eµ,g = 2µ. Then keepingg solelyafunctionofr,wecutoffg(r,θ)smoothly
m m
starting at some point past the point where H = H + g to give Hamiltonian H . As
2 sing 3
longas ∂g > −2µr,therearenonewcriticalpoints. Wecan makeitinsuchawaythatH
∂r 3
agrees with H outsideasmallball.
sing
(3) Recall thatH = H +f +g near theoriginand g(r,θ) = 2µ > 0. Notethat g(r,θ)
3 sing exp m
is small for large m. Now keeping g constant we modify H +f +g near the origin
sing exp
to give us H which is Br2 − C near the origin (for B > 0), which corresponds to the
4
Hamiltonian flow rotating at a constant angular rate. Since ∂H3 = ∂(Hsing+fexp) > 0 for
∂r ∂r
r < r , we can patch together Br2 − C near the origin with H outside a small ball of
c 2
radius less than r in a radially symmetric manner to get H such that ∂H4 > 0 for r < r
c 4 ∂r c
(wedothisbychoosingC sufficientlylarge). NotethatH hasacriticalpointattheorigin.
4
(4) Finally, to ensure no fixed points of the time-1 flow of the Hamiltonian vector field of H,
we let H be H multiplied by a radially symmetric function which is ǫ for r < R (for ǫ
4
sufficientlysmallthattheonlyfixedpointsofthetime-1flowinsideradiusRarethecritical
points and for R large enough that H agrees with H for r > R) and 1 for r > 2R.
4 sing
This creates no new fixed points in the region R < r < 2R because H and ∂H4 have the
4 ∂r
samesignthere. Nowtherearenofixedpointsofthetime-1flowoftheHamiltonianvector
field ofH,exceptforthen|k|+1criticalpointsofH becauseoutsideradiusR thereareno
compact flowlines.
Let p ,...,p denote the equally spaced saddle points of H ordered counterclockwise, i.e.,
1 nk
R (p ) = p | |, where R corresponds tothe 2π -rotationaround thecenter ofD(r ).
nk i i+1 nk nk sing
| | | | | |
Remark 3.1. We first note that H(p ) = 0 for s = 1,...,n|k|. Hence, by Morse lemma (arguing
s
the same way as in Lemma 3.2 in [8]) we get that there is a neighborhood U of p such that
s s
H = axy on U , where s = 1,...,n|k| and a > 0. In addition, observe that H is 2π -symmetric
s nk
with respect to θ. Therefore, U ’s togetherwith coordinates (x,y)are 2π -symmetric| w| ith respect
s nk
to θ, i.e., R (U ) = U and coordinates on U maps to the coordin| a|te on U , where R
nk s s+1 s s+1 nk
denotes 2π -r|o|tation with respect to θ. Finally, note that H = B˜r2 −C˜ on a neighborhood of th|e|
nk
center of |D|(r ), which we call U, where C˜ > 0 and B˜ is a small positive number and hence
sing
Hamiltonianflow rotatesat aconstantratenear theorigin.
3.2. Sutured contact solidtori. In thissection, we constructthe sutured contact solidtorus with
2nsuturesofslope k,where n ∈ N,(k,l) = 1,|k| > l > 0.
l
Let γ be an embedded curve in R2 which starts at p and ends at p for s = 1,...,n|k|. For
p,ps s
thetimebeing,wecan thinkaboutγ as aboutthesegmentconnectingp and p .
p,ps s
Lemma 3.2. There existsa 1-formβ onR2 satisfyingthefollowing:
THECYLINDRICALCONTACTHOMOLOGYOFUNIVERSALLYTIGHTSUTUREDCONTACTSOLIDTORI 9
(1) dβ > 0;
(2) itssingularfoliationgiven bykerβ hasisolatedsingularitiesand noclosed orbits;
(3) β = εcr2dθ on U with respect to the polar coordinates whose origin is at the center of
2
D(r ); β = εsym(xdy − ydx) on U with respect to the coordinates from Remark 3.1,
sing 2 s
wheres ∈ {1,...,n|k|};β = 1r2dθonR2\D(r )withrespecttothepolarcoordinates
2 sing
whoseoriginisatthecenter ofD(r );here 0 < ε ≪ ε ≪ 1;
sing c sym
(4) the set of hyperbolic points of the singular foliation of β is given by {qs}ns=|k1| such that qs
lieson γ outsideof U andU;
p,ps s
(5) β is 2π -symmetric, i.e., R (β) = β, where R : R2 → R2 is a 2π -rotation with
nk n∗ k nk nk
respec|t t|othecenter ofD(r | |). | | | |
sing
Proof. ConsiderasingularfoliationF on R2 whichsatisfies thefollowing:
(1) F isMorse-Smaleandhas no closedorbits.
(2) The singular set of F consists of elliptic points and hyperbolic points. The elliptic points
aretheequallyspacedsaddlepointsofH andthecenterofD(r ). Thesetofhyperbolic
sing
pointsofthesingularfoliationofβ isgivenby {qs}ns=|k1| such that qs lieson γp,ps outsideof
U and U.
s
(3) F is oriented and for one choice of orientation the flow is transverse to and exits from
∂D(r ).
sing
(4) F is 2π -symmetricwithrespect to θ.
nk
| |
Next, we modify F near each of the singular points so that F is given by β = 1(xdy − ydx)
0 2
on U with respect to the coordinates from Remark 3.1 and β = 2xdy + ydx near a hyperbolic
s 0
point. On R2 \ D(r ), β = 1r2dθ with respect to the polar coordinates whose origin is at the
sing 0 2
center of D(r ). In addition, on U, β = 1r2dθ with respect to the polar coordinates whose
sing 0 2
origin is at the center of D(r ). From Remark 3.1 it follows that we can do it in such a way
sing
that the modification of F is still 2π -symmetric. Finally, we get F given by β , which satisfies
nk 0
dβ > 0 near the singular points an|d| on R2 \ D(r ). Now let β = gβ , where g is a positive
0 sing 0
function with dg(X) ≫ 0 outsideof U ∪(∪ns=|k1|Us)∪(R2 \D(rsing)), g|∪ns=|k1|Us = εsym, g|U = εc,
g| = 1 and X is an oriented vector field for F (nonzero away from the singularpoints).
R2 D(rsing)
Here\0 < ε ≪ ε ≪ 1. Sincedβ = dg∧β +g∧dβ ,dg(X) ≫ 0guaranteesthatdβ > 0. (cid:3)
c sym 0 0
Remark 3.3. From thepreviouslemmawe getβ defined onR2 withthefollowingproperties:
(i) dβ > 0on R2;
(ii) β = εcr2dθandH = B˜r2−C˜ onU,whereC˜ > 0;hereε isasmallpositiverealnumber;
2 c
(iii) β = εsym(xdy −ydx) andH = axy onU fors = 1,...,n|k|;here0 < ε ≪ ε ≪ 1;
2 s c sym
(iv) β = 1r2dθ and H = µr2cos(nθ) onR2 \D(r ).
2 sing
For the comparison of the level sets of H with the singular foliation of β in the case n = 1,
|k| = 3werefer to Figure2.
Lemma 3.4. Let β be a 1-form from Lemma 3.2. The Hamiltonian vector field X of H with
H
respect to the area form dβ satisfiesβ(XH) = H on (∪ns=|k1|Us)∪(R2 \D(rsing)). In addition,the
Hamiltonianvector field X of H with respect to the area form dβ satisfies β(X )−H = C˜ on
H H
U.
10 ROMANGOLOVKO
FIGURE 2. The level sets of H (left) and the characteristic foliationof β (right) in
thecasen = 1, |k| = 3
Proof. First, Remark 3.3 implies that β = εcr2dθ, H = B˜r2 −C˜ on U and ε is a small positive
2 c
number. NowweshowthatX = 2B˜ ∂ isasolutionofβ(X )−H = C˜ on U. Wecalculate
H εc ∂θ H
2B˜ ∂
i (dβ) = y(ε rdr∧dθ) = −2B˜rdr = −dH,
XH ε ∂θ c
c !
and
ε 2B˜ ∂
β(X )−H = cr2dθ −B˜r2 +C˜ = C˜.
H
2 ε ∂θ
c !
(cid:16) (cid:17)
Next,wework on U , wheres = 1,...,n|k|. From Remark 3.3 it followsthatβ = εsym(xdy −
s 2
ydx)and H = axy on U . LetX bea Hamiltonianvectorfield defined by i dβ = −dH.
s H XH
Weshowthat
ax ∂ ay ∂
X = − +
H
ε ∂x ε ∂y
sym sym
isasolutionoftheequation
(3.2.1) β(X ) = H
H
on U . We calculate
s
ax ∂ ay ∂
i (dβ) = − + y(ε dx∧dy) = −axdy −aydx = −dH
XH ε ∂x ε ∂y sym
(cid:18) sym sym (cid:19)