aa r X i v : . [ m a t h . S G ] D ec ( RP n − , ξ std ) IS NOT EXACTLY FILLABLE FOR n = 2 k ZHENGYI ZHOU
Abstract.
We prove that ( RP n − , ξ std ) is not exactly fillable for any n = 2 k and there exist strongly fillablebut not exactly fillable contact manifolds for all dimension ≥ Introduction
One fundamental principle in contact topology is the dichotomy between overtwisted and tight contactstructures discovered by Eliashberg [13] in dimension 3. The dichotomy was generalized to all higher dimen-sions recently by Borman, Eliashberg, and Murphy [2]. The h -principle for overtwisted contact structuresimplies that they are governed by their underlying formal data. On the other hand, the more mysterioustight contact structures can be roughly categorized into the following classes based on their fillability. { Weinstein fillable } ⊆ {
Exactly fillable } ⊆ {
Strongly fillable } ⊆ {
Weakly fillable } ⊆ {
Tight } . It is an interesting question to study differences between these classes. In dimension three, those inclusionswere shown to be proper by Bowden [6], Ghiggini [18], Eliashberg [15], Etnyre and Honda [17] respectively.In higher dimensions, the first, third and fourth inclusions were shown to be proper by Bowden, Crowley,and Stipsicz [7], Bowden, Gironella, and Moreno [8], Massot, Niederkr¨uger, and Wendl [21], who also firstshowed the properness of the third inclusion in dimension five. See also [28] for exactly fillable, almostWeinstein fillable, but not Weinstein fillable examples. The situation in dimension three differs from higherdimensions in the sense that we have gauge theoretic tools as well as better holomorphic curve theories,but also face more topological constraints. In higher dimensions, we have fewer tools but more flexibility inconstructions. The first, third and fourth inclusions can be studied from more structured perspectives. Thechallenges in those cases are finding examples and executing the machineries. On the other hand, it seemsthat we are poorly equipped to study the second inclusion in higher dimensions. Fortunately, we have simplepotential examples, i.e. real projective spaces with the standard contact structure induced from the doublecover. It was conjectured by Eliashberg [11, § Theorem 1.1.
For n = 2 k , ( RP n − , ξ std ) is not exactly fillable. Note that ( RP n − , ξ std ) admits a strong filling O ( − − CP n − , which isnot exact since the zero section CP n − is a symplectic submanifold. The condition n ≥ RP , ξ std ) is exactly (Weinstein) fillable by T ∗ S . Moreover, it is known from [16, § RP n − , ξ std )admits a Weinstein filling iff n ≤ n ≥
3, see Remark 2.16 for how our proof sees the exceptionwhen n = 2. The n = 2 k condition is only used in the topological argument in Proposition 2.2. Our proofalso shows that ( RP n − , ξ std ) , n = 2 k admits no symplectically aspherical filling and no Calabi-Yau filling(i.e. strong fillings W such that c ( W ) = 0 ∈ H ( W ; Q )), see Remark 3.6. Remark 1.2.
The n = 3 case was also announced by Ghiggini and Niederkr¨uger using a different method. More specifically, our strategy of proof is the following.Symplectic part: ( RP n − , ξ std ) has a very nice Reeb dynamics, moreover, its double cover is the standardcontact sphere. In the latter, a short Reeb orbit will bound a rigid holomorphic curve with a point constraint.For a generic point or a generic almost complex structure, the curve lives completely in the symplectization.In particular, one can push the curve down to the hypothetical exact filling of RP n − . Such curve kills aunit hence the whole symplectic cohomology for any exact filling of ( S n − , ξ std ). Then we argue that thisis also the case for ( RP n − , ξ std ) for n ≥ H i ( W ) → H i ( RP n − ). We can study this map bycomputing Chern classes of ξ std since c i ( W ) | RP n − = c i ( ξ std ). The n = 2 k condition is used to ensure thetotal Chern class of ξ std is not trivial.Note that ( RP n − , ξ std ) can be viewed as the link of the quotient singularity C n / Z . Our method isadaptable to other quotient singularities as well. We use ( S n − / Z k , ξ std ) to denote the link of C n / Z k , where Z k acts on C n by multiplication by e πik . Then we prove the following. Theorem 1.3.
Let p be an odd prime, assume n has the p -adic representation n = P ks =0 a s p s . Then ( S n − / Z p , ξ std ) has no exact filling if P ks =0 a s > p − . Note that the n = 2 case is again exactly fillable, see [14]. The threshold is by no means sharp, andthe p -adic information is very likely unnecessary. We will speculate in the proof that the symplectic partworks for n ≥ p + 1. While the p -adic information of n is used in the topological argument to guaranteethat the total Chern class of ( S n − / Z p , ξ std ) has enough nontrivial terms. Note that n = p is the thresholdfor the quotient singularity to be canonical, or admit a crepant resolution, and when n > p the singularitybecomes terminal. As explained in [23], being terminal is equivalent to having positive minimal SFT degreefor some contact form, which is closely related to the concept of asymptotically dynamical convexity [20] . Inparticular, the result here bears certain similarity with [28]. However, we do not assume the exact filling tohave any topological properties (e.g. vanishing first Chern class and π -injectivity) as in [28]. Our approachcan be adapted to study more general quotient singularity C n /G . The relation between exact fillability andits algebro-geometric properties is an interesting question, we wish to study it in the future.Combining with the Z and Z quotient singularities, we will show that the second inclusion is proper forall dimension ≥
5, hence complete the question of proper inclusions for all dimension > Theorem 1.4.
For every n ≥ , then there exists a n − dimensional contact manifold which is stronglyfillable but not exactly fillable. The main difference between [23] and [20] lies in the treatment of non-contractible orbits, which play important roles inthis paper. More precisely, [23] assigned a rational SFT degree to every Reeb orbits in the case of c ( ξ ) = 0 ∈ H ( Y, Q ) and H ( Y, Q ) = 0 and being terminal is equivalent to that there exists a contact form such that all Reeb orbits have positive SFTdegree. On the other hand, [20] only considered contractible Reeb orbits in the case of c ( ξ ) = 0 and asymptotically dynamicalconvexity is roughly admitting a contact form such that all contractible Reeb orbits have positive (integer) SFT degree. In thecase of ( RP n − , ξ std ) , n ≥
3, since those non-contractible simple Reeb orbits play an important role, the relevant concept is theformer one. RP n − , ξ std ) IS NOT EXACTLY FILLABLE FOR n = 2 k Acknowledgement.
The author is supported by the National Science Foundation Grant No. DMS1638352.It is a great pleasure to acknowledge the Institute for Advanced Study for its warm hospitality. The authoris in debt to Paolo Ghiggini for very helpful comments and pointing out a gap in an earlier draft. Theauthor is deeply grateful to anonymous referees for many helpful comments and suggestions. This paper isdedicated to the memory of Chenxue. 2.
Proof of Theorem 1.1
In the following, the coefficient is Z unless otherwise specified. The contradiction leading to the proof ofTheorem 1.1 is the following. Proposition 2.1.
For n ≥ , if ( RP n − , ξ std ) has an exact filling W , then the following holds.(1) If n is odd, then the total cohomology H ∗ ( W ; R ) = R , supported in degree .(2) If n is even, then H ∗ ( W ; R ) = R or R ⊕ R , and in the latter case, the cohomology is preciselysupported in degree and n . We will first prove Theorem 1.1 assuming Proposition 2.1. First of all, we observe the following fact.
Proposition 2.2.
Let W be a strong filling of ( RP n − , ξ std ) for n = 2 k p for odd p ≥ , then H k +1 ( W ) → H k +1 ( RP n − ) = Z and H n − k +1 ( W ) → H n − k +1 ( RP n − ) = Z , induced by the inclusion RP n − ֒ → W ,are both surjective.Proof. Since c i ( W ) | RP n − = c i ( ξ std ), we can prove the claim if both c k ( ξ std ) and c n − k ( ξ std ) are nonzero.In the following, we will compute the total Chern class of ξ std from the standard filling O ( − O ( −
2) can be computed from the total Chern class of T O ( − | CP n − , wherethe bundle splits into T CP n − ⊕O ( − T CP n − is (1+ u ) n and the total Chernclass of the bundle O ( −
2) is 1 − u [24, Theorem 14.4], where u is the generator of H ( CP n − ) = H ( O ( − O ( −
2) is (1 + u ) n (1 − u ).Using the fact that ( P a i ) = P a i mod 2, we have(1 + u ) n (1 − u ) = (1 + u ) p k = (1 + pu + . . . + pu p − + u p ) k = 1 + p k u k + . . . + p k u ( p − k + u n mod 2 . Therefore we have both c k ( O ( − c n − k ( O ( − Z . By the Gysin exact sequence, therestriction map Z = H i ( O ( − → H i ( RP n − ) = Z is the mod 2 map for 1 ≤ i ≤ n −
1. Then the claimfollows. (cid:3)
Remark 2.3.
When n = 2 k , the total Chern class of ξ std is trivial.Proof of Theorem 1.1. Assume ( RP n − , ξ std ) has an exact filling W and n = 2 k p for odd p ≥
3. Then byProposition 2.1, H k +1 − ( W ), H k +1 ( W ), H k +1 +1 ( W ), H n − − k +1 ( W ), H n − k +1 ( W ), H n +1 − k +1 ( W ) areall torsions. By looking at the long exact sequence of ( W, RP n − ), we have the following,0 → H k +1 ( W, RP n − ) → H k +1 ( W ) → Z → H k +1 +1 ( W, RP n − ) → H k +1 +1 ( W ) → , → H n − k +1 ( W, RP n − ) → H n − k +1 ( W ) → Z → H n +1 − k +1 ( W, RP n − ) → H n +1 − k +1 ( W ) → , which also implies that all groups above are torsions. Then by Lefschetz duality and universal coefficienttheorem, we have H k +1 ( W, RP n − ) ≃ H n − k +1 ( W ) ≃ H n +1 − k +1 ( W ) ,H k +1 +1 ( W, RP n − ) ≃ H n − − k +1 ( W ) ≃ H n − k +1 ( W ) , ZHENGYI ZHOU H n − k +1 ( W, RP n − ) ≃ H k +1 ( W ) ≃ H k +1 +1 ( W ) ,H n +1 − k +1 ( W, RP n − ) ≃ H k +1 − ( W ) ≃ H k +1 ( W ) . Therefore the two long exact sequences become0 → H n +1 − k +1 ( W ) → H k +1 ( W ) → Z → H n − k +1 ( W ) → H k +1 +1 ( W ) → , → H k +1 +1 ( W ) → H n − k +1 ( W ) → Z → H k +1 ( W ) → H n +1 − k +1 ( W ) → . By Proposition 2.2, H k +1 ( W ) → Z and H n − k +1 ( W ) → Z above are surjective. Then the long exactsequences above imply that H n − k +1 ( W ) ≃ H k +1 +1 ( W ). Since all those groups are finite groups, we have | H n − k +1 ( W ) | = | H k +1 +1 ( W ) | and also | H n − k +1 ( W ) | = 2 | H k +1 +1 ( W ) | , which is a contradiction. (cid:3) The rest of the section is devoted to the proof of Proposition 2.1.2.1.
Setup of symplectic cohomology.
Note that ( RP n − , ξ std ) is equipped with the Boothby-Wangcontact form α std such that the Reeb vector gives the Hopf fibration, and the periods of Reeb orbits aregiven by N + . We can choose a C -small perfect Morse function f on CP n − . Let q , . . . , q n − denote thecritical points of f ordered by their critical values. Let π : RP n − → CP n − denote the projection, thenthe r = 1 + π ∗ f hyperspace in the symplectization RP n − × ( R + ) r gives a perturbed contact form α f , suchthat the Reeb vector field for α f is R f = 11 + π ∗ f R std + Z, with Z ∈ ξ std , ι Z dα std | ξ std = − π ∗ f ) π ∗ df | ξ std . (2.1)In other words, Z is the horizontal lift of the Hamiltonian vector of π ∗ f on ( CP n − , ω F S ) using α std as aconnection on RP n − . We may fix a small ǫ > nǫ <
1, such that f ( q i ) = iǫ . Since f is C -small,we may assume that the Hamiltonian vector of π ∗ f , i.e. π ∗ Z , has no non-constant orbit of period smallerthan 3. As a consequence of (2.1), we have the following.(1) There is a simple Reeb orbit γ i , such that π ( γ i ) = q i and the period of γ i is 1 + iǫ = 1 + f ( q i ).(2) All Reeb orbits of period smaller than 2(1 + nǫ ) are non-degenerate and is either γ i or its doublecover γ i .Let p : S n − → RP n − denote the double cover. Then ( S n − , p ∗ α f ) is roughly an ellipsoid, in particular,all Reeb orbits of period smaller than 2(1 + nǫ ) are the lifts of γ i . Among them, the lift of γ has theminimal period and minimal Conley-Zehnder index n + 1.In the following, we will fix a specific choice of f . Let ǫ be a small positive number, we choose f to be ǫ n − X i =0 i iǫ | z i | ! , n − X i =0 | z i | iǫ ! . Then the function f satisfies all conditions above. Moreover, ( S n − , p ∗ α f ) is indeed the ellipsoid givenby P n − i =0 | z i | iǫ ) = 1 in ( C n , i π P n − i =0 d z i ∧ d z i ). Here when we identify the symplectic manifold ( RP n − × R + , d ( rα std )) with ( { C n − { }} / Z , i π P n − i =0 dz i ∧ dz i ), we have r = P n − i =0 | z i | . An ellipsoid P n − i =0 | z i | a i = 1is called non-degenerate iff a i /a j Q for any i = j . If the ellipsoid is non-degenerate, the induced contactform is non-degenerate and all Reeb orbits are circles in each coordinate plane. In our case, if we pick ǫ / ∈ Q ,( S n − , p ∗ α f ) is a non-degenerate ellipsoid. RP n − , ξ std ) IS NOT EXACTLY FILLABLE FOR n = 2 k Let (
W, λ ) be an exact filling of ( RP n − , α f ), i.e. the Liouville form λ restricted to RP n − is α f . Let c W := W ∪ ∂W × (1 , ∞ ) denote the completion. We will set up our moduli spaces for symplectic cohomologyfollowing [28] combined with the autonomous setting in [5]. Instead of using a single Hamiltonian as in [28],we will phrase symplectic cohomology as a direct limit over Hamiltonians with finite slopes like the classicalconstruction in [26]. We refer to them as well as references therein for details of symplectic cohomology. Inparticular, we will use Hamiltonians and almost complex structures satisfies the following.(1) H = 0 on W and H = h ( r ) on ∂W × (1 , ∞ ) with h ′ ( r ) = a for r ≫ h ′′ ( r ) > h = 0 or h ′ ( r ) = a . We will be only interested in the case a ∈ (0 , nǫ )) and is not the period of a Reeborbit. The class of Hamiltonians with slope a is denoted as H a . Since h ′ ( r ) = a iff r ≥ w forsome w >
0, we will call the minimum w with such property the width of the Hamiltonian H .(2) The almost complex structure J t is independent of t on W ∪ ∂W × (1 , r ], where h ′ ( r ) = 1 + ( n − ǫ ,i.e. r is the last level containing a simple Hamiltonian orbit. J t is compatible with symplecticstructure and is cylindrical convex near every r such that h ′ ( r ) is the period of a Reeb orbit, i.e. J t ξ = ξ and J t ( r∂r ) = R α f . This guarantees the integrated maximum principle [1] can be appliedto obtain compactness of moduli spaces, see [28, Lemma 2.5] for details.We also choose a Morse function g on W , such that ∂ r g > ∂W and g has a unique minimum. Theextra requirement on g will be specified later. We use γ to denote the S family of Hamiltonian orbitscorresponding to γ . Then we pick two different generic points ˆ γ and ˇ γ on im γ , this is equivalent to choosinga Morse function g γ with one maximum and one minimum on im γ in [5, § g γ can be used to perturb the Hamiltonian H to get two non-degenerate orbits from γ , which areoften denoted by ˆ γ and ˇ γ in literatures with µ CZ (ˆ γ ) = µ CZ ( γ ) + 1 and µ CZ (ˇ γ ) = µ CZ ( γ ).Then we have a Floer cochain complex C ( H ), which is a free R -module generated by critical points of g with Morse index as grading, and two generators ˆ γ, ˇ γ for each Reeb orbit γ of period smaller than a ,with Z gradings n − µ CZ ( γ ) − n − µ CZ ( γ ). The differential is defined by counting rigid cascades[5, (39),(40)]. Moreover, we have a subcomplex ( C ( H ) , d ), which is the Morse cochain complex of g and a quotient complex ( C + ( H ) , d + ) := C ( H ) /C ( H ) generated by the generators from Reeb orbits. Thedifferential on C ( H ) also has a component d + , : C + ( H ) → C ( H ), which induces the connecting map H ∗ ( C + ( H )) → H ∗ +1 ( C ( H )) in the tautological long exact sequence. We can achieve transversality usingour almost complex structure, because on r ≤ r all orbits are simple [5, Proposition 3.5]. The differentialcan be described in a pictorial way as follows. u u ∇ gu u Figure 1. d + and d + , from 2 level cascades(1) Each unlabeled horizontal arrow is a negative gradient flow of g γ in im γ , i.e. flow toward ˇ γ .(2) u is a solution to the Floer equation ∂ s u + J t ( ∂ t u − X H ) = 0 modulo R translation. ZHENGYI ZHOU (3) Every intersection point of line with surface satisfies the obvious matching condition.More formally, the differential is defined by counting the following compactified moduli spaces.(1) For p, q ∈ Crit ( g ), M p,q := { γ : R s → W | γ ′ + ∇ g = 0 , lim s →∞ γ = p, lim s →−∞ γ = q } / R . (2) For γ + , γ − ∈ { ˇ γ, ˆ γ |∀ S family of orbits γ } , a k -cascade from γ + to γ − is a tuple ( u , l , . . . , l k − , u k ),such that(a) l i are positive real numbers.(b) nontrivial u i ∈ { u : R s × S t → c W | ∂ s u + J t ( ∂ t u − X H ) = 0 , lim s →∞ u ∈ γ i − , lim s →−∞ u ∈ γ i } / R such that γ + ∈ γ and γ − ∈ γ k , where the R action is the translation on s .(c) φ l i −∇ g γi (lim s →−∞ u i ( s, s →∞ u i +1 ( s,
0) for 1 ≤ i ≤ k − γ + = lim t →∞ φ − t −∇ g γ (lim s →∞ u ( s, γ − = lim t →∞ φ t −∇ g γk (lim s →−∞ u k ( s, φ t −∇ g γ is the time t flow of −∇ g γ on im γ .Then we define M γ + ,γ − to be the compactification of the space of all cascades from γ + to γ − .The compactification involves the usual Hamiltonian-Floer breaking of u i as well as degenerationcorresponding to l i = 0 , ∞ . The l i = 0 degeneration is equivalent to a Hamiltonian-Floer breakinglim s →−∞ u i = lim s →∞ u i +1 . In particular, they can be glued or paired, hence do not contribute(algebraically) to the boundary of M γ + ,γ − . The l i = ∞ degeneration is equivalent to a Morsebreaking for g γ i , which will contribute to the boundary of M γ + ,γ − .(3) For γ + ∈ { ˇ γ, ˆ γ |∀ S family of orbits γ } and q ∈ Crit ( g ), a k -cascades from γ + to q is a tuple( l , u , l , . . . , u k , l k ) as before, except(a) u k ∈ { u : C → c W | ∂ s u + J t ( ∂ t u − X H ) = 0 , lim s →∞ u ∈ γ k − , u (0) ∈ W ◦ } / R , where we use theidentification R × S → C ∗ , ( s, t ) e π ( s + it ) . W ◦ is the interior of W , where the Floer equationis ∂ J u = 0, hence the removal of singularity implies that u (0) is a well-defined notation.(b) q = lim t →∞ φ t ∇ g ( u k (0)).Then M γ + ,q is defined to be the compactification of the space of all cascades from γ + to q . Remark 2.4.
It is important to note that in M x,y , x is the asymptotic orbit at the positive and y is theasymptotic orbit at the negative end, which is opposite to the notation in [28] . All of the moduli spaces above can be equipped with coherent orientations. In view of the notation above,the differentials are defined as follows. d ( p ) = X q, dim M p,q =0 M p,q q,d + ( γ + ) = X γ − , dim M γ + ,γ − =0 M γ + ,γ − γ − ,d + , ( γ + ) = X q, dim M γ + ,q =0 M γ + ,q q, where M denotes the signed count of the zero-dimensional compact moduli space M . Our symplecticaction follows the cohomological convention A ( γ ) = − Z γ λ + Z γ H, RP n − , ξ std ) IS NOT EXACTLY FILLABLE FOR n = 2 k where λ is a Liouville form such that λ | ∂W = α f . Our convention for X H is d λ ( · , X H ) = d H . Every non-constant Hamiltonian orbit is contained in a level set ∂W × { r } , such that h ′ ( r ) is the period of a Reeb orbit.The symplectic action of this Hamiltonian orbit is given by − rh ′ ( r ) + h ( r ). Since for any non-trivial solution u solving ∂ s u + J t ( ∂ t u − X H ) = 0, we have A ( u ( ∞ )) < A ( u ( −∞ )), hence if u ( ∞ ) ∈ γ and u ( −∞ ) ∈ β , thenthe period of γ is larger than the period of β . Remark 2.5.
A few remarks on different models of symplectic cohomology are in order.(1) The most classical construction is using linear time-dependent Hamiltonians, which is also C smallMorse on W , the total symplectic cohomology is the direct limit of the Hamiltonian-Floer cohomologywhen the slope converges to ∞ , c.f. [26] .(2) One can also use Hamiltonians that is autonomous. In particular, H = h ( r ) on the cylindrical end ∂W × (1 , ∞ ) with h ′′ > can be used iff the contact form is non-degenerate. Then cascades modelis needed to deal with the S -family of Hamiltonian orbits. This is the construction in [5] .(3) One can use time-dependent Hamiltonians that are zero on W and are small perturbations to theHamiltonian in (2) on the cylindrical end. Then it is very close to a Morse-Bott situation and weneed to introduce an auxiliary Morse function g on W such that ∂ r g > on ∂W . This is the approachtaken in [28] . The vanishing of the Hamiltonian on W makes it easier to apply neck-stretching inthis setup.(4) When the contact form is only Morse-Bott non-degenerate in the sense of [3] , if we take Hamiltoniansin the form of h ( r ) . The Hamiltonian orbits will come in more general family than S -family. Thenwe can pick an auxiliary Morse function on the space parameterizing the family and apply a cascadeconstruction. Moreover, one can also choose H to be zero on W and pick another auxiliary Morsefunction on W . This is the approach taken in [12, § .The approach taken in this paper is a mixture of (2) and (3) for moduli spaces setup but with finite slopeHamiltonians as in (1) , and can also be viewed as a special case of (4) with a Hamiltonian of finite slope. Thecompactness of relevant moduli spaces follows from [28, Proposition 2.6.] and [5, § . The transversalityessentially follows from somewhere injectivity of Floer cylinders, see [27, Proposition 2.8.] and [5, § .One way to relate the cascades construction with the classical construction is through a gluing analysis fordegeneration [5] , another approach is via cascades continuation/Viterbo transfer maps used in [12, § . Remark 2.6.
To define a global Conley-Zehnder index, we need to choose a trivialization of the determinantline bundle det C ξ std . In our case, c ( ξ std ) is not always in H ( RP n − ) for any n , therefore, we may notbe able to trivialize det C ξ std globally. In this case, we can assign a Conley-Zehnder index for each orbit γ after fixing a trivialization of γ ∗ det C ξ std . The parity of the Conley-Zehnder index does not depend on thetrivialization. One way to get a natural trivialization of det C ξ std is by choosing a bounding disk u of γ eitherin RP n − or some symplectic filling W of RP n − . Since u ∗ det C ξ std /u ∗ det C T W is uniquely trivialized, itinduces a trivialization of γ ∗ det C ξ std . When using Conley-Zehnder index from bounding disks to computevirtual dimensions of moduli spaces, it is crucial to check the bounding disks are compatible via gluing.The Conley-Zehnder index of Hamiltonian orbits has the same property. By computing the indexes in thestandard filling O ( − , we have that all check generators ˇ γ have odd grading and all hat generators ˆ γ haveeven grading.As another important ingredient to our proof, any holomorphic curve in the symplectization RP n − × R + has a well-defined index depending only on its asymptotics. This is because c ( ξ std ) is torsion (which is called This complex (called presplit Floer complex in [12]) does not require the monotonicity condition in [12].
ZHENGYI ZHOU numerically Q -Gorenstein in the context of singularity theory [23] ). When we use the trivialization inducedby the obvious disk bounded by γ i and γ i in O ( − , then the SFT grading is given by µ CZ ( γ i ) + n − i, µ CZ ( γ i ) + n − i + 2 . (2.2) As an example, we compare indexes from different trivializations as follows, the disk in O ( − bounded by γ differs from the contraction of γ in RP n − by a generator A of H ( O ( − . Therefore if we use suchtrivialization induced by the contraction, we have µ CZ ( γ )+ n − c ( A )+ 2 = 2 n − , i.e. µ CZ ( γ ) = n + 1 which is same as the Conley-Zehnder index of the shortest Reeb orbits on an ellipsoid. Remark 2.7.
To elaborate the compatibility of trivializations, if we consider curve u in the symplectization RP n − × R + with one positive puncture asymptotic to γ i and two negative punctures asymptotic to γ j , γ k ,then the trivialization from the bounding disks in O ( − are compatible. This is because: if we glue thebounding disks of γ k , γ i to u , we get a disk which relatively homotopic to the bounding disk of γ i in O ( − .One can see this by checking the intersection number with CP n − . On the other hand, if we change thepositive asymptotics to γ i , then the natural bounding disks from O ( − are no longer compatible with adifference from the generator of H ( CP n − ) . Moreover, inside H a , we have a partial order given by increasing homotopies. Every increasing homotopyinduces a continuation map, which also preserves the splitting into C and C + . Therefore we can define thefiltered symplectic cohomology as follows, SH ∗ , ≤ a ( W ; R ) = lim −→ H ∈H a H ∗ ( C ( H )) , SH ∗ , ≤ a + ( W ; R ) = lim −→ H ∈H a H ∗ ( C + ( H )) . And we have a tautological long exact sequence (or circle, since they are only Z graded in general), . . . → H ∗ ( W ; R ) → SH ∗ , ≤ a ( W ; R ) → SH ∗ , ≤ a + ( W ; R ) → H ∗ +1 ( W ; R ) → . . . . (2.3)The continuation maps also gives ι a,b : SH ≤ a ( W ; R ) → SH ≤ b ( W ; R ) for a ≤ b , similarly for the positivesymplectic cohomology. They are compatible with tautological long exact sequence. To avoid using directlimit, we will use the following. Proposition 2.8.
For H ∈ H a , the natural morphisms H ∗ ( C ( H )) → SH ∗ , ≤ a ( W ; R ) and H ∗ ( C + ( H )) → SH ∗ , ≤ a + ( W ; R ) are both isomorphism.Proof. We will prove that if H ≤ H ′ ∈ H a , then the continuation map induces isomorphism H ∗ ( C ( H )) ≃ H ∗ ( C ( H ′ )) and H ∗ ( C + ( H )) ≃ H ∗ ( C + ( H ′ )). Note that there exists a positive number c such that H ′ ≤ H + c .Although H + c is no longer admissible in our sense, the Hamiltonian-Floer cohomology can still be definedand there is a continuation map C ( H ′ ) → C ( H + c ). Moreover the composition C ( H ) → C ( H ′ ) → C ( H + c )is homotopic to the continuation map from H to H + c , which is identity. In particular, H ∗ ( C ( H )) → H ∗ ( C ( H ′ )) is injective and H ∗ ( C ( H ′ )) → H ∗ ( C ( H + c )) is surjective. We can apply the same argumentto the composition C ( H ′ ) → C ( H + c ) → C ( H ′ + c ) to conclude that H ∗ ( C ( H ′ )) → H ∗ ( C ( H + c )) isinjective. Therefore both H ∗ ( C ( H ′ )) → H ∗ ( C ( H + c ) and H ∗ ( C ( H )) → H ∗ ( C ( H ′ )) are isomorphism. Since H ∗ ( C ( H )) → H ∗ ( C ( H ′ )) is an isomorphism if we use the same Morse function g on W , the five lemmaimplies that H ∗ ( C + ( H )) → H ∗ ( C + ( H ′ )) is also an isomorphism. (cid:3) The index is computed in a similar way to [28, Theorem 6.3]. One way to explain is writing µ CZ ( γ ji ) = (2 i − n + 1) + 2 j ,then 2 i − n + 1 = ind q i − dim CP n − is the Conley-Zehnder index comes from the Hamiltonian of π ∗ f and 2 j is theConley-Zehnder index comes from wrapping around the disk j times. RP n − , ξ std ) IS NOT EXACTLY FILLABLE FOR n = 2 k Due to the fact that α f is a small perturbation of the Morse-Bott contact form α std , we can use theMorse-Bott spectral sequence [26, (3.2)] to estimate the filtered positive symplectic cohomology . In fact,by a compactness and gluing argument similar to [5], one can show that for ǫ sufficiently small, we have d + ˆ γ i = 2ˆ γ i − , which is from the Gysin sequence of the degree 2 circle bundle RP n − → CP n − . To avoid thegluing analysis overhead, in the following, we give a weaker result that is sufficient for our purpose, whichonly uses compactness argument in the spirit of [9, Lemma 2.1] and the Viterbo transfer map. Proposition 2.9.
Assume there is an exact filling W of ( RP n − , α f ) . For ǫ sufficiently small, we have thefollowing.(1) d + ˆ γ i = a i ˇ γ i − for a i = 0 .(2) For < a < ǫ , we have im( SH ∗ , ≤ a + ( W ; R ) → H ∗ +1 ( W ; R )) is at most rank and is supportedin even degrees.Proof. For ǫ sufficiently small, we can find a small δ such that α f < (1 + δ ) α std , i.e. there exists a function h > RP n − , such that (1 + δ ) α std = hα f . Then we have two exact (trivial) cobordisms X from( RP n − , (1 − δ ) α std ) to ( RP n − , α f ) and X from ( RP n − , α f ) to ( RP n − , (1 + δ ) α std ). Then we have twoViterbo transfer maps from SH ∗ , ≤ a + ( W ; R ) → SH ∗ , ≤ a + ( W std − δ ; R ) and SH ∗ , ≤ a + ( W std δ ; R ) → SH ∗ , ≤ a + ( W ; R ),c.f. [10, Definition 5.2] , where W std ± δ is the exact filling of ( RP n − , (1 ± δ ) α std ) modified from W . Thecomposition is the Viterbo transfer maps SH ∗ , ≤ a + ( W std δ ) → SH ∗ , ≤ a + ( W std − δ ) by the functorial property ofViterbo transfer maps [10, Proposition 5.4]. Then for (1 + δ ) < a < − δ , we have the SH ∗ , ≤ a + ( W std δ ) → SH ∗ , ≤ a + ( W std − δ ) is an isomorphism by Proposition A.4 in the appendix. If we use W δ − δ to denote W ∪ ∂W × (1 , δ − δ ], then for δ − δ (1 + ( n − ǫ ) < a <
2, we have the Viterbo transfer SH ∗ , ≤ a + ( W δ − δ ) → SH ∗ , ≤ a + ( W ) isan isomorphism by Proposition A.4. That is the two compositions in the following are both isomorphism, SH ∗ , ≤ a + ( W δ − δ ) → SH ∗ , ≤ a + ( W std δ ) → SH ∗ , ≤ a + ( W ) → SH ∗ , ≤ a + ( W std − δ ) . As a consequence, we have SH ∗ , ≤ a + ( W ) → SH ∗ , ≤ a + ( W std − δ ) is an isomorphism.By the same argument of [9, Lemma 2.1], we have that all Floer trajectories as well as the continuationtrajectories in the Viterbo transfer maps are contained in a tubular neighborhood of the boundary ∂W (containing ∂W std − δ , ∂W std δ ) for ǫ, δ small enough . Since all curves are contained in this neighborhood, wecan assign a “local Z grading” as c ( ξ std ) is torsion, although c ( W ) may not be torsion. Note that all ofgenerators are in the same homotopy class, after fixing a trivialization of γ ∗ i det C ξ std and using that c ( ξ std ) istorsion, we can assign a Z -grading | γ | = n − µ CZ ( γ ) to every generator. Moreover, we have | ˇ γ |−| ˇ γ i | = 2 i and | ˇ γ |−| ˆ γ i | = 2 i +1, e.g. we can use the one from Remark 2.6. The Viterbo transfer preserves this local gradingas the trajectories in the Viterbo transfer are contained in this tubular neighborhood, and SH ∗ , ≤ a + ( W std − δ ; R )is the real cohomology of the critical submanifold RP n − of parameterized simple Reeb orbits, which is The statement of Morse-Bott spectral sequence in [26] requires the vanishing of the first Chern class, the absence of thiscondition will not effect the validity of the spectral sequence but will cost us the grading in [26]. But we will not use the gradingin this paper. Note that we need to adapt a cascades construction for the Viterbo transfer in the sense of (4) of Remark 2.5, since α std isMorse-Bott. This is just a cascades continuation map for some special Hamiltonians. Since for ǫ = δ = 0, all Floer trajectories/continuation trajectories become trivial cylinder on ( RP n − , α std ). Note that thisonly uses compactness of trajectories. supported in grading | ˇ γ | and | ˆ γ n − | . In view of the isomorphism SH ∗ , ≤ a + ( W ; R ) → SH ∗ , ≤ a + ( W std − δ ; R ) andthe grading, we must have d + ˆ γ i = a i ˇ γ i − for a i = 0. If we use Z -coefficient, we can conclude that a i = ± SH ∗ , ( a,b ]+ ( W ; R ) to denote the cohomology of the quotient complex generated bygenerators corresponding to Reeb orbits of period in ( a, b ]. Then for 2 < a < ǫ , δ − δ (2 + (2 n − ǫ )
Proposition 2.10.
Let W be an exact domain and A ∈ ⊕ i> H i ( W ; R ) . If A is mapped to zero in ι ,a : H ∗ ( W ; R ) → SH ∗ , ≤ a ( W ; R ) , then H ∗ ( W ; R ) → SH ∗ , ≤ a ( W ; R ) is zero and connecting map SH ∗ , ≤ a + ( W ; R ) → H ∗ +1 ( W ; R ) in the tautological long exact sequence is surjective .Proof. We have the following commutative diagram, e.g. see [19, Lemma 2.8], H ∗ ( W ; R ) ⊗ H ∗ ( W ; R ) id ⊗ ι ,a / / (cid:15) (cid:15) H ∗ ( W ; R ) ⊗ SH ≤ a ( W ; R ) ∪ / / SH ≤ a ( W ; R ) (cid:15) (cid:15) H ∗ ( W ; R ) ⊗ H ∗ ( W ; R ) ∪ / / H ∗ ( W ; R ) ι ,a / / SH ≤ a ( W ; R ) The proposition holds for A with positive degree, the emphasis on even degree only makes 1 + A degree 0 in the Z grading,which we have on SH ∗ ( W ; R ). RP n − , ξ std ) IS NOT EXACTLY FILLABLE FOR n = 2 k Since A is nilpotent in H ∗ ( W ; R ), we have that 1 + A is a unit in H ∗ ( W ; R ). Then for any x ∈ H ∗ ( W ; R ),we have 0 = ( x ∪ (1 + A ) − ) ∪ ι ,a (1 + A ) = ι ,a ( x ), i.e. H ∗ ( W ; R ) → SH ∗ , ≤ a ( W ; R ) is zero. Then by thetautological long exact sequence, we have SH ∗ , ≤ a + ( W ; R ) → H ∗ +1 ( W ; R ) is surjective. (cid:3) Vanishing of symplectic cohomology.
The key ingredient in our proof is that ˇ γ up to certainerror kills the symplectic cohomology as it does for the double cover whenever n ≥
3. If this was proven,Proposition 2.10 can be used to estimate the rank of cohomology of the filling by filtered positive sym-plectic cohomology. To show that ˇ γ essentially kills the unit, we will study the map SH ∗ , ≤ ǫ + ( W ; R ) d + , −→ H ∗ +1 ( W ; R ) projection −→ H ( W ; R ), in particular, we are interested to understand if 1 is in the image, i.e. weare interested in the contribution d + , (ˇ γ ) to the minimum of g . The choice of action threshold is to includeˇ γ in the cochain complexes but nothing else with larger periods.By our setup of symplectic cohomology, one part of this contribution is the count of the following modulispace (i.e. 1 level cascades). M (ˇ γ , q ) := { u : C → c W | ∂ s u + J t ( ∂ t u − X H ) = 0 , lim s →∞ u ( s,
0) = ˇ γ , u (0) = q } / R , (2.4)where q is a fixed point inside W , which is the unique minimum of the Morse function g and ( s, t ) is the polarcoordinate on C ∗ by ( s, t ) e π ( s + it ) . Since q is the minimum and ˇ γ is a check generator, both the Morseflow lines degenerate to point constraints. Since ˇ γ is on im γ , we will call the constraint from ˇ γ an orbitpoint constraint to differentiate it from the point constraint from q . We can choose q to be arbitrarily closeto ∂W . We will perform a neck stretching along Y ⊂ W , which is a slight push-in along the − r directionand strictly contactomorphic to ( RP n − , (1 − δ ) α f ) for δ small. We use X to denote the cobordism from( RP n − , (1 − δ ) α f ) to ( RP n − , α f ). We may assume q is an interior point of X . We use ∂ + X to denotethe positive boundary and ∂ − X to denote the negative boundary.We first recall the setup of neck-stretching for general case following [28, § W, λ ) be a exactdomain and ( Y , α := λ | Y ) be a contact type hypersurface inside W . The hyperplane divides W intoa cobordism X union with a domain W ′ . Then we can find a small slice ( Y × [1 − η, η ] r , d( rα ))symplectomorphic to a neighborhood of Y in W . Assume J | Y × [1 − η, η ] r = J , where J is independentof S and r and J ( r∂ r ) = R α , J ξ = ξ for ξ := ker α . Then we pick a family of diffeomorphism φ R :[(1 − η ) e − R , (1 + η ) e R − ] → [1 − η, η ] for R ∈ (0 ,
1] such that φ = id and φ R near the boundaryis linear with slope 1. Then the stretched almost complex structure N S R ( J ) is defined to be J outside Y × [1 − η, η ] and is ( φ R × id) ∗ J on Y × [1 − η, η ]. Then N S ( J ) = J and N S ( J ) gives almostcomplex structures on the completions b X , c W ′ and Y × R + , which we will refer as the fully stretched almostcomplex structure.We will consider the degeneration of curves solving the Floer equation with one positive cylindrical endasymptotic to a non-constant Hamiltonian orbit of X H . Since either the orbit is simple or J depends onthe S coordinate near non-simple orbits, the topmost curve in the SFT building, i.e. the curve in b X , hasthe somewhere injectivity property. In particular, we can find regular J t on b X such that all relevant modulispaces, i.e. those with point constraint from q (which is in b X ), or with negative cylindrical ends asymptoticto non-constant Hamiltonian orbits of X H , possibly with negative punctures asymptotic to Reeb orbits of Y and multiple cascades levels, are cut out transversely. We say a almost complex structure on W is genericiff the fully stretched almost complex structure N S ( J ) is regular on b X . The set of generic almost complex structures form an open dense subset in the set of compatible almost complex structures that are cylindricalconvex and S , r independent on Y × [1 − η, η ] r .For the compactification of curves in the topmost SFT level, in addition to the usual SFT building in thesymplectization ∂ + X × R + = Y × R + stacked from below [4], we also need to include Hamiltonian-Floerbreakings near the cylindrical ends. In our context, since we use autonomous Hamiltonians and cascades,we need to include curves with multiple cascades levels and their degeneration, e.g. l i = 0 , ∞ in the cascadesfor some horizontal level i . A generic configuration is described in the top-right of the figure below, but wecould also have more cascades levels with the connecting Morse trajectories degenerate to 0 length or brokenMorse trajectories. qu u qu u qu ∞ u ∞ b XY × R + c W ′ Figure 2.
Neck-stretchingIn the figure, we use (cid:13) to indicate the puncture that is asymptotic to a Reeb orbit. We call u is thetopmost level of the cascades and all curves in b X in the fully stretched picture curves in the topmost (SFTbuilding) level. We call u ∞ the topmost cascades level in the topmost level. u is the bottom cascades leveland u ∞ is the bottom cascades level in the topmost SFT level.The benefit of neck-stretching is two-fold. (1) After neck-stretching along Y inside the hypothetical exactfilling W of ( RP n − , α f ), the virtual dimension of the topmost level can be computed using only theirasymptotic orbits. This is because c ( ξ std ) is torsion. While we can not do this in the filling, since c ( W ) isnot necessarily zero in H ( W ; Q ), in particular, we need to keep track of the relative homology class of thecurve. (2) In the topmost level, γ , γ are in different homology classes, which may not be the case in thefilling W . The price we pay is that we need to analyze more configurations. This is because there are only finitely many moduli spaces that can have positive energy. RP n − , ξ std ) IS NOT EXACTLY FILLABLE FOR n = 2 k In the fully stretched case, one particularly important moduli space is the following 1-level cascades M ∞ (ˇ γ , q ) := { u : C → b X | ∂ s u + J t ( ∂ t u − X H ) = 0 , lim s →∞ u ( s,
0) = ˇ γ , u (0) = q } / R , (2.5)which is closely related to (2.4), as we shall see in Proposition 2.12. In the following, we derive an actionconstraint. Let ˜ ω denote the smooth 2-form on b X , such that it is the symplectic form on X and ∂ + X × (1 , ∞ )and d( gα f ) on ∂ − X × (0 , g is a strictly increasing positive function on (0 ,
1) such that lim x → g ( x ) =(1 − δ ) η and lim x → g = (1 − δ ) for η <
1. It is clear that we can find such g , so that ˜ ω is smooth and exacton b X with a smooth primitive ˜ α which is rα f on X ∪ ∂ + X × (1 , ∞ ) and gα f on ∂X − × (0 , J t iscylindrical convex on ∂ − X × (0 , J t is also compatible with d( gα f ) on ∂ − X × (0 , u solvingthe Floer equation in (2.5) but possibly with negative punctures asymptotic to Reeb orbits on ∂X − , we have˜ ω ( ∂ s u, ∂ t u − X H ) ≥
0. We use g X H to denote the Hamiltonian vector field of H using e ω . Since H = 0 below ∂ − X , we have X H = g X H . Then by Stokes’ theorem, the integration of ˜ ω ( ∂ s u, ∂ t u − X H ) = ˜ ω ( ∂ s u, ∂ t u − g X H )implies 0 ≤ Z C \{ z ,...,z | Γ |} ˜ ω ( ∂ s u, ∂ t u − g X H )d s ∧ d t = Z C \{ z ,...,z | Γ |} d( u ∗ ˜ α ) − Z C ∂ s u ∗ H d s ∧ d t = 2 r − X γ ∈ Γ (1 − δ ) η Z γ α f − h ( r )where r is the unique value such that h ′ ( r ) = 2 and Γ is set of negative asymptotic orbits of u viewed as theReeb orbits of α f . The first equality follows from that u ∗ H is zero near punctures z i . Let η →
1, we have2 r − h ( r ) | {z } negative symplecticaction of γ − X γ ∈ Γ (1 − δ ) Z γ α f | {z } contact action of Γ ≥ , (2.6)When the width of H converges to zero, the unique value r such that h ′ ( r ) = 2 converges to 1. In themeantime, h ( r ) converges to 0. Therefore if we choose H to have arbitrarily small width and δ arbitrarilysmall, we have 2 r − h ( r ) → X γ ∈ Γ Z γ α f < ǫ ′ = Z γ α f + ǫ ′ , for ǫ ′ > C ( γ ) = R γ α f denote the contact action. In general, for a curve u in b X ,possibly with multiple cascades levels, with topmost positive cylindrical end asymptotic to γ + and bottomnegative cylindrical end asymptotic to γ − and a collection of negative punctures asymptotic to Γ, in additionto the usual symplectic action relation A ( γ + ) < A ( γ − ), we also have C ( γ + ) − C ( γ − ) ≥ X γ ∈ Γ C ( γ ) , (2.7) A priori, the negative puncture of u is asymptotic to Reeb orbits of ( RP n − , (1 − δ ) α f ) = Y = ∂ − X . That is why we have(1 − δ ) in the expression when we view the Reeb orbits as in ( RP n − , α f ). for suitable choice of H and δ by the same argument as above. Of course, the choice of H and δ dependson γ + , γ − . In the case of H ∈ H a , there are finitely many families of orbits. In the following we fix H and δ such that (2.7) holds for any γ + , γ − in our setup of symplectic cohomology and neck-stretching. We referto the C ( γ + ) − C ( γ − ) − P γ ∈ Γ C ( γ ) as the contact energy, which is non-negative.In the following, we first state a key property for the double cover ( S n − , ξ std ), which will supply us withthe holomorphic curve we need for ( RP n − , ξ std ) for Proposition 2.12. The following result follows from atailored proof of [28, Theorem A]. Proposition 2.11.
Let ( S n − , α ) be the standard contact sphere with a non-degenerate ellipsoid contactform that is close to a round sphere with n ≥ . Then for any small enough positive number δ . Let X ′ denote the (trivial) symplectic cobordism from ( S n − , (1 − δ ) α ) to ( S n − , α ) . Let q be a interior point in X ′ , let H be an admissible Hamiltonian on the completion c X ′ in the sense of last subsection, in particular, H is zero below ( S n − , α ) and is linear on the positive end of c X ′ . Let γ be the Reeb orbit with minimalperiod of α , we define M ∞ (ˇ γ, q ) := { u : C → c X ′ | ∂ s u + J t ( ∂ t u − X H ) = 0 , lim s →∞ u ( s,
0) = ˇ γ, u (0) = q } / R . (2.8) We say J t is nice, if every curve in M ∞ (ˇ γ, q ) is cut out transversely and there is no curve in form of thosein M ∞ (ˇ γ, q ) with one extra negative puncture asymptotic to a simple Reeb orbit. Then the set of nice J t is not empty, and for any nice J t , M ∞ (ˇ γ, q ) is compact and the algebraic count is after we choose anappropriate orientation of the determinant line bundle associated to γ .Proof. Assume J t is nice but the moduli space is not compact, then we have a SFT building breaking.First of all, there are no multiple cascades level in the topmost SFT level, i.e. no configuration in the fullystretched case of Figure 2. This is because γ already has the maximal symplectic action. If there was amulti-level cascades, the negative cylindrical end must be asymptotic to a non-constant Hamiltonian orbitswith larger symplectic action, which is impossible. Then we only need to rule out the case of 1-level cascadeswith negative punctures for the topmost SFT level. By action reasons explained in (2.7), there is at mostone negative puncture asymptotic to a simple Reeb orbit, but such configuration is ruled out since J t is nice.Next we will show the set of nice J t is not empty, in fact, a generic J t is nice. Since (1 − δ ) α is a non-degenerate ellipsoid, the minimal Conley-Zehnder index is n + 1. Then the virtual dimension of the topmostcurve, i.e. a curve in M ∞ (ˇ γ, q ) with possible negative punctures asymptotic to Γ − , is − P γ − ∈ Γ − ( µ CZ ( γ − ) + n − ≤ − (2 n −
2) as long as Γ − = ∅ , where Γ − is the set of negative asymptotic Reeb orbits of the topmostcurve. Since the transversality of the topmost curve is guaranteed by the genericity of J t , there is no suchSFT building.To prove the algebraic count is 1, we consider the filling of ( S n − , (1 − δ ) α ) by the standard ball, whichunion with the cobordism X is a filling D of ( S n − , α ), i.e. the standard ball. We can use H and aMorse function g with unique minimum at q to define symplectic cohomology of D . Since SH ∗ ( D ) = 0with Z coefficient, and there is only one generator with degree −
1, that is exactly ˇ γ . Therefore we musthave d + , (ˇ γ ) = ± q , and the coefficient can be fixed to 1 after we choose an appropriate orientation of the A priori, H has a finite slope, hence only defines a filtered symplectic cohomology. However we can modify H outside alarge r to be a small perturbation of the quadratic Hamiltonian, which would define the full symplectic cohomology. Since weare only interested in the moduli space asymptotic to an interior point and γ . The integrated maximal principle implies thatany change outside a large region does not affect our curve. RP n − , ξ std ) IS NOT EXACTLY FILLABLE FOR n = 2 k determined line of γ . Note that d + , (ˇ γ ) = q implies that { u : C → b D | ∂ s u + J t ( ∂ t u − X H ) = 0 , lim s →∞ u ( s,
0) = ˇ γ, u (0) = q } / R = 1 , (2.9)for any regular admissible J t . Then we can apply neck-stretching along ( S n − , (1 − δ ) α ). If the fullystretched almost complex structure is nice, then we have the moduli space (2.9) is contained completelyoutside ( S n − , (1 − δ ) α ) and regular for sufficiently stretched almost complex structure, since there can notbe any breaking. And in the fully stretched case, it is identified with (2.8). Hence the claim follows. (cid:3) In the following, we use h d + , ( a ) , b i to denote the coefficient of b in d + , ( a ). Since only bottom cascadeslevel can have the point constraint u (0) = q , which makes the bottom cascades level has a relative low virtualdimension, we will focus on analyzing the bottom cascades level in the following proposition. Proposition 2.12.
For sufficiently stretched generic almost complex structure J t , we have the algebraiccount of M (ˇ γ , q ) is and h d + , (ˇ γ ) , q i = 2 .Proof. The proof is divided into three parts.
Step 1:
For sufficiently stretched generic almost complex structure J t , we have M (ˇ γ , q ) = M ˇ γ ,q .We need to rule out multiple level cascades in order to prove the equality. Suppose we have a multiplelevel cascade, since each curve increases the symplectic action, then the negative cylindrical ends of thetopmost cascades level is asymptotic to an orbit in γ i for some i . If we apply neck-stretching to Y = ∂ − X ,since γ and γ i are in different homology classes, we must have an extra negative puncture in the limit ofneck-stretching. Then by the action reason (2.7), we must have i = 0 for sufficiently stretched J t . Thereforethe bottom level of the cascades is a map u : C → c W with ∂ s u + J t ( ∂ t u − X H ) = 0 and lim s →∞ u ( s, · ) ∈ γ and u (0) = q . Then in the full neck-stretching, by action and homology class reason, we end up a map withan extra negative puncture asymptotic to γ . Since c ( ξ std ) is torsion and the trivializations from the obviousdisk in O ( −
2) are compatible by Remark 2.7, the virtual dimension of such space (the positive cylindricalend has no orbit point constraint) is µ CZ ( γ ) − n − ( µ CZ ( γ ) + n −
3) = µ CZ (ˆ γ ) − n − − ( µ CZ ( γ ) + n −
3) = 3 − n < . Therefore for generic and sufficient stretched J t , there is no such configuration hence no multi-level cascades. Step 2:
For sufficiently stretched generic almost complex structure J t , we have M ∞ (ˇ γ , q ) is identifiedwith M (ˇ γ , q ) and both of them are compact.We first argue for generic fully stretched J t , M ∞ (ˇ γ , q ) is compact. By the same argument in step 1,there is no multi-level cascades like Figure 2 in the compactification of M ∞ (ˇ γ , q ). Then we need to rule outthe case of 1-level cascades with negative punctures. The virtual dimension of the moduli space of curvessolving (2.5) with negative punctures asymptotic asymptotic to Reeb orbits in Γ is the following µ CZ (ˇ γ ) − n − | {z } virtual dimension of (2.5) − X γ ∈ Γ ( µ CZ ( γ ) + n −
3) = ( µ CZ (ˇ γ ) + n − − X γ ∈ Γ ( µ CZ ( γ ) + n −
3) + 2 − n. (2.10)We have to make sure the Conley-Zehnder index are computed using compatible trivializations. By actionreason explained above and homology class of the Reeb orbits, we know the only SFT building breakingconfigurations for the compactification of M ∞ (ˇ γ , q ) that we can have contain either two negative puncturesboth asymptotic to γ or one negative puncture asymptotic to γ . The trivializations from the obvious diskin O ( −
2) are compatible, hence the virtual dimensions are well-defined and they are 4 − n <
0, 2 − n < J t . That is M ∞ (ˇ γ , q ) is compact for generic J t . By the similar argument in Proposition 2.11 and the dimension computation above, we know thatfor sufficiently stretched generic J t , M (ˇ γ , q ) is contained outside ∂X − and is identified with M ∞ (ˇ γ , q ). Step 3:
For generic almost complex structure J t , we have M ∞ (ˇ γ , q ) = 2.Let γ be the lift of γ in ( S n − , p ∗ α f ). Let ˇ γ ± , q ± denote the two lifts of ˇ γ and q in S n − . Then it isclear that we have a map P : M := [ ♦ , ♥∈{±} M ∞ (ˇ γ ♦ , q ♥ ) → M ∞ (ˇ γ , q ) , induced by the projection p : S n − → RP n − . Since C is simply connected, every curve in M ∞ (ˇ γ , q )has two lifts in M . Therefore P is a two-to-one surjective map. We know that p ∗ α f is a non-degenerateellipsoid close to a round sphere, to apply Proposition 2.11, we need to show that p ∗ J t is nice. First weverify every curve in M is cut out transversely. It is clear that a non-zero vector in the kernel of thelinearized perturbed Cauchy-Riemann operator D for M will be pushed down by p ∗ (note that p is a localdiffeomorphism) to a non-zero vector in the kernel of the linearized perturbed Cauchy-Riemann operator D for M ∞ (ˇ γ , q ). Therefore dim ker D ≤ dim ker D . Since J t is generic, we have dim ker D = 1 generated bythe R translation. Hence dim ker D = 1, since ker D always has the vector generated by the R translation.Since both the expected dimension of M and M ∞ (ˇ γ , q ) are zero, we have dim coker D = dim coker D = 0,i.e. M is cut out transversely. To prove p ∗ J t is nice, we still need to show that there is no curve with oneextra negative puncture asymptotic to a simple Reeb orbit. Any such curve can be pushed to RP n − via p to a curve with a negative puncture asymptotic to γ i . However, such configuration is ruled out in theprevious step for generic J t .As a consequence, by Proposition 2.11, we know that each of the four components of M has an algebraiccount of 1 with an appropriate orientation of the determinant line bundle over γ . This choice of orientationis consistent for all four components of M as ˇ γ ± , q ± are connected to each other respectively in the spaceof (orbit) point constraints. We can push the orientation of the determinant bundle of γ to an orientationof the determinant bundle of γ because γ is a good orbit. Using this orientation structure for M ∞ (ˇ γ , q ),we know that P preserves orientations and M = 4. Therefore we have M ∞ (ˇ γ , q ) = 2. This finishes theproof of the proposition, since h d + , (ˇ γ ) , q i = M ˇ γ ,q = M (ˇ γ , q ) = M ∞ (ˇ γ , q ). (cid:3) Remark 2.13.
Here we use n ≥ to rule out the other potential configuration from neck-stretching in step2. However this is just a convenient argument and it is not the reason our proof breaks down when n = 2 .In fact, if we use a pure symplectic field theory setup, then the curve is necessarily a double cover of a trivialcylinder, that lives over the critical point q of f . Then we choose q such that π ( q ) = q , then there is nosuch configuration. Proposition 2.14.
When n ≥ , for a generic and sufficiently stretched almost complex structure, we have h d + , ˇ γ , q i = 0 .Proof. We first argue that for generic and sufficiently stretched almost complex structure, there is no contri-bution from multiple level cascades. If there is a multiple level cascades, then by symplectic action reason,the topmost cascades’ negative end must be asymptotic to γ . As a consequence, the bottom level of thecascades must have positive cylindrical end asymptotic to γ . However such configuration was ruled out inthe step 1 of Proposition 2.12.Next we argue that it is impossible to have a single level cascades contributing to h d + , ˇ γ , q i . Since γ isnot contractible in RP n − , we know that in the fully stretched configuration, we must have breaking intoholomorphic buildings. Since q is in X and by contact action reasons, the topmost curve in the SFT building RP n − , ξ std ) IS NOT EXACTLY FILLABLE FOR n = 2 k must be a curve u : C \{ z } → b X such that ∂ s u + J t ( ∂ t u − X H ) = 0 , lim s →∞ u ( s,
0) = ˇ γ , u (0) = q, with z is a negative puncture, where u is asymptotic to γ or γ . Since the trivializations from the disks in O ( −
2) are compatible for our moduli space, we have the moduli space of the above curve has a well-definedvirtual dimension 4 − n for γ puncture and 2 − n for γ puncture. Then for n ≥
3, we can assume theconfiguration is empty. (cid:3)
Proposition 2.15. If W is an exact filling of ( RP n − , ξ std ) for n ≥ , then SH ∗ , ≤ ǫ + ( W ; R ) → H ∗ +1 ( W ; R ) is surjective.Proof. In view of Proposition 2.10, it is sufficient to prove that there is a class 1 + A ∈ ⊕ i ≥ H i ( W ; R ) for A ∈ ⊕ i> H i ( W ; R ) is mapped to zero in SH ∗ , ≤ ǫ ( W ; R ). Since we have a Z grading, it is sufficientto show that the composition SH ∗ , ≤ ǫ + ( W ; R ) → H ∗ +1 ( W ; R ) projection −→ H ( W ; R ) = R is nonzero by thetautological long exact sequence (2.3).We consider the generator ˇ γ , it is not necessarily a closed class in the positive cochain ( C + , d + ). However,we claim that d + (ˇ γ ) can only have nonzero components in ˆ γ for a sufficiently stretched J t . Again by contactaction and homology reason, the only possible configuration after the neck-stretching is with negative endasymptotic to either ˇ γ or ˆ γ and one negative puncture asymptotic to γ . Since µ CZ (ˇ γ ) and µ CZ (ˇ γ ) has thesame parity, then we have the only contribution is to ˆ γ . . By Proposition 2.9, we have that d + (ˇ γ ) = a ˆ γ for a = 0. If we write d + (ˇ γ ) = k ˆ γ , then ˇ γ − ka ˇ γ is closed in the positive cochain complex. Then byProposition 2.12 and Proposition 2.14, we have SH ∗ , ≤ ǫ ( W ; R ) → H ∗ +1 ( W ; R ) projection −→ H ( W ; R ) = R isnonzero. (cid:3) Remark 2.16.
The reason that our proof does not work for n = 2 is because Proposition 2.14 does not holdfor n = 2 . Indeed, for the fully stretched almost complex structure, the algebraic count of the top curve is .Hence the contribution M ˇ γ ,q is reduced to the augmentation to γ . Then we can discuss the following twocases,(1) When W is the exact filling T ∗ S , then the augmentation is . One can see it from the completionof T ∗ S into S × S . Moreover, one can show that d + (ˇ γ ) = 2ˆ γ by the neck-stretching argumentand augmentation. Then using d + (ˆ γ ) = 2ˇ γ (c.f. discussion before Proposition 2.9), one sees that is not killed at least in SH ≤ ǫ ( W ; R ) . The full computation in this spirit was carried out in [12] .(2) When W is the strong filling O ( − , then the augmentation is t − , where t is the formal variable ofdegree to keep track of the intersection with divisor CP in the Novikov field. Then d + (ˇ γ ) = t − ˆ γ .As a consequence ˇ γ − t − ˆ γ is closed in the positive symplectic cohomology and is mapped to − t − .Then SH ∗ ( W ; Λ) = 0 , this coincides with the result in [25] . Remark 2.17.
Ritter [25] showed that for n ≥ , SH ∗ ( O ( − ω ] / ( ω n − − t ) , where t is the formalvariable in the Novikov field Λ and ω is the generator of H ( CP n − ; R ) . On the other hand, the quantumcohomology QH ∗ ( O ( − ω ] / ( ω n − tω ) . Therefore the positive symplectic cohomology is the quotient It indeed contributes to the differential in any case there are rigid holomorphic plane bounded by γ in W , see Remark2.16 The t in [25] is different from the t in Remark 2.16, in the sense that t in this remark is the generator of H ( CP n − ), whichintersects CP n − n − QH ∗ ( O ( − /SH ∗ ( O ( − , which can be viewed as generated by ω n − − tω and ω n − − t . In the Morse-Bott spectral sequence, the former is represented by multiples of ˇ γ and the latter is represented by multiplesof ˇ γ . Moreover, ω n − − t projected to H ( W ; Λ) is indeed a unit in Λ . However, ω n − − t is a zero divisorin the quantum cohomology, hence it does not lead to the vanishing of symplectic homology.Proof of Proposition 2.1. By Proposition 2.9 and 2.15, we have P dim H ∗ ( W ; R ) ≤ H n ( W ; R ) to be nonzero. For otherwise, if we have H k ( W ; R ) = R for 0 < k = n , then H k ( W ) contains a Z summand. Then from the long exact sequence,we know that H k ( W, RP n − ) also contains Z summand. Therefore, by Lefschetz duality and universalcoefficient theorem, we have H n − k ( W ) also has a Z summand, which contradicts that the total rank is ≤ (cid:3) Generalizations
In this section, we prove Theorem 1.3 using the same argument. The threshold is not optimal. The upshotis for n > k , the cohomology of any exact filling of ( S n − / Z k , ξ std ) will have a bounded free part, which willlead to a contradiction. We first note that ( S n − / Z k , ξ std ) has a strong filling O ( − k ), i.e. the total space ofthe degree − k line bundle over CP n − . Proposition 3.1.
Let W be an exact filling of ( S n − / Z k , ξ std ) for n > k , then we have P i ∈ N dim H i ( W ; R ) ≤ k and P i ∈ N dim H i +1 ( W ; R ) ≤ k − . Moreover, H n − i ( W ; R ) = H i ( W ; R ) for every < i < n .Proof. Similar to the proof of Theorem 1.1, we perturb the standard Boothby-Wang contact form to α f using a C -small perfect Morse function f on CP n − , such that the following holds.(1) Reeb orbits of period smaller than k + 1 are γ ji for 0 ≤ i ≤ n − , ≤ j ≤ k , where γ ji is the j -multiplecover of γ i and γ i projects to the i th critical point q i of f .(2) The period of γ j is 1 + ǫ j .(3) ǫ j < ǫ j +1 k , ǫ j ≪ d + . In view of these conditions, we can choosethe Morse function f to be the following, n − X i =0 ǫ i | z i | ǫ i ! , n − X i =0 | z i | ǫ i ! . Then the pull back of the contact form α f back to S n − is the one given by the ellipsoid P n − i =0 | z i | k (1+ ǫ i ) = 1.With suitable choice of ǫ i , we may assume it is a non-degenerate ellipsoid.Similar to the proof of Proposition 2.12, we separate the proof into several steps. Step 1:
For a generic and sufficiently stretched almost complex structure, M (ˇ γ k , q ) = M ˇ γ k ,q .Assume there are multi-level cascades, then the negative cylindrical end of the top cascades level mustbe asymptotic to γ ji for j < k by symplectic action. After fully stretching the almost complex structure,the top cascades level must develop a set of negative punctures asymptotic to Γ = { γ j i , . . . , γ j m i m } , then wehave P ms =1 j s + j = 0 mod k by homology reasons. Among all such configurations, the only cases withnon-negative contact energy are i = i = . . . = i m = 0 and P ms =1 j s + j = k . If the next cascades level isnot the bottom level, then by the same argument, the negative cylindrical end must be asymptotic to γ j ′ for j ′ < j . We can keep the argument going and conclude that for sufficiently stretched almost complexstructure, the bottom cascade level must have a positive cylindrical end asymptotic to γ s for s < k . Then in RP n − , ξ std ) IS NOT EXACTLY FILLABLE FOR n = 2 k the fully stretched picture, this bottom level must have negative punctures asymptotic to γ j , . . . γ j m with j + . . . + j m = s by the same argument as before. Note that the Conley-Zehnder index of γ ji using thebounding disk in O ( − k ) is µ CZ ( γ ji ) + n − i + 2 j − . Note the bottom curve has the point constraint from q . The virtual dimension of this configuration (thepositive cylindrical end has no orbit point constraint) is µ CZ ( γ s ) − n − m X i =1 ( µ CZ ( γ j i ) + n −
3) = 2 s − n + 1 − m X i =1 (2 j i −
2) = 2 m − n + 1 ≤ s − n + 1 < . Hence for generic and sufficiently stretched J t , we do not have any multi-level cascades contributing to h d + , (ˇ γ k ) , q i . Step 2:
For a generic and sufficiently stretched almost complex structure, we have that moduli space of1-level cascades M (ˇ γ k , q ) is identified with the fully stretched moduli space of 1-level cascades M ∞ (ˇ γ k , q )and both are compact.Similar to the step 2 of Proposition 2.12, it is enough to prove the compactness of M ∞ (ˇ γ k , q ). The multi-level cascades are ruled out by step 1. We only need to rule out the case with negative punctures. Againby action and homology reason, the potential breakings are those with negative punctures γ k , . . . , γ k j for P k i = k . But the expected dimension of this moduli space is µ CZ ( γ k ) − n − − j X s =1 ( µ CZ ( γ k s ) + n −
3) = 2 k − n − j X s =1 (2 k s −
2) = 2 j − n < . Hence such moduli space is empty for generic J t . Step 3:
For a generic almost complex structure, M ∞ (ˇ γ k , q ) = k .Using Proposition 2.11 and the fact that γ k is a good Reeb orbit, this claim follows from the sameargument in step 3 of Proposition 2.12. So far we have proven that h d + , (ˇ γ k ) , q i = M ˇ γ k ,q = M (ˇ γ k , q ) = M ∞ (ˇ γ k , q ) = k for generic and sufficiently stretched almost complex structures. Step 4:
For a sufficiently stretched almost complex structure, we have d + (ˇ γ k ) = P k − i =1 b i ˆ γ i .For this we use the similar neck-stretching argument as in Proposition 2.15. By parity of generators,we only need to consider h d + (ˇ γ k ) , ˆ γ ij i . In the fully stretched picture, the curve in b X (could have multiplecascades levels) with maximal contact energy is the one with k − i negative punctures asymptotic to γ . Inthis case, we have C ( γ k ) − C ( γ ij ) − ( k − i ) C ( γ ) = k (1 + ǫ ) − i (1 + ǫ j ) − ( k − i )(1 + ǫ ) = i ( ǫ − ǫ j ) < , if j = 0 . As a consequence of (2.7), we have that d + (ˇ γ k ) = P k − i =1 b i ˆ γ i . Step 5:
For ǫ i sufficiently small and sufficiently stretched almost complex structure and i < k, j >
0, wehave d + (ˇ γ ij ) = k ˆ γ ij − + X m j. Therefore the claim follows from (2.7).
Step 6 : We claim that d + (ˇ γ k + k − X i =1 k − i X j =1 c i,j ˇ γ ij ) = 0 , (3.2)where c i,j ∈ R is defined recursively by a m,l .To see (3.2), first note that by (3.1), we have d + (ˇ γ k − k − X i =1 b i k ˇ γ i ) = k − X i =1 1 X j =0 d i,j ˆ γ ij , Then we have d + (ˇ γ k − k − X i =1 b i k ˇ γ i − k − X i =1 1 X j =0 d i,j k ˇ γ ij +1 ) = k − X i =1 2 X j =0 e i,j ˆ γ ij . Then we can keep applying the argument to obtain (3.2).
Step 7:
For a generic and sufficiently stretched almost complex structure, we have h d + , (ˇ γ ij ) , q i = 0 for i + j ≤ k, j ≥ h d + , (ˇ γ ij ) , q i . Assume otherwise, thetop cascades level’s negative cylindrical end is asymptotic to γ ml for m ≤ i . In the fully stretched picture,the curve with maximal contact energy are those with i − m (which could be zero, when i = m ) negativepunctures asymptotic to γ . The contact energy is given by C ( γ ij ) − C ( γ ml ) − ( i − m ) C ( γ ) = iǫ j − mǫ l − ( i − m ) ǫ < , if l > j. Therefore we must have l ≤ j . Of course, the top cascades level’s negative cylindrical end cannot beasymptotic to γ ij , as this would force the curve to be a trivial cylinder. We can keep the argument goingand conclude the bottom level of the cascades must have the positive cylindrical end asymptotic to γ ml with m ≤ i, l ≤ j and the equality does not holds simultaneously. In particular, m + l < i + j . We consider thisbottom cascades level in the fully stretched picture, since m < i + j ≤ k , we must have negative punctures by RP n − , ξ std ) IS NOT EXACTLY FILLABLE FOR n = 2 k homology reason. Then the maximal virtual dimension of the curve in b X is from the curve with m negativepunctures asymptotic to γ , which is µ CZ ( γ ml ) − n − m ( µ CZ ( γ ) + n −
3) = 2 l + 2 m + 1 − n < i + 2 j − n ≤ k − n < . In particular, there is no multi-level contribution. Next we will rule out the single level cascades. In thefully stretched situation, since j ≥
1, we have i < i + j ≤ k and the curve must have negative puncturesin the topmost SFT level by homology reason. The maximal virtual dimension of the topmost level is fromthe curve with i negative punctures asymptotic to γ . Hence the virtual dimension of the top level curve isat most µ CZ ( γ ij ) − n − − i ( µ CZ ( γ ) + n −
3) = 2 i + 2 j − n ≤ k − n < . Assembling the results above, we know that the closed cochain ˇ γ k + P k − i =1 P k − ij =1 c i,j ˇ γ ij is mapped to k in H ( W ; R ). As a result, we have SH ∗ , ≤ k + ǫ + ( W ; R ) → H ∗ +1 ( W ; R ) is surjective by Proposition 2.10. Step 8:
For ǫ i sufficiently small im( SH ∗ , ≤ k + ǫ + ( W ; R ) → H ∗ +1 ( W ; R )) has rank at most 2 k − SH ∗ , ≤ k + ǫ + ( W ; R ) can be assembled from k filtered symplec-tic cohomology with action window around 1 , . . . , k by iterating the tautological long exact sequences.More precisely, the cochain complex of the first k − γ i , ˆ γ i , . . . , ˇ γ in − , ˆ γ in − for 1 ≤ i ≤ k −
1, and the cohomology is H ∗ ( S n − / Z k ; R ) and generated by ˇ γ i , ˆ γ in − .The cochain complex of the last filtered symplectic cohomology is generated by ˇ γ k and ˆ γ k , which also gener-ate the cohomology. By the same argument in Proposition 2.9, ˆ γ k will be killed when we increase the actionthreshold and ˆ γ n − does not map to nontrivial class in H ∗ ( W ; R ) by S symmetry which is guaranteed bythe S -equivariant transversality. Therefore im( SH ∗ , ≤ k + ǫ + ( W ; R ) → H ∗ +1 ( W ; R )) has rank at most 2 k − k are from check orbits and at most k − (cid:3) Proposition 3.2.
Let p be an odd prime, then for any strong filling W of ( S n − / Z p , ξ std ) , we have H i ( W ) → H i ( S n − / Z p ) = Z p is surjective if < i < n and in p -adic representation, i is digit-wiseno larger than n , i.e. if we write n = P ∞ s =0 a s p s , i = P ∞ s =0 b s p s then b s ≤ a s for all s ≥ .Proof. The proof is similar to Proposition 2.2. We first compute the Chern classes of ξ std using the standardfilling O ( − p ), i.e. the total space of degree − p line bundle over CP n − . The total Chern class of the totalspace O ( − p ) is (1 + u ) n (1 − pu ), where u is generator of H ( O ( − p )) = H ( CP n − ). We write n in p -adic as P ks =0 a s p s . Then using the fact ( P x ∗ ) p = P x p ∗ mod p and (cid:0) ml (cid:1) = 0 mod p whenever 0 ≤ l ≤ m < p . Wehave the following(1 + u ) n (1 − pu ) = (1 + u ) n = k Y s =0 (1 + u p s ) a s = k Y s =0 (1 + a s X j =1 c s,j u p s j ) mod p, (3.3)for c s,j = (cid:0) a s j (cid:1) = 0 mod p . That is in (3.3), the monomials with non-constant coefficient are those u P ks =0 p s j s for j s ≤ a s , i.e. the degree is digit-wise smaller or equal to n in p -adic representation. In other words, the i th Chern class of the total space O ( − p ) mod p is nonzero iff in p -adic representation, i is digit-wise nolarger than n . By the Gysin sequence, we know H i ( O ( − p )) → H i ( S n − / Z p ) = Z p is the mod p map for0 < i < n and c i ( O ( − p )) | S n − / Z p = c i ( ξ std ). Therefore we have c i ( ξ std ) = 0 for any such i . Since for anystrong filling W , we have c i ( W ) | S n − / Z p = c i ( ξ std ), the claim follows. (cid:3) Proof of Theorem 1.3.
Let I be the set of i such that 0 < i < n and is digit-wise no larger than n in p -adicrepresentation. A basic observation is that if i ∈ I then n − i ∈ I . It is clear that | I | = P a s −
2. Then I := I ∩ (0 , n ) has at least P a s − P a s > p −
3, then n > p . Then we can applyProposition 3.1 to get that P ≤ j
3, we have | I | > P ≤ j If one uses the polyfold technique in [27] to achieve S -equivariant transversality. We canbring the rank of H ∗ ( W ; R ) down to k , since those hat orbits will not contribute to H ∗ ( W ; R ) as in the proofof Proposition 2.1. Observe that the check orbit will be mapped to even degree cohomology of W . Then we canimprove Theorem 1.3 by a factor to P a s > p + 3 . It is interesting to note that n ≥ k + 1 in Proposition 3.1is the threshold for C n / Z k to be a terminal singularity. By [23] , the terminality of a singularity is equivalentto that the link has a contact form with positive rational SFT degrees.The key observation in this paper is that SH ∗ + ( W ; R ) → H ∗ +1 ( ∂W ; R ) contains in the image for thehypothetical exact filling W , which bears a lot similarity with results in [28] . In view of Ritter results [25] , SH ∗ + ( O ( k ); Λ) → H ∗ +1 ( S n − / Z k ; Λ) is very likely to be isomorphic to the hypothetical SH ∗ + ( W ; Λ) → H ∗ +1 ( S n − / Z k ; Λ) for n ≥ k + 1 . The invariance phenomenon here has gone beyond those in [28] as wehave multiple augmentations. On the other hand, when n ≤ k , as we seen from the n = k = 2 case, themap from positive symplectic cohomology to the cohomology of boundary depends on the filling. For higher n ≤ k examples, although we do not know if there are more fillings, but there are algebraic augmentationswhich would change whether is in the image of the map from linearized non-equivariant contact homologyto the cohomology of the boundary. The n = k case is indeed the limit of our method, as our symplectic partdoes not differentiate exact fillings with Calabi-Yau fillings (see Remark 3.6) and C n / Z n indeed carries aCalabi-Yau filling with the right rank of cohomology. In case of strong fillings, the sequence (2.3) still holds after we replace H ∗ ( W ; R ) by the quantum coho-mology QH ∗ ( W ; Λ). As a group, QH ∗ ( W ; Λ) ≃ H ∗ ( W ; Λ), but the map QH ∗ ( W ; Λ) → SH ∗ ( W ; Λ) is aunital ring map if we use the deformed quantum ring structure on QH ∗ ( W ; Λ).In view of the proof of Proposition 2.10, to imply the vanishing of symplectic cohomology and then acontradiction, we need to show that 1 + A is never a zero divisor. Hence we have the following. Corollary 3.4. Let W be any (semi-positive [22] ) strong filling of the contact manifolds in Theorem 1.1 and1.3, then the quantum cohomology QH ∗ ( W ; Λ) has a zero divisor in the form of A for A ∈ ⊕ i> H i ( W ; Λ) . RP n − , ξ std ) IS NOT EXACTLY FILLABLE FOR n = 2 k Remark 3.5. Symplectic cohomology for general strong fillings requires virtual techniques to deal with spherebubbles in general. If one wishes to avoid the technical overhead, W should be restricted to the case ofsemi-positive strong fillings. Then the symplectic cohomology can be defined as usual. The filtered positivesymplectic cohomology SH ∗ , ≤ a + ( W ; Λ) can also be defined, based on the asymptotic behavior lemma [10,Lemma 2.3] instead of the action filtration . See [28, § for a setup for the Calabi-Yau case, the generalsemi-positive case is similar.Proof of Corollary 3.4. By the argument for Theorem 1.1 and 1.3, we have SH ∗ , ≤ a + ( W ; Λ) → QH ∗ +1 ( W ; Λ) projection −→ H ( W ; Λ) hits 1 for a suitable a for any (semi-positive) strong filling W . Note that for any x ∈ QH ∗ ( W ; Λ)the quantum product x ∪ · : QH ∗ ( W ; Λ) → QH ∗ + | x | ( W ; Λ) is Λ-linear map between finite dimensional Λ-spaces. Therefore x is either an invertible element or a zero divisor. If there is no zero divisor in the formof 1 + A for A ∈ ⊕ i> H i ( W ; Λ), we know that SH ∗ , ≤ a + ( W ; Λ) → QH ∗ +1 ( W ; Λ) hits an invertible element.Then the proof of Proposition 2.10 goes through and SH ∗ , ≤ a + ( W ; Λ) → QH ∗ +1 ( W ; Λ) is surjective. Thenwe can derive a topological contradiction as before. (cid:3) Remark 3.6. Note that the exactness is used to get H ∗ ( W ; R ) → SH ∗ ( W ; R ) is a unital ring map. Thenwe use the fact that A is a unit in H ∗ ( W ; R ) for A ∈ ⊕ i> H i ( W ; R ) , as A is a nilpotent element. Thisproperty also holds for symplectically aspherical filling or fillings with undeformed quantum cohomology, asshowed by Corollary 3.4. On the other hand, if the filling is Calabi-Yau, i.e. c ( W ) = 0 in H ( W ; Q ) , thenwe have a Z grading and A is necessarily . Although the multiplicative structure might be deformed, isalways a unit in QH ∗ ( W ; Λ) , hence we have surjectivity of SH ∗ , ≤ a + ( W ; Λ) → QH ∗ +1 ( W ; Λ) for a suitable a .In other words, our proof shows that contact manifolds in Theorem 1.1 and 1.3 do not have symplecticallyaspherical or Calabi-Yau fillings. Combining with Z and Z quotient singularities, we can prove Theorem 1.4. Proof of Theorem 1.4. In view of Theorem 1.1, we only need to prove the case for n = 2 k ≥ 4. In thiscase, we will use ( S n − / Z , ξ std ). Let W be an exact filling of ( S n − / Z , ξ std ). By Proposition 3.1, weknow that 1 ≤ P i ∈ N dim H i ( W ; R ) ≤ P i ∈ N dim H i +1 ( W ; R ) = 0. If H ( W ; R ) = 0, then we know H ( W ) , H ( W ) , H ( W ) , H n − ( W ) , H n − ( W ) , H n − ( W ) are all torsions. Moreover by Proposition 3.2,we have H ( W ) → H ( S n − / Z ) and H n − ( W ) → H n − ( S n − / Z ) are surjective. Then we arrive at acontradiction by the same argument in Theorem 1.1 and 1.3.In the case of dim H ( W ; R ) ≥ 1, we must have H n ( W ; R ) = 0 with n = 2 k even. Then we have thefollowing long exact sequence0 → H n ( W, S n − / Z ) → H n ( W ) → Z → H n +1 ( W, S n − / Z ) → H n +1 ( W ) → . Since H n − ( W ) , H n ( W ) , H n +1 ( W ) are all torsions, then Lefschetz duality and universal coefficient theoremimply that H n ( W, S n − / Z ) ≃ H n +1 ( W ) and H n +1 ( W, S n − / Z ) ≃ H n ( W ). The long exact sequencethen becomes 0 → H n +1 ( W ) → H n ( W ) → Z → H n ( W ) → H n +1 ( W ) → . which will contradict that they are all torsions. (cid:3) The filtered symplectic cohomology is actually filtered by the contact action, which roughly coincides with the negativesymplectic action when the filling is exact. We use H ( W ; Λ) here to stand for the Λ-space spanned by 1 ∈ H ( W ; Λ). Note that the degree 0 part QH ( W ; Λ) maybe different from H ( W ; Λ). Inspired by Remark 3.3, we end this paper with the following conjecture. Conjecture 3.7. If the isolated quotient singularity C n /G for finite nontrivial G ∈ U ( n ) is terminal, thenthe link of the singularity does not have symplectically aspherical fillings or Calabi-Yau fillings. Appendix A.In the following, we prove the property for Viterbo transfer map used in Proposition 2.9. We first recallan alternative way of defining the filtered positive symplectic cohomology SH ∗ , ≤ a + ( W ) following [10] . Let H denote the set of admissible Hamiltonians with slope that is not the period of a Reeb orbit on ∂W . For H ∈ H , we define can define C ≤ a ( H ) to be the subcomplex generated by critical point of g (i.e. those withzero symplectic action) and ˇ γ, ˆ γ with A H ( γ ) ≥ − a . We define C ≤ a + ( H ), or equivalently, C (0 ,a ] ( H ), to bethe quotient complex of C ≤ a ( H ) generated only by ˇ γ and ˆ γ . For H ≤ K in H , the continuation mapinduces maps f ≤ aHK : C ≤ a ( H ) → C ≤ a ( K ) and f ≤ a + ,HK C ≤ a + ( H ) → C ≤ a + ( K ), both satisfy the obvious functorialproperty . Then we define SH ≤ a ( W ) := lim −→ H ∈H H ∗ ( C ≤ a ( H )) , SH ≤ a ( W ) + := lim −→ H ∈H H ∗ ( C ≤ a + ( H )) Proposition A.1. The above definition is equivalent to the definition in § We consider a special class of Hamiltonians H ′ , such that H ∈ H ′ iff H on ∂W × (1 , w ) coincideswith a Hamiltonian in H a with width w and H ∈ H b for some b > a . Then H ′ is cofinal in H . Let H ∈ H ′ ,we define H a to the Hamiltonian equals to H on W ∪ ∂W × (1 , w ) and then extends linearly withslope a to c W . Note that C ≤ a + ( H ) = C ≤ a ( H a ), as R -module they are same since all other orbits of H havesymplectic action < − a . Then by the integrated maximal principle, we have the differentials for C ≤ a + ( H )and C ≤ a ( H a ) can be identified for suitable choice of almost complex structures. For w sufficiently small,we have C ≤ a ( H a ) = C ( H a ). Moreover for H ≤ K ∈ H ′ , we have H a ≤ K a ∈ H a . The functorial propertyof continuation maps implies that the compositions C ≤ a ( H a ) → C ≤ a ( K a ) → C ≤ a ( K ) and C ≤ a ( H a ) → C ≤ a ( H ) → C ≤ a ( K ) are homotopic to each other. Therefore we havelim −→ H ∈H H ∗ ( C ≤ a ( H )) = lim −→ H ∈H ′ H ∗ ( C ≤ a ( H )) = lim −→ H ∈H ′ H ∗ ( C ( H a )) = lim −→ H ∈H a H ∗ ( C ( H )) , where the last isomorphism is by Proposition 2.8. The proof for SH ≤ a + ( W ) is identical. (cid:3) Let V ⊂ W be an exact subdomain, then we can define H W ( V ) to be set of Hamiltonians that is 0on V and is linear on ∂W × (1 , ∞ ) r for r big with slope not a period of Reeb orbits on ∂W . Then suchHamiltonians can be used to compute SH ∗ ( V ) by the following. Lemma A.2 ([10, Lemma 5.1]) . For any positive real number a that is not a period of a Reeb orbit on ∂V ,we have SH ∗ , ≤ a ( V ) = lim −→ H ∈H W ( V ) H ∗ ( C ≤ a ( H )) , SH ∗ , ≤ a + ( V ) = lim −→ H ∈H W ( V ) H ∗ ( C (0 ,a ]) ( H )) . Strictly speaking, [9] uses the Hamiltonians that is C small Morse on W , the equivalence of these two models can beobtained by an argument similar to [28, Proposition 2.10]. Note that we have a sign difference in the convention of symplectic action compared to [9]. Our f HK is the f KH in [9]. We choose this convention, so that everything is parsed from left to right, i.e. f HK is from H to K . Similarly, M a,b is this paper counts differential from a to b . RP n − , ξ std ) IS NOT EXACTLY FILLABLE FOR n = 2 k Then the Viterbo transfer map for filtered symplectic cohomology is defined as follows [10, Definition 5.2], ι ≤ aW,V : SH ≤ a ( W ) → SH ≤ a ( V ) , ι ≤ aW,V := lim −→ H ≤ k,H ∈H ( W ) ,K ∈H W ( V ) f ≤ aHK ,ι ≤ a + ,W,V : SH ≤ a + ( W ) → SH ≤ a + ( V ) , ι ≤ a + ,W,V := lim −→ H ≤ k,H ∈H ( W ) ,K ∈H W ( V ) f ≤ a + ,HK , where f ≤ aHK : C ≤ a ( H ) → C ≤ a ( K ) , f ≤ a + ,HK : C (0 ,a ] ( H ) → C (0 ,a ] ( K ) are the continuation maps.The following proposition shows that the Viterbo transfer map is an isomorphism for trivial cobordism ifthere is no difference in the set of Reeb orbits that generate the filtered cochain complexes. Proposition A.3. Let W be an exact domain, such that the contact form on ∂W is Morse-Bott. For δ > we use W δ to denote W ∪ ∂W × (1 , δ ] . Assume ∂W has no Reeb orbit with period in [ a δ , a ] . Thenthe Viterbo transfer maps SH ∗ , ≤ a ( W δ ; R ) → SH ∗ , ≤ a ( W ; R ) is an isomorphism.Proof. Let H ∈ H a ( W δ ). Note that when we view H as a function on c W , i.e. using the r -coordinatefrom W , we have the slope of H is a δ . We consider H ′ ∈ H a δ ( W ), which is a shift of H . We claim theViterbo transfer map SH ∗ , ≤ a ( W δ ) → SH ∗ , ≤ a ( W ) can be computed by the continuation map f HH ′ . Wefirst take H ′′ ∈ H a ( W ) such that H ′′ ≥ H ′ , since there is no Reeb orbits of ∂W with period in [ a δ , a ]. Thecombination of arguments in Proposition 2.8 and A.1 yield that f H ′ H ′′ is an quasi-isomorphism. Hence it issufficient to prove f HH ′′ computes the Viterbo map. By Proposition A.1, the Viterbo transfer map can becompute by lim −→ H ≤ K,K ∈H W δ ( W ) f ≤ aHK . By the argument in [10, Lemma 5.1], we can find a cofinal family of functions K ∈ H W ( W ), such that allgenerators with symplectic action > − a are contained in an arbitrarily small neighborhood of W where K behave like a function in H b ( W ) form some b ≫ a . We use K b to denote the truncation as in the proofof Proposition A.1. We can choose the cofinal family has the property that K b ≥ K , see the figure below.The integrated maximal principle implies that f ≤ aKK b is a quasi-isomorphism. Since f ≤ aHK b = f ≤ aKK b ◦ f ≤ aHK , theViterbo transfer map can be computed by lim −→ H ≤ K b ,b ≫ aK b ∈H b ( W ) f ≤ aHK b . HH ′ H ′′ KK b W W δ Again by the argument in Proposition A.1, we have f ≤ aH ′′ K b is an quasi-isomorphism for K b ∈ H b ( W ) ≥ H ′′ with b ≫ a . Therefore the Viterbo transfer is computed by f HH ′′ , hence also f HH ′ . Then we can use theshift trick in Proposition 2.8 to show f HH ′ is a quasi-isomorphism, which concludes the proof. (cid:3) As an application of the above proposition, we have the following. Proposition A.4. Assume ∂W has no Reeb orbit with period in [ a δ , a ] , then SH ∗ , ≤ a + ( W δ ) → SH ∗ , ≤ a + ( W ) is an isomorphism. 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