A landscape of contact manifolds via rational SFT
AA LANDSCAPE OF CONTACT MANIFOLDS VIA RATIONAL SFT
AGUSTIN MORENO, ZHENGYI ZHOU
Abstract.
We define a hierarchy functor from the exact symplectic cobordism category to a totally orderedset from a BL ∞ (Bi-Lie) formalism of the rational symplectic field theory (RSFT). The hierarchy functorconsists of three levels of structures, namely algebraic planar torsion , order of semi-dilation and planarity ,all taking values in N ∪ {∞} , where algebraic planar torsion can be understood as the analogue of Latschev-Wendl’s algebraic torsion [48] in the context of RSFT. The hierarchy functor is well-defined through a partialconstruction of RSFT and is within the scope of established virtual techniques. We develop computation toolsfor those functors and prove all three of them are surjective. In particular, the planarity functor is surjectivein all dimension ≥
3. Then we use the hierarchy functor to study the existence of exact cobordisms. Wediscuss examples including iterated planar open books, spinal open books, Bourgeois contact structures, affinevarieties with uniruled compactification and links of singularities.
Contents
1. Introduction 12. L ∞ algebras and BL ∞ algebras 73. Rational symplectic field theory 184. Semi-dilations 445. Lower bounds for planarity 476. Upper bounds for planarity 517. Examples and applications 55References 741. Introduction
One central subject in symplectic and contact topology is the study of symplectic cobordisms. Unlikethe usual cobordism relation in differential topology, a symplectic cobordism is asymmetric; the collectionof such cobordisms endows the collection of contact manifolds with a structure similar to a partial order.The fundamental dichotomy between overtwisted contact structures and tight contact structures discoveredby Eliashberg in dimension 3 [29] and Borman-Eliashberg-Murphy in higher dimensions [6] is reflected bythe fact that overtwisted contact structures behave like minimal elements. Namely, there is always an exactcobordism from an overtwisted contact manifold to any other contact manifold in dimension 3 [32] andthe same holds for higher dimensions when the obvious topological obstructions vanish [30]. To explorethe realm of the more mysterious class of tight contact structures, the hierarchy imposed by the existenceof symplectic cobordisms is a useful guiding principle, as the complexity of contact topology should notdecrease in a cobordism. In dimension 3, a further hierarchy in the world of tight contact manifolds was a r X i v : . [ m a t h . S G ] D ec AGUSTIN MORENO, ZHENGYI ZHOU discovered by Giroux [41] and Wendl [73]. In higher dimensions, the notion of Giroux torsion was generalizedby Massot-Niederkr¨uger-Wendl [53].On the other hand, since contact manifolds and (exact) symplectic cobordisms form a natural category,which we will refer to as the (exact) symplectic cobordism category
Con , one natural approach to study
Con is by understanding functors from
Con to some algebraic category, a.k.a. a field theory.
Symplectic fieldtheory (SFT), as proposed by Eliashberg-Givental-Hofer in [28], is a very general framework for defining suchfunctors, and many invariants of contact manifolds and symplectic cobordisms can be defined via suitablecounts of punctured holomorphic curves which approach Reeb orbits at their punctures. The formidablealgebraic richness of the general theory, together with the serious technical difficulties arising in buildingits analytical foundations, conspire to make explicit computations a complicated matter. Therefore, ratherthan focusing on computing the full SFT invariant, one could focus on extracting simpler invariants fromthe general theory whose computation is in principle approachable via currently available techniques. Anexample of this philosophy is the notion of algebraic torsion introduced by Latschev-Wendl in [48], whichassociates to every contact manifold a number in N ∪ {∞} and can be viewed as the algebraic interpretationof the geometric concept of planar torsion defined by Wendl [73].In this paper, we follow the same methodology of Latschev-Wendl to study the structure of Con . Insteadof the full SFT, we use the rational SFT (RSFT), i.e. we only consider genus 0 curves, to construct a functorfrom
Con to a totally ordered set. Our main theorem is the following.
Theorem A.
There is a covariant monoidal functor H cx from Con to H , where H is the totally ordered set { APT < APT < . . . < ∞ APT (cid:124) (cid:123)(cid:122) (cid:125) P < SD < SD < . . . < ∞ SD (cid:124) (cid:123)(cid:122) (cid:125) P < P < . . . < ∞ P } . Here, functoriality of H cx means that H cx ( Y ) ≤ H cx ( Y (cid:48) ) whenever there exists an exact cobordism from Y to Y (cid:48) . The monoidal structure on Con is given by disjoint union and the monoidal structure on H is moreinvolved; it will be explained in § §
4. In particular, one can compute H cx of a disjoint union from itscomponents. The hierarchy functor H cx of contact complexity is assembled from three functors: algebraicplanar torsion APT, planarity
P and order of semi-dilation
SD, all of them taking values in N ∪ {∞} . Thereare overlaps between those invariants: when APT is finite, it is necessary to have P = 0; and if SD is finite,it is necessary to have P = 1. Therefore in the definition of the totally ordered set H , k APT stands for thecase where APT = k and P = 0, k SD stands for the case where SD = k and P = 1, and k P stands for the casewhere P = k . APT can be viewed as the analogue of algebraic torsion in the context of RSFT. In particular,finiteness of APT( Y ) implies that Y has no strong filling just like algebraic torsion. However, H cx goeswell beyond non-fillable contact manifolds, i.e. SD , P provide measurements for fillable contact manifolds.Roughly speaking, APT looks for rational curves without negative punctures and P looks for rational curveswith a point constraint in symplectizations. And SD is defined using the Q [ u ]-module structure on linearizedcontact homology introduced by Bourgeois-Oancea [12]. APT measures the obstruction to augmentationsof RSFT, while SD , P can be phrased in the linearized theory, hence require the existence of augmentations.To make SD , P independent of the augmentation, we need to define SD and P via traversing the set of allpossible augmentations of the RSFT.1.1.
Rational SFT.
The original algebraic formalism of SFT in [28] packaged the full SFT into a superWeyl algebra with a distinguished odd degree Hamiltonian H such that H (cid:63) H = 0. Cieliebak-Latschevreformulated the algebra into a BV ∞ algebra [22], which was used in the definition of algebraic torsion There is a (unique) morphism a → b iff a ≤ b , a, b ∈ H . LANDSCAPE OF CONTACT MANIFOLDS VIA RATIONAL SFT 3 [48]. The BV ∞ algebra structure was further refined to an IBL ∞ ( I nvolutive B i- L ie infinity ) algebra byCieliebak-Fukaya-Latschev [21], which roughly speaking, is precisely the boundary combinatorics for theSFT compactification [9]. For rational SFT, the original algebraic formalism was a Poisson algebra with adistinguished odd degree Hamiltonian h such that { h , h } = 0. Analogous reformulations of the algebraicstructure of RSFT can be found in Hutchings’ “ q -variable only RSFT” [45], and an L ∞ formalism of RSFTby Siegel [70]. In this paper, we introduce a notion of BL ∞ ( B i- L ie infinity ) algebra to describe RSFT,which precisely describes the boundary combinatorics for rational curves in the SFT compactification and isa specialization of the IBL ∞ formalism. By building functors from the category of BL ∞ algebras to totallyordered sets, we can build the hierarchy functor in Theorem A by a composition(1.1) H cx : Con
RSFT −→ BL ∞ (with additional structures up to homotopy) −→ H . On the other hand, the general holomorphic curve theory in manifolds with contact boundaries facesserious analytical challenges, which makes a complete construction of the first functor in (1.1) a difficulttask. To obtain a construction of SFT/RSFT, one needs to deploy more powerful virtual techniques, e.g.either polyfold approaches [33, 44] by Fish-Hofer and Hofer-Wysocki-Zehnder, implicit atlases and virtualfundamental cycles by Pardon [65, 66], or
Kuranishi approaches by Ishikawa [47]. However, for the purposeof defining H cx , it is sufficient to build RSFT partially. In particular, we do not need to discuss compositionsand homotopies for BL ∞ algebras as H is a totally ordered set, where there is no ambiguity for compositionsand homotopy equivalences. This greatly simplifies our demands for virtual machinery, as homotopies inSFT is a subtle subject. Moreover, the combinatorics for a BL ∞ algebra is “tree-like”, which is very similarto the combinatorics for contact homology. As a consequence, we can use Pardon’s construction of contacthomology [66] to provide all the analytic foundation of the functor H cx . In particular, Theorem A is well-posed without any hidden hypotheses on virtual machinery. Moreover, it is expected that any other virtualtechnique will suffice for Theorem A. We will also explain how to obtain another construction of H cx froma small part of the polyfold construction of SFT [33].In general, a full computation of RSFT and SFT is very difficult, as we need to understand many modulispaces. On the other hand, the hierarchy functor H cx extracts partial information from BL ∞ algebras, sothat only partial knowledge of the moduli spaces are needed. In particular, H cx is relatively computable.It is a nontrivial question whether H cx is independent of the choice of virtual technique. However, sinceevery virtual technique has the property that we can count a compactified moduli space geometrically if itis cut out transversely in the classical sense, the following theorem does not depend on the choice of virtualtechnique: Theorem B.
The functors above have the following properties.(1) If Y has planar k -torsion [73] , then APT( Y ) ≤ k .(2) If Y is overtwisted then APT( Y ) = 0 .(3) If Y has (higher dimensional) Giroux torsion [53] , then APT( Y ) ≤ .(4) If APT( Y ) < ∞ , then Y is not strongly fillable. If Y admits an exact filling then P( Y ) ≥ .(5) If Y is an iterated planar open book [2] where the initial page has k -punctures, then P( Y ) ≤ k .(6) If Y has an exact filling that is not k -uniruled [55] , then P( Y ) ≥ k + 1 .(7) APT , SD , P are all surjective. In particular, P is surjective in all odd dimension ≥ . The relation between APT and algebraic torsion AT [48] is not direct. In fact, they are both impliedby a stronger notion of torsion, which is implied by planar k -torsion [73], defined through an alternativerepresentation of the IBL ∞ algebra of the full SFT. There is a grid of torsions serving as obstructions to AGUSTIN MORENO, ZHENGYI ZHOU strong fillings, where algebraic planar torsion and algebraic torsion are two axes. The only common groundis that 0-algebraic planar torsion is equivalent to 0-algebraic torsion and algebraically overtwistedness [11],which is implied by overtwistedness [15, 77]. Moreover, there are 5-dimensional examples with underlyingsmooth manifold Y = S ∗ X × Σ g (where S ∗ X is the unit cotangent bundle of a hyperbolic surface X andΣ g is the orientable surface of genus g ≥ Y ) > Y ) = 1; see [58, Sec. 6.5]. This would provide concrete examples on which thetwo notions of torsion strictly differ.1.2. Applications.
Since H cx is a measurement of the complexity of contact topology, the main applicationof H cx is obstructing the existence of exact cobordisms. The following theorem answers a conjecture ofWendl [73] affirmatively, although the invariant we use is P instead of algebraic torsion (as opposed to theoriginal conjecture). Theorem C.
For any dimension ≥ , there exists an infinite sequence of contact manifolds Y , Y . . . , suchthat there is an exact cobordism from Y i to Y i +1 , but there is no exact cobordism from Y i +1 to Y i . The above result was obtained in dimension 3 in [48]. In fact, there are many examples of Y i , the simplestexample being the boundary of the product of n copies of a k -punctured sphere S k , as we will show in § ∂ ( S k ) n ) = k for n ≥
2. There are many more examples for Theorem C to hold; see e.g. Theorem Lbelow.Following the definition of P, it is easy to see that if Y admits a contact structure without Reeb orbits,then P( Y ) = ∞ . Therefore, as a corollary, we have the following. Corollary D. If P( Y ) < ∞ , then the Weinstein conjecture holds for Y . In other words, counterexamples to the Weinstein conjecture (if any) should be looked for in the highestcomplexity level ∞ P . In particular, the combination of (5) in Theorem B and Corollary D yields a proofof the Weinstein conjecture for iterated planar open books, which was previously obtained for dimension3 in [1] and higher dimensions in [2, 5]. In some sense, the proof of (5) of Theorem B endows the rulingholomorphic curve in [1, 2, 5] with a homological meaning, i.e. the ruling curve defines a map that is visibleon homology; in particular, such curve can not be eliminated by perturbing the contact form. On the otherhand, not every contact manifold with finite planarity is iterated planar: for example P( T , ξ std ) = 2 byCorollary 6.9, while ( T , ξ std ) is not supported by a planar open book by [31] (it is, however, supportedby a planar spinal open book [72, 52]). By functoriality, if there is an exact cobordism from Y to Y (cid:48) withP( Y (cid:48) ) < ∞ , then the Weinstein conjecture holds for Y .The study of planar open book in dimension 3 has a very long history, since they enjoy nice properties likeequivalence of weak fillability and Weinstein fillability [62, 72]. We refer readers to the introduction of [3]for a comprehensive summary on the subject. Obstructions to planar open book structures were obtainedin [31, 64]. In higher dimensions, obstructions to supporting an iterated planar open book were found in [5].By (5) of Theorem B, infinite planarity is an obstruction to an iterated planar structure. In particular, weanswer [4, Question 1.14] negatively by the following general result. Corollary E.
In all dimension ≥ , consider ( Y, J ) an almost contact manifold which has an exactly fillablecontact representative ( Y, ξ ) . Then there is a contact structure ξ (cid:48) in the homotopy class of J , such that ( Y, ξ (cid:48) ) is not iterated planar. In particular, since every simply connected almost contact 5-manifold is almost Weinstein fillable [39],there is a contact structure in each homotopy class of almost complex structures that is not iterated planarfor every simply connected 5-manifold.
LANDSCAPE OF CONTACT MANIFOLDS VIA RATIONAL SFT 5
Examples.
In addition to Theorem B, there are many situations where we can compute or estimateH cx . By (6) of Theorem B, it is natural to look at affine varieties with a uniruled projective compactification.One special case is affine varieties with a CP n compactification. Theorem F.
Let D be k generic hyperplanes in CP n for n ≥ , then we have the following.(1) P( ∂D c ) ≥ k + 1 − n for k > n + 1 .(2) P( ∂D c ) = k + 1 − n for n + 1 < k < n − and n odd.(3) P( ∂D c ) = 2 for k = n + 1 .(4) H cx ( ∂D c ) = 0 SD for k ≤ n . The condition on n being odd (also for Theorem H, I below) is not essential. We use it to obtain automaticclosedness of a chain in the computation of planarity for any augmentation. In Remark 7.20, we explain howone can drop this condition using polyfold techniques in [82]. On the other hand, the role of k < n − ismore mysterious. Although it is unlikely to be optimal, whether an upper bound is necessary is unclear. Onedifficulty of computing P and obtaining cobordism obstructions is that we need to carry out computationfor all hypothetical “fillings”, i.e. augmentations. Indeed, different choice of augmentation will affect thecomputation dramatically. For example, there exists affine varieties with a CP n compactification whosecontact boundary has infinite planarity, cf. Theorem 7.12. However, if we use the augmentation from theaffine variety, then the planarity is finite.On the other hand, D c k embeds exactly into D c k +1 , which follows from a general construction, as follows.Let L be a very ample line bundle over a smooth projective variety X . Then for any nonzero holomorphicsection s ∈ H ( L ), X \ s − (0) is an affine variety whose contact boundary is denoted by Y s . The projectivespace P H ( L ) should be stratified by the singularity type of s − (0) , with the top stratum correspondingto the case where s − (0) is smooth with multiplicity 1. We say that there is a morphism from stratum A to stratum B , if we can change s − (0) from A to B by an arbitrarily small perturbation of the section s ,i.e. A is contained in the closure of B . Moreover, one obtains an exact cobordism from Y s to Y s (cid:48) , where s (cid:48) is the perturbed section. Then we have a natural functor from the category of strata to Con . As a concreteexample, consider L = O (2) on CP , then the category of stratification is the graph A → A → A ,where A , A , A correspond to a double line, two generic lines and a smooth quadratic curve as the divisor,respectively. The corresponding affine varieties as exact domains are C , C × T ∗ S , T ∗ RP , which clearlyhave the exact embedding relations as claimed.In view of this, when we view one of the k hyperplanes in D k as having multiplicity 2 (a “double”hyperplane), we can get an exact cobordism from ∂D c k to ∂D c k +1 , by perturbing the double hyperplane totwo distinct hyperplanes. Then Theorem F asserts that a reversed exact cobordism can not be found if n ≤ k < n − and n is odd. Note that the natural inclusion D c k +1 ⊂ D c k is symplectic, hence we always havea strong cobordism from ∂D c k +1 to ∂D c k , which shows the essential difference between these two notions ofcobordisms and the obstruction from P is not topological. When k ≤ n , D c k is in fact subcritical and theycan be embedded exactly into each other regardless of k .As a concept closely related to Con , we introduce
Con ∗ as the under category of Con under ∅ , i.e. theobjects of Con ∗ are pairs of contact manifolds with exact fillings and morphisms are exact embeddings.Then SD , P can be defined on
Con ∗ using the augmentation from the given exact filling. Moreover, we It is quite a nontrivial task to make this stratification precise, as in general we do not have a classification of the possiblesingularities of the divisor. The functorial property of those two functors requires a full construction of RSFT, including compositions and homotopies.In particular, it makes more demands for virtual constructions than explained in this paper.
AGUSTIN MORENO, ZHENGYI ZHOU recall another functor U, called the order of uniruledness , which is defined to be the minimal k such thatan exact domain W is k -uniruled in the sense of McLean [55]. That U is a functor from Con ∗ → N + ∪ {∞} was proven in [55]. By (6) of Theorem B, P( Y ) is bounded below by U( W ) for an exact filling W of Y . Thefunctor U measures the complexity of exact domains and serves as an exact embedding obstruction. Aninteresting aspect of U is that the well-definedness and basic properties of U do not depend on any Floertheory. As a byproduct of the proof of Theorem F, we have following for any n ≥ Theorem G.
Let D k denote the divisor of k generic hyperplanes in CP n for n ≥ and D c k denote thecomplement affine variety. Then U( D c k ) = max { , k + 1 − n } . In particular, D c k +1 can not be embedded into D c k exactly for k ≥ n . Remark 1.1.
The same embedding question is studied independently by Ganatra and Siegel [37] , where moregeneral normal crossing divisors in CP n are studied. The planarity for exact domains mentioned above isequivalent to G (cid:104) p (cid:105) in [37] . The authors of [37] also consider holomorphic curves with local tangent constraintsto define functors G (cid:104) T m p (cid:105) on Con ∗ , which is briefly explained in § [55] . Suchfunctor can also serve as an embedding obstruction as U in Theorem G. It is an interesting question onwhether those geometric invariants are the same as the algebraic invariants (defined via RSFT in [37] ),which is the case for Theorem G. In view of (6) of Theorem B, one can also consider affine varieties with uniruled compactification, in par-ticular those affine varieties with Fano hypersurfaces as compactification. In general, we have the following.
Theorem H.
Let X be a smooth degree m hypersurface in CP n +1 for ≤ m ≤ n and D be k ≥ n generichyperplanes, i.e. D = ( H ∪ . . . ∪ H k ) ∩ X for H i is a hyperplane in CP n +1 in generic position with eachother and X , then P( ∂D c ) = k + m − n for n odd and k + m < n +12 . When m = n + 1, X is again uniruled and results with similar nature should hold. However, X is notuniruled by the degree 1 curves, while our proof uses somewhere injectivity of degree 1 curves to obtaintransversality in various places, hence is only applicable to m ≤ n . A more systematic way to study H cx isderiving formulas for RSFT of affine varieties with normal crossing divisor complement using log/relativeGromov-Witten invariants similar to the formula for symplectic cohomology in [27].The following results provide affine variety examples with nontrivial SD. Theorem I.
Assume D s is a smooth degree ≤ k < n +12 hypersurface in CP n for n ≥ , then P( ∂D c s ) = 1 and H cx ( ∂D c s ) ≤ ( k − SD . When n is odd, then same holds for ≤ k < n , and moreover we have H cx ( ∂D c s ) ≥ ( k − SD . Another rich class of contact manifolds comes from links of isolated singularities. They provide exampleswith every order of semi-dilation.
Theorem J.
Let LB ( k, n ) denote the contact link of the Brieskorn singularity x k + . . . + x kn = 0 . Then H cx ( LB ( k, n )) is(1) ( k − SD if k < n ;(2) ≥ ( k − SD if k = n and > P if k = n + 1 ;(3) ∞ P , if k > n + 1 . LANDSCAPE OF CONTACT MANIFOLDS VIA RATIONAL SFT 7
Another type of singularity is the quotient singularity, whose contact links are not exact fillable in manycases [79]. In fact, the symplectic aspect of the proof in [79] can be restated as a computation of H cx asfollows. Theorem K.
Let Y be the quotient ( S n − / Z k , ξ std ) by the diagonal action of e πik for n ≥ .(1) If n > k , we have H cx ( Y ) = 0 SD .(2) If n ≤ k , we have SD ≤ H cx ( Y ) ≤ ( n − SD . When n = k , we have H cx ( Y ) ≥ SD . The second case of the above theorem is another situation where the computation depends on the aug-mentation. Roughly speaking, H cx ( Y ) = 0 SD means that any exact filling of Y has vanishing symplecticcohomology. And if there is a (possibly strong) filling with vanishing symplectic cohomology, then the orderof semi-dilation using the induced augmentation from the filling is 0. The natural prequantization bun-dle filling provides augmentations such that the symplectic cohomology vanishes [67]. On the other hand,there are other augmentations with positive orders of semi-dilation. For example the exact filling T ∗ S of( RP , ξ std ) has order of semi-dilation 1, such phenomenon was also explained in [79, Remark 2.16]. Theorem L.
Let V be an exact domain and S k be the k -punctured sphere. Then(1) P( ∂ ( V × S k )) ≤ k .(2) If V is an affine variety that is not ( k − -uniruled, then P( ∂ ( V × S k )) = k .(3) H cx ( ∂ ( V × D )) = 0 SD . In particular, (2) in Theorem L provides many examples to Theorem C and (3) is a reformulation of thesymplectic step in [80] to obtain uniqueness results on fillings of ∂ ( V × D ). We also discuss Bourgeois contactstructures and cosphere bundles in § § Organization of the paper.
We introduce the concept of BL ∞ algebra in § §
3, we implement Pardon’s VFC [66] to defineAPT and P. We then discuss their properties and generalizations including the related formalism on thefull SFT. We recall in § Q [ u ] module structure on linearized contact homology following [12] to defineSD and finish the proof of Theorem A. We give a lower bound for P in § § § Acknowledgments.
We would like express our gratitude to Helmut Hofer, Mark McLean, John Pardon andChris Wendl for helpful conversations. A.M. acknowledges the support by the Swedish Research Councilunder grant no. 2016-06596, while the author was in residence at Institut Mittag-Leffler in Djursholm,Sweden. Z.Z. is supported by National Science Foundation under Grant No. DMS-1926686 and is pleasedto acknowledge the Institute for Advanced Study for its warm hospitality.2. L ∞ algebras and BL ∞ algebras In this section, we recall the basics of L ∞ algebras and introduce BL ∞ algebras, which serve as the un-derlying algebraic structures for rational symplectic field theory. The algebraic formalism here is essentiallythe q -variable only reformulation in [45] and the L ∞ algebra formalism on contact homology algebra in [70],but we make the compatibility of the algebraic structure on the contact homology algebra with the L ∞ structure more precise and define such an object as a BL ∞ algebra, which is a specialization of the IBL ∞ algebra in [21] and the homotopic version of bi-Lie algebras (with curvature). The algebraic relations in Assuming the filling is monotone, so that we can evaluate T = 1 in the Novikov coefficient. AGUSTIN MORENO, ZHENGYI ZHOU BL ∞ algebra are precisely the boundary combinatorics of the moduli spaces of rational curves in the SFTcompactification. We then introduce algebraic planar torsion and planarity at the algebraic level.2.1. L ∞ algebras. Throughout this section, we assume k is a field with characteristic 0. Let V be a Z -graded k -vector space. Then we have the Z -graded symmetric algebra SV := ⊕ k ≥ S k V and the non-unitalsymmetric algebra SV = ⊕ k ≥ S k V , where S k V = ⊗ k V /Sym k in the graded sense. In particular, we have ab = ( − | a || b | ba for homogeneous elements a, b in SV, SV . Therefore S k V is spanned by vectors of the form v . . . v k for v i ∈ V . However, to introduce the L ∞ algebra, we will view SV, SV as coalgebras by the following co-product operation: ∆( v . . . v k ) = k − (cid:88) i =1 (cid:88) σ ∈ Sh ( i,k − i ) ( − (cid:5) ( v σ (1) . . . v σ ( i ) ) ⊗ ( v σ ( i +1) . . . v σ ( k ) ) , where Sh ( i, k − i ) is the subset of permutations σ such that σ (1) < . . . < σ ( i ) and σ ( i + 1) < . . . < σ ( k ) and (cid:5) = (cid:88) ≤ i
We use sV to denote V [1] . An L ∞ algebra on V is a degree coderivation (cid:98) (cid:96) on SsV satisfying (cid:98) (cid:96) = 0 . Note that we have a well-defined degree − s : V → sV . The coderivation property of (cid:98) (cid:96) implies thatit is determined by maps (cid:96) k : S k sV → sV defined by the composition S k sV (cid:44) → SsV (cid:98) (cid:96) → SsV → sV , wherethe first map is the natural inclusion and the last map is the natural projection, satisfying the quadraticrelation(2.1) n (cid:88) k =1 (cid:88) σ ∈ Sh ( k,n − k ) ( − (cid:5) (cid:48) (cid:96) n − k +1 ( (cid:96) k ( sv σ (1) . . . sv σ ( k ) ) sv σ ( k +1) . . . sv σ ( n ) ) = 0 , where (cid:5) (cid:48) = (cid:88) ≤ i
Given φ i : S k i sV → sV (cid:48) , ≤ i ≤ n and m = (cid:80) ni =1 k i , we define φ . . . φ n : S m sV → S n sV (cid:48) by sending sv . . . sv m to π (cid:32)(cid:88) σ ( − (cid:5) (cid:48) k ! . . . k n ! ( φ ⊗ . . . ⊗ φ n )(( sv σ (1) . . . sv σ ( k ) ) ⊗ . . . ⊗ ( sv σ ( m − k n +1) . . . sv σ ( m ) )) (cid:33) . Here π is the natural map ⊗ k sV → S k sV . By the coalgebra property, if (cid:98) φ is an L ∞ morphism, we know that (cid:98) φ is determined by { φ k : S k sV → SsV (cid:98) φ → SsV (cid:48) → sV (cid:48) } k ≥ . More explicitly, (cid:98) φ is defined by the followingformula, (cid:98) φ ( sv . . . sv n ) = (cid:88) k ≥ i + ... + i k = n k ! ( φ i . . . φ i k )( sv . . . sv n ) . The relation (cid:98) φ ◦ (cid:98) (cid:96) = (cid:98) (cid:96) (cid:48) ◦ (cid:98) φ can be written as (cid:88) p + q = n +1 (cid:88) σ ∈ Sh ( q,n − q ) ( − (cid:5) (cid:48) φ p ( (cid:96) q ( sv σ (1) . . . sv σ ( q ) ) sv σ ( q +1) . . . sv σ ( n ) ) = (cid:88) k ≥ i + ... + i k = n k ! (cid:96) k ( φ i . . . φ i k )( sv . . . sv n ) . In particular, (cid:98) φ preserves the word length filtration. The composition of L ∞ homomorphism is the naivecomposition (cid:98) φ ◦ (cid:98) ψ , which is clearly a coalgebra chain map. Unwrapping the definition, we have( φ ◦ ψ ) n = (cid:88) k ≥ i + ... + i k = n k ! φ k (( ψ i . . . ψ i k )( sv . . . sv n )) . BL ∞ algebras. In this section, we define the BL ∞ (bi-Lie-infinity) algebra on a Z graded vector space V , which will govern the rational symplectic field theory. Let EV denote SSV . Given a linear operator p k,l : S k V → S l V for k ≥ , l ≥
0, we will define a map (cid:98) p k,l : S k SV → SV . To emphasize the differencesbetween products on two symmetric algebras, we use (cid:12) for the product on the outside symmetric product S and ∗ for the product on the inside symmetric product S when it can not be abbreviated. We will firstdescribe the definition using formulas and then introduce a graph description, which is very convenient todescribe BL ∞ algebras as well as various related structures and also governs all the signs and coefficients.Let w , . . . , w k ∈ SV , then (cid:98) p k,l is defined by the following properties.(1) (cid:98) p k,l | (cid:12) k V ⊂ S k SV is defined by p k,l .(2) If w i ∈ k , then (cid:98) p k,l ( w (cid:12) . . . (cid:12) w k ) = 0.(3) (cid:98) p k,l satisfies the Leibniz rule in each argument, i.e. we have (cid:98) p k,l ( w (cid:12) . . . (cid:12) w k ) = m (cid:88) j =1 ( − (cid:3) v . . . v j − (cid:98) p k,l ( w (cid:12) . . . (cid:12) v j (cid:12) . . . (cid:12) w k ) v j +1 . . . v m . Here w i = v . . . v m and (cid:3) = i − (cid:88) s =1 | w s | · j − (cid:88) s =1 | v s | + j − (cid:88) s =1 | v i || p k,l | + n (cid:88) s = i +1 | w s | · m (cid:88) s = j +1 | v s | . It is clear from the definition that (cid:98) p k,l is determined uniquely by the above three conditions. More explicitly, (cid:98) p k,l is defined by the following: w (cid:12) . . . (cid:12) w k (cid:55)→ (cid:88) ( i ,...,i k )1 ≤ i j ≤ n j ( − (cid:13) p k,l ( v i (cid:12) . . . (cid:12) v ki k ) ˇ w . . . ˇ w k , where w j = v j . . . v jn j , ˇ w j = v j . . . ˇ v ji j . . . v jn j and w . . . w k = ( − (cid:13) v i . . . v ki k ˇ w . . . ˇ w k . Then we define (cid:98) p k : S k SV → SV by (cid:76) l ≥ (cid:98) p k,l . To assure it is well-defined, we need to assume for any v , . . . , v k ∈ V , thereare at most finite many l such that p k,l ( v (cid:12) . . . (cid:12) v k ) (cid:54) = 0. Then we can define (cid:98) p : EV → EV by w (cid:12) . . . (cid:12) w n (cid:55)→ n (cid:88) k =1 (cid:88) σ ∈ Sh ( k,n − k ) ( − (cid:5) (cid:98) p k ( w σ (1) (cid:12) . . . (cid:12) w σ ( k ) ) (cid:12) w σ ( k +1) (cid:12) . . . (cid:12) w σ ( n ) , i.e. following the same rule of (cid:98) (cid:96) from (cid:96) k . Definition 2.3. ( V, { p k,l } k ≥ ,l ≥ ) is a BL ∞ algebra if (cid:98) p ◦ (cid:98) p = 0 and | (cid:98) p | = 1 . To explain the terminology, assume p , = 0 , p , = 0. Then p , defines a differential on V , such that p , defines a Lie bracket on the homology of ( V, p , ) and p , defines a Lie cobracket on the homology. Thecompatibility is that p , ◦ p , = 0 on the homology level. The main difference with the IBL ∞ algebra[21] is that we will not consider the compatibility condition on p , ◦ p , , which will increase genus . Adirect consequence of the definition is that ( SV, (cid:98) p ) is a chain complex and the (cid:98) p k define an L ∞ structure on( SV )[ − SV carries a natural commutative algebra structure, the Leibnizrule in the definition of (cid:98) p k,l implies the L ∞ structure is compatible with the algebra structure, and ( SV )[ − G ∞ algebra. Definition 2.3 can be viewed as making the compatibility precise. Remark 2.4. BL ∞ algebra is not a “direct” specialization of the IBL ∞ algebra as introduced in [21] .However, there is an equivalent reformulation of the IBL ∞ relations, from which one can see that an IBL ∞ algebra contains a BL ∞ algebra, as well as algebras with any genus upper bound, see § A useful way to explain the combinatorics of operations is the following description using graphs, whichappeared in [70, § § w ∈ S k V by a graph with k + 1 vertices, where k vertices are connected to the remainingvertex. Those k vertices are placeholders for elements from V and are not ordered, which reflects that theinput is from S k V rather than from ⊗ k V . We use • to represent those placeholders and the other vertex willnot be emphasized, whose only role is connecting all other vertices. To represent p k,l , we use a graph with k + l + 1 vertices, k top vertices representing the placeholders for k inputs and l bottom vertices representingthe placeholders for l outputs and one middle vertex representing the operation type. And the k inputvertices and l output vertices are connected to . To represent (cid:98) p on S i V (cid:12) . . . (cid:12) S i n V , we first place the n clusters on the top row representing S i V, . . . , S i n V , then we add on second level a graph representing p k,l with k inputs glued to k placeholder vertices on the top row, such that the glued graph has no cycle.Then we add a vertical dashed edge to every top row placeholder that is not glued to represent the identitymap. Then each connected component of the glued graph represents a (cid:12) -component in the output, and eachbottom row vertex represent a placeholder for a ∗ -component in the corresponding (cid:12) -component represented The other difference is that the
IBL ∞ algebra in [21] describes the algebra for linearized SFT, where p k, ,g = 0 for anynumber of positive punctures k and genus g . LANDSCAPE OF CONTACT MANIFOLDS VIA RATIONAL SFT 11 by the connected component. Then (cid:98) p is the sum of all such glued graphs. In particular, (cid:98) p k,l is the case ofattaching p k,l to get a connected graph. To compute (cid:98) p , we just plug in vectors in V at the placeholders witha sign by the Koszul-Quillen convention. Figure 1.
A component of (cid:98) p from S V (cid:12) S V (cid:12) S V to S V (cid:12) S V using p , . Example 2.5.
If we use ( v v v ) (cid:12) ( v v v ) (cid:12) ( v v ) as an input into the configuration in Figure 1, thenthe output is π (cid:0) ( ⊗ id ⊗ ( p , ◦ π ) ⊗ id)( v ⊗ . . . ⊗ v ) (cid:1) = ( − | p , | ( | v | + | v | ) ( v v p , ( v v ) v v ) (cid:12) ( v v ) , where π is the projection from tensor product to symmetric product. It is easy to check that, because of thesign convention, when we “reorder” the same graph the output does not depend on the order. For example,the following two presentations of the same graph will give the same output up to a sign ( − (cid:13) , where v . . . v = ( − (cid:13) v . . . v , which is the sign difference from reordering the input. v v v v v v v v v v v v v v v v We use p k,l : S k V → S l V for k ≥ , l ≥ k input vertices and l output vertices. Note that p k,l can be viewed as the codimension 1 boundaryof the rational SFT moduli space. The following proposition shows that the BL ∞ algebra structure capturesexactly such combinatorics. Proposition 2.6. { p k,l } k ≥ ,l ≥ forms a BL ∞ algebra iff p k,l = 0 for k ≥ , l ≥ .Proof. Let π ,l denote the projection EV → S SV → S l V , then we have p k,l = π ,l ◦ (cid:98) p | (cid:12) k V . Therefore if { p k,l } k ≥ ,l ≥ forms a BL ∞ algebra then p k,l = 0 for k ≥ , l ≥
0. Now assume p k,l = 0 for k ≥ , l ≥
0, thenwe have (cid:98) p | (cid:12) k V = k (cid:88) i =1 ∞ (cid:88) l =0 p i,l (cid:12) k − i id = 0 . Then we will argue inductively on i , . . . , i k and k such that (cid:98) p | S i V (cid:12) ... (cid:12) S ik V = 0. When we consider (cid:98) p (( ab ) (cid:12) w (cid:12) . . . (cid:12) w n ), if the two p ∗ , ∗ does not connect to the cluster representing ab , then it reduces a k − ab , then it reduces to a smaller i case.If both of them are connected to the cluster representing ab and two p ∗ , ∗ are connected, then either a or b does not get glued. In particular, it reduces to a case with smaller i . Finally if two p ∗ , ∗ are disconnected,and one of them is glued to a component of a and the other is glued to a component of b . In this case,such configurations can be paired up by switching the two component of p ∗ , ∗ (i.e. switching the compositionorder in a configuration like such). They will cancel out since | p ∗ , ∗ | = 1. Then the claim follows frominduction. (cid:3) In the following, we define morphisms between BL ∞ algebras. Given a family of operators { φ k,l : S k V → S l V (cid:48) } k ≥ ,l ≥ , such that for any v . . . v k ∈ S k V , there are at most finitely many l , such that φ k,l ( v . . . v k ) (cid:54) = 0.To explain the map (cid:98) φ : EV → EV (cid:48) , we will use the description of graphs. To represent φ k,l , we use a graphsimilar to the one representing p k,l but replace by to indicate that they are maps of different roles. Torepresent a component configuration in the definition of (cid:98) φ on S i V (cid:12) . . . (cid:12) S i n V , we first place n clustersrepresenting input placeholders on the top row, then we glue in a family of graphs representing φ k,l suchthat input vertices and the top row placeholder vertices are completely paired and glued and the resultedgraph has no cycles. The rule to determine the output space is the same as before. Then (cid:98) φ is the sum of allpossible configurations. Figure 2.
A component of (cid:98) φ from S V (cid:12) S V (cid:12) S V to S V (cid:48) In terms of formulas, we define (cid:98) φ k := π ◦ (cid:98) φ | S k SV : S k SV → SV (cid:48) , which is represented by all glued graphthat is connected and has k components in the top row. In particular, it is determined by the following.(1) (cid:98) φ k +1 ( w (cid:12) . . . (cid:12) w k (cid:12)
1) = 0 for k ≥ (cid:98) φ (1) = 1.(2) (cid:98) φ k : (cid:12) k V ⊂ S k SV → SV (cid:48) is defined by (cid:80) l ≥ φ k,l .(3) Let { i j } ≤ j ≤ k be a sequence of positive integers. We define N := (cid:80) kj =1 i j and N i := (cid:80) ij =1 i j . Let w i = v N i − +1 . . . v N i . The following sum is over all partitions J (cid:116) . . . (cid:116) J b = { , . . . , N } , such that thegraph with k + b + N vertices A , . . . , A k , B , . . . , B b , v , . . . , v N with A i connected to v N i − +1 , . . . , v N i and B i connected to v j iff j ∈ J i , has no circle: (cid:98) φ k ( w (cid:12) . . . (cid:12) w k ) = (cid:88) admissible partitions J (cid:116) ... (cid:116) J b ( − (cid:13) b ! ∞ (cid:88) l =0 φ | J | ,l ( v J ) (cid:12) . . . (cid:12) ∞ (cid:88) l =0 φ | J b | ,l ( v J b ) , where w . . . w k = ( − (cid:13) v J . . . v J b .The reduction by b ! is a consequence of the fact that we are counting over different graphs with unorderedvertices. Then we define (cid:98) φ from (cid:98) φ k just like the L ∞ morphism (cid:98) φ built from φ k . LANDSCAPE OF CONTACT MANIFOLDS VIA RATIONAL SFT 13 v v v v v v v v A A A B B B B B B Figure 3.
An admissible partition
Definition 2.7. { φ k,l } k ≥ ,l ≥ is a BL ∞ morphism from ( V, p ) to ( V (cid:48) , p (cid:48) ) if (cid:98) φ ◦ (cid:98) p = (cid:98) p (cid:48) ◦ (cid:98) φ and | (cid:98) φ | = 0 . The composition of φ : V → V (cid:48) and ψ : V (cid:48) → V (cid:48)(cid:48) is defined by the following. Let I = { , . . . , k } and I (cid:116) . . . (cid:116) I a be an admissible partition of I as above, then( ψ ◦ φ ) k,l ( v . . . v k ) = π l (cid:88) admissible partitions I (cid:116) ... (cid:116) I a (cid:98) ψ a (cid:32) ( − (cid:13) a ! ∞ (cid:88) l =0 φ | I | ,l ( v I ) (cid:12) . . . (cid:12) ∞ (cid:88) l =0 φ | I a | ,l ( v I a ) (cid:33) , where v . . . v k = ( − (cid:13) v I . . . v I a and π l is the projection SV (cid:48)(cid:48) → S l V (cid:48)(cid:48) . It is clear that the graph repre-senting (cid:98) ψ ◦ (cid:98) φ has no cycle. Then ( ψ ◦ φ ) k,l is represented by connected graphs without cycles glued from onelevel from φ and one level from ψ . It is clear that (cid:91) ψ ◦ φ = (cid:98) ψ ◦ (cid:98) φ by construction. Remark 2.8. An L ∞ algebra can be described by special graphs such that each cluster has one placeholderand we only have p k, . Similarly for L ∞ morphisms and their compositions. Remark 2.9.
There are different notions of homotopies between BL ∞ morphisms if we wish to definenotions of homotopy equivalences of BL ∞ algebras. In practice, we can not associate a conical BL ∞ algebrato a contact manifold but one depends on various choices and is only well-defined up to homotopy. However,for the purpose of this paper, we are constructing functors from Con to a totally ordered set, homotopyinvariance is not needed. Nevertheless, we have the following brief remarks on homotopy.(1) One can define a notion of homotopy, which is a homotopy on the bar/cobar complex. That is one candefine a map by counting rigid but disconnected curves in a one-parameter family. One advantage ofsuch definition is that it is easier to construct as we will neglect the structures from each connectedcomponent. Any homological structure on the level of bar/cobar complex will be an invariant. Forexample, the contact homology in [66] used this notion of homotopy.(2) Another notion of homotopy is defined through the notion of path objects, e.g. [21, Definition 4.1] , seealso [70, Definition 2.9] for the homotopy in the L ∞ context with a specific path object model. Thisdefinition is the right one to discuss linearized theory but is more involved to get in the constructionof SFT. In particular, homotopic augmentations give rise to homotopic linearized theories with suchnotion of homotopy. Such homotopy is expected to be derived from the homotopy used in [28] .However, from the curve counting point of view, such construction is more subtle.A detailed construction of RSFT in terms of BL ∞ algebras up to homotopy will appear in a future work. Augmentations.
When V = { } , it has a unique trivial BL ∞ algebra structure by p k,l = 0. We use to denote this trivial BL ∞ algebra. Note that is the initial object in the category of BL ∞ algebras, with → V defined by φ k,l = 0. Definition 2.10. A BL ∞ augmentation is a BL ∞ morphism (cid:15) : V → , i.e. a family of operators (cid:15) k : S k V → k satisfying Definition 2.7. For a BL ∞ algebra V , we define E k V = B k SV , which is a filtration on EV compatible with the differential (cid:98) p . Note that E = k ⊕ S k ⊕ + . . . with (cid:98) p = 0, we have H ∗ ( E ) = E . Similarly we have H ∗ ( E k ) = E k for all k ≥
1. We define 1 be the generator in E , then 1 (cid:54) = 0 ∈ H ∗ ( E k ) for all k ≥
1. Then wedefine 1 V ∈ H ∗ ( E k V ) to be the image of 1 under the chain map E k → E k V induced by the trivial BL ∞ morphism → V . Proposition 2.11.
If there exists k ≥ , such that V ∈ H ∗ ( E k V ) is zero, then V has no BL ∞ augmenta-tion.Proof. If there is an augmentation (cid:15) : V → , then the sequences of BL ∞ morphisms → V (cid:15) → inducea chain morphism E k (cid:55)→ E k V (cid:55)→ E k . It is direct to check the composition is identity by definition. If1 V ∈ H ∗ ( E k V ) is zero, then we have a contradiction since 1 (cid:54) = 0 ∈ H ∗ ( E k ). (cid:3) Definition 2.12.
We define the torsion of a BL ∞ algebra V to be T( V ) := min { k − | V = 0 ∈ H ∗ ( E k V ) , k ≥ } . Here the minimum of an empty set is defined to be ∞ . By definition, we have that T( V ) = 0 iff 1 V ∈ H ∗ ( SV, (cid:98) p ) is zero. Since H ∗ ( SV, (cid:98) p ) is an algebra with1 V a unit, we have H ∗ ( SV, (cid:98) p ) = 0. In the context of SFT, T( V ) = 0 iff the contact homology vanishes, i.e.algebraically overtwisted [11].Since a BL ∞ morphism preserves the word filtration on the bar complex, we know that if there is a BL ∞ morphism from V to V (cid:48) then T( V ) ≥ T( V (cid:48) ). Therefore we have the following obvious property, which iscrucial for the invariant property for our applications in symplectic topology. Proposition 2.13.
If there are BL ∞ morphisms between V, V (cid:48) in both directions, then we have T( V ) =T( V (cid:48) ) . Given a BL ∞ augmentation (cid:15) , we can linearize w.r.t. (cid:15) by the following procedure. More precisely, thereis a change of coordinate to kill off all constant terms p k, . We define F , (cid:15) = id V and F k, (cid:15) = (cid:15) k and all other F k,l(cid:15) = 0. Then following the recipe of constructing (cid:98) φ from φ k,l , we can define (cid:98) F (cid:15) on EV . Then (cid:98) F (cid:15) preservesthe word length filtration and on the diagonal π k ◦ (cid:98) F (cid:15) | S k SV is (cid:12) k (cid:98) F (cid:15) , where (cid:98) F (cid:15) is an algebra isomorphismdetermined by (cid:98) F (cid:15) ( x ) = x + (cid:15) ( x ) and π k is the projection EV → S k SV . Indeed the inverse is given by thefollowing proposition. Proposition 2.14.
Let (cid:98) F − (cid:15) denote the map on EV defined by F , − (cid:15) = id V and F k, − (cid:15) = − (cid:15) k, and all other F k,l − (cid:15) = 0 , then (cid:98) F − (cid:15) is the inverse of (cid:98) F (cid:15) Proof.
We use the graph representation to prove the claim. Note that in the composition (cid:98) F − (cid:15) ◦ (cid:98) F (cid:15) , we canfind pairs of configuration as follows as long as there are some components from (cid:15) . LANDSCAPE OF CONTACT MANIFOLDS VIA RATIONAL SFT 15 . . . . . . . . . . . .(cid:15) . . . . . . . . . . . .. . . . . . . . . . . .. . . . . . . . . . . . . . . . . . id id − (cid:15) . . . . . . . . . . . .. . . . . . . . . . . .. . . . . . Figure 4.
An example with (cid:15) It is clear those pairs will cancel with each other. Hence the only remaining term is those only has identitymap component. Hence (cid:98) F − (cid:15) ◦ (cid:98) F (cid:15) = id, similarly, we have (cid:98) F (cid:15) ◦ (cid:98) F − (cid:15) = id (cid:3) We use (cid:98) F (cid:15) as a change of coordinate on EV and consider (cid:98) p (cid:15) := (cid:98) F (cid:15) ◦ (cid:98) p ◦ (cid:98) F − (cid:15) : EV → EV , then (cid:98) p (cid:15) = 0. Wecan define p k,l(cid:15) := π ,l ◦ (cid:98) F (cid:15) ◦ (cid:98) p ◦ (cid:98) F − (cid:15) | (cid:12) k V , where π ,l is the projection EV → S S l V . Since (cid:98) p (1 (cid:12) . . . ) = 0, we have p k,l(cid:15) = π ,l ◦ (cid:98) F (cid:15) ◦ (cid:98) p | (cid:12) k V and itis represented by the sum of following connected graphs without cycle with k inputs, l outputs, one (cid:13) component and possibly several components from (cid:15) . (cid:15) (cid:15) p , Figure 5.
A component of p , (cid:15) Proposition 2.15. (cid:98) p is determined by p k,l(cid:15) following the same recipe for (cid:98) p from p k,l . Moreover, we have p k, (cid:15) = 0 for all k .Proof. In the graph configuration of (cid:98) F (cid:15) ◦ (cid:98) p ◦ (cid:98) F − (cid:15) on S i V (cid:12) . . . (cid:12) S i k V , there is exactly one componentcontaining a p k,l(cid:15) as a subgraph. In the connected component containing this subgraph, we might haveother components containing ± (cid:15) and we have other connected components that are not purely identities,i.e. containing ± (cid:15) . To prove that (cid:98) p is determined by p k,l(cid:15) , we need to rule out all those other componentscontaining ± (cid:15) . This follows from the same argument in Proposition 2.14, as in both case they will pair up and cancel each other. Because (cid:15) is an augmentation, we have π , ◦ (cid:98) F (cid:15) ◦ (cid:98) p = (cid:98) (cid:15) ◦ (cid:98) p = 0. Therefore p k, (cid:15) = 0for all k . (cid:3) As a corollary of Proposition 2.15, (cid:96) k(cid:15) := p k, (cid:15) defines an L ∞ structure on V [ − p k,l • : S k V → S l V, k ≥ , l ≥ (cid:98) p k,l • and (cid:98) p • just like (cid:98) p k,l and (cid:98) p by component-wise Leibniz rule and coLeibnizrule with the modification that | p k,l • | is not necessarily 1 ∈ Z . Definition 2.16.
We say { p k,l • } is a pointed map, iff (cid:98) p • ◦ (cid:98) p = ( − | (cid:98) p • | (cid:98) p ◦ (cid:98) p • . In applications, p k,l • will come from counting holomorphic curves with one interior marked point subjectto a constrain from H ∗ ( Y ). The degree of p • is same as the degree of the constraint. Typically we willonly consider a point constraint, then the degree is 0. Note that it does not define BL ∞ morphisms asthe combinatorics for packaging (cid:98) p • is different from (cid:98) φ . Nevertheless, (cid:98) p • still defines a morphism on the barcomplex and preserves the word length filtration.Then by the same argument in Proposition 2.15, we can define (cid:98) p • ,(cid:15) := (cid:98) F (cid:15) ◦ (cid:98) p • ,(cid:15) ◦ (cid:98) F − (cid:15) and (cid:98) p • ,(cid:15) is determined by p k,l • ,(cid:15) , which is defined similarly to p k,l(cid:15) . Note that we also have (cid:98) p (cid:15) ◦ (cid:98) p • ,(cid:15) = (cid:98) p • ,(cid:15) ◦ (cid:98) p (cid:15) . However, it is not necessarilytrue that p k, • ,(cid:15) = 0. In fact, the failure of this property on homological level will be another hierarchy thatwe are interested in. We define (cid:96) k, • ,(cid:15) by p k, • ,(cid:15) . Then (cid:98) (cid:96) • ,(cid:15) := (cid:80) k ≥ (cid:96) k, • ,(cid:15) defines a chain morphism ( SV, (cid:98) (cid:96) (cid:15) ) → k .That (cid:98) (cid:96) • ,(cid:15) commutes with (cid:98) (cid:96) (cid:15) follows from π k ◦ (cid:98) p (cid:15) ◦ (cid:98) p • ,(cid:15) = π k ◦ (cid:98) p • ,(cid:15) ◦ (cid:98) p (cid:15) restricted to SV = SS V ⊂ EV and π k is the projection from EV to k ⊂ S SV ⊂ EV . Definition 2.17.
Given a BL ∞ augmentation and a pointed map p • , the ( (cid:15), p • ) order of V is defined to be O ( V, (cid:15), p • ) := min (cid:110) k (cid:12)(cid:12)(cid:12) ∈ im (cid:98) (cid:96) • ,(cid:15) | H ∗ ( B k V, (cid:98) (cid:96) (cid:15) ) (cid:111) , where the minimum of an empty set is defined to be ∞ . There is another notion of order, which is related to O ( V, (cid:15), p • ). We use EV to denote SSV , then EV has a splitting EV = (cid:0) EV ⊕ (cid:12) k (cid:1) ⊕ (cid:0) ( EV (cid:12) k ) ⊕ (cid:12) k (cid:1) ⊕ (cid:0) ( EV (cid:12) k ) ⊕ (cid:12) k (cid:1) ⊕ . . . = ⊕ ∞ k =1 ( EV ⊕ k )Since p k, (cid:15) = 0 for all k , we know that (cid:98) p is a differential on EV . Moreover, note that (cid:98) p (cid:15) ( k (cid:12) . . . ) = 0 bydefinition, we have the homology of EV respects the splitting, i.e. H ∗ ( EV, (cid:98) p ) = ⊕ ∞ k =1 (cid:0) H ∗ ( EV , (cid:98) p ) ⊕ k (cid:1) . Onthe other hand, (cid:98) p • ,(cid:15) does not respect such splitting. In particular, one interesting portion is the restriction of π k ◦ (cid:98) p • ,(cid:15) to the first copy of H ∗ ( EV , (cid:98) p ), where π k is the projection to the first k in the splitting H ∗ ( EV, (cid:98) p ) = ⊕ ∞ k =1 (cid:0) H ∗ ( EV, (cid:98) p ) ⊕ k (cid:1) . Definition 2.18.
Given a BL ∞ augmentation and a pointed map p • , we define ˜ O ( V, (cid:15), p • ) := min (cid:110) k (cid:12)(cid:12)(cid:12) ∈ π k ◦ (cid:98) p • ,(cid:15) | H ∗ ( B k SV, (cid:98) p (cid:15) ) (cid:111) . Proposition 2.19. O ( V, (cid:15), p • ) ≤ ˜ O ( V, (cid:15), p • ) .Proof. We use B k ⊂ B k SV to denote the subspace ( ⊕ ∞ i =2 S i V ) (cid:12) k − SV . Since p k, (cid:15) = 0, we have B k is asubcomplex. Moreover, the quotient complex B k SV /B k is exactly ( B k V, (cid:98) (cid:96) (cid:15) ). Since π k ◦ (cid:98) p • ,(cid:15) on B k is zero.We know that 1 ∈ π k ◦ (cid:98) p • ,(cid:15) | H ∗ ( B k SV, (cid:98) p (cid:15) ) implies that 1 ∈ im (cid:98) (cid:96) • ,(cid:15) | H ∗ ( B k V, (cid:98) (cid:96) (cid:15) ) . Hence the claim follows. (cid:3) LANDSCAPE OF CONTACT MANIFOLDS VIA RATIONAL SFT 17
The inequality in Proposition 2.19 is likely to be necessary. The main issue is that a closed class in H ∗ ( B k V ) may not be closed in H ∗ ( B k SV ). For the definition of planarity, we will use O ( V, (cid:15), p • ), which hasthe benefit of the existence of another hierarchy when O ( V, (cid:15), p • ) = 1. Remark 2.20.
We can similarly define a more generalized order as min (cid:110) k (cid:12)(cid:12)(cid:12) ∈ π k ◦ ( (cid:98) p • ,(cid:15) ) l | H ∗ ( B k SV, (cid:98) p (cid:15) ) (cid:111) . It it easy to show that the number is non-decreasing w.r.t. l , where the l = 1 case is ˜ O ( V, (cid:15), p • ) . See § [70] . Next we need to compare the construction under BL ∞ morphisms. Given a BL ∞ morphism φ : ( V, p ) → ( V (cid:48) , q ) and a family of morphisms φ k,l • : S k V → S l V , then we can define (cid:98) φ • : EV → EV (cid:48) by the samerule of (cid:98) φ with exactly one φ k,l • component and all the others are φ k,l components with the exception that (cid:98) φ • ( k (cid:12) . . . ) = 0. Definition 2.21.
Assume p • , q • are two pointed maps of ( V, p ) , ( V (cid:48) , q ) respectively of the same dergee. Wesay p • , q • , φ are compatible, if there is a family of φ k,l • such that (cid:98) q • ◦ (cid:98) φ − ( − | (cid:98) q • | (cid:98) φ ◦ (cid:98) p • = (cid:98) q ◦ (cid:98) φ • − ( − | (cid:98) φ • | (cid:98) φ • ◦ (cid:98) p and | (cid:98) φ • | = | (cid:98) p • | + 1 . In practice, φ k,l • is defined by counting connected rational holomorphic curves in the cobordism with amarked point passing through a cobordism between the constraints in the definition of p • , q • . In our typicalcase of point constraint, the cobordism will be a path connecting the point constraints, where we have | (cid:98) φ • | = 1. In principle, we can consider the category consists of pairs ( p, p • ) with morphisms given by pairs( φ, φ • ) with a suitable definition of composition. Then the definition of orders is functorial. For our purpose,we only need the following property without the precise definition of a composition. Proposition 2.22.
Assume φ is a BL ∞ -morphism from ( V, p ) to ( V (cid:48) , q ) with pointed maps p • , q • of degree respectively, such that p • , q • , φ are compatible. Then for any BL ∞ augmentation (cid:15) of V (cid:48) , we have O ( V, (cid:15) ◦ φ, p • ) ≥ O ( V (cid:48) , (cid:15), q • ) and ˜ O ( V, (cid:15) ◦ φ, p • ) ≥ ˜ O ( V (cid:48) , (cid:15), q • ) .Proof. By the definition of compatibility, we have (cid:98) q • ,(cid:15) ◦ (cid:98) F − (cid:15) ◦ (cid:98) φ ◦ (cid:98) F − (cid:15) ◦ φ − (cid:98) F (cid:15) ◦ (cid:98) φ ◦ (cid:98) F − (cid:15) ◦ φ ◦ (cid:98) p • ,(cid:15) ◦ φ = (cid:98) F (cid:15) ◦ (cid:98) φ • ◦ (cid:98) F − (cid:15) ◦ φ ◦ (cid:98) p (cid:15) ◦ φ + (cid:98) q (cid:15) ◦ (cid:98) F (cid:15) ◦ (cid:98) φ • ◦ (cid:98) F − (cid:15) ◦ φ . From (cid:98) φ ◦ (cid:98) p = (cid:98) q ◦ (cid:98) φ , we have (cid:98) F (cid:15) ◦ (cid:98) φ ◦ (cid:98) F − (cid:15) ◦ φ ◦ (cid:98) p (cid:15) ◦ φ = (cid:98) q (cid:15) ◦ (cid:98) F (cid:15) ◦ (cid:98) φ ◦ (cid:98) F − (cid:15) ◦ φ . Therefore we have the followingcommutative diagram of complexes up to homotopy, where we define (cid:98) φ (cid:15) = (cid:98) F (cid:15) ◦ (cid:98) φ ◦ (cid:98) F − (cid:15) ◦ φ EV (cid:98) p • ,(cid:15) ◦ φ (cid:47) (cid:47) (cid:98) φ (cid:15) (cid:15) (cid:15) EV (cid:48) (cid:98) φ (cid:15) (cid:15) (cid:15) EV (cid:98) q • ,(cid:15) (cid:47) (cid:47) EV (cid:48) Similar to Proposition 2.14, (cid:98) φ (cid:15) is determined by φ k,l(cid:15) following a similar rule to (cid:98) φ and φ k,l , where φ k,l(cid:15) isdefined similar to p k,l(cid:15) . Moreover, φ k, (cid:15) = 0, as nontrivial contributions pair up and cancel each other similarto Proposition 2.14. As a consequence, (cid:98) φ (cid:15) preserves the splitting in the definition of ˜ O . Since all themaps including the homotopy preserve the word length filtration, if 1 ∈ im π k ◦ (cid:98) p • ,(cid:15) ◦ φ | H ∗ ( B k SV, (cid:98) p (cid:15) ◦ φ ) , then ∈ im π k ◦ (cid:98) q • ,(cid:15) | H ∗ ( B k SV (cid:48) , (cid:98) q (cid:15) ) . Therefore ˜ O ( V, (cid:15) ◦ φ, p • ) ≥ ˜ O ( V (cid:48) , (cid:15), q • ). In order to prove the other inequality,we have a diagram SV (cid:98) (cid:96) • ,(cid:15) ◦ φ (cid:47) (cid:47) (cid:98) φ (cid:15) (cid:15) (cid:15) k id (cid:15) (cid:15) SV (cid:48) (cid:98) (cid:96) • ,(cid:15) (cid:47) (cid:47) k where (cid:98) φ (cid:15) is determined by the L ∞ morphism φ k, (cid:15) since φ k, (cid:15) = 0. The diagram is in fact commutativeup to homotopy (cid:98) φ • ,(cid:15) : SV → k defined by (cid:80) k ≥ φ k, • ,(cid:15) . Therefore 1 ∈ im (cid:98) (cid:96) • ,(cid:15) ◦ φ | H ∗ ( B k V, (cid:98) (cid:96) (cid:15) ◦ φ ) implies that1 ∈ im (cid:98) (cid:96) • ,(cid:15) | H ∗ ( B k V (cid:48) , (cid:98) (cid:96) (cid:15) ) . Hence O ( V, (cid:15) ◦ φ, p • ) ≥ O ( V (cid:48) , (cid:15), q • ) (cid:3) Rational symplectic field theory
In this section, we explain the construction of rational symplectic field theory (RSFT) as BL ∞ algebras.RSFT was original packaged into a Poisson algebra with a distinguished odd degree class h such that { h , h } = 0 in [28]. However for the purpose of building hierarchy functors from contact manifolds, it isuseful to reformulate RSFT as BL ∞ algebras. It is important to note that we will use same moduli spacesof holomorphic curves as the original RSFT but reinterpret the relations as other algebraic structures.3.1. Notations on symplectic topology.
We first briefly recall the basics of symplectic and contacttopology. A (co-oriented) contact manifold (
Y, ξ ) is a 2 n − ξ such that there is a one form α with ξ = ker α and α ∧ (d α ) n − (cid:54) = 0. Such oneform α is called a contact form and we will call ( Y, α ) a strict contact manifold. Given a contact form α , theReeb vector field R α is characterized by α ( R α ) = 0 , ι R α d α = 0. We say a contact form α is non-degenerateiff all Reeb orbits are non-degenerate. Any contact form can be perturbed into a non-degenerate contactform and in particular, every contact manifold admits non-degenerate contact forms. Throughout this paper( Y, α ) is always assumed to be a strict contact manifold with a non-degenerate contact form unless specifiedotherwise.
Definition 3.1.
A symplectic manifold ( X, ω ) with ∂W = Y − (cid:116) Y + is(1) a strong cobordism from ( Y − , ξ − ) to ( Y + , ξ + ) iff ω = d λ ± near Y ± with ξ ± = ker λ ± such that wedefine V ± by ι V ± ω = λ ± , then V + points out on Y + and V − points in on Y − ;(2) an exact cobordism from ( Y − , ξ − ) to ( Y + , ξ + ) if moreover ω = d λ on X . The vector field V definedby ι V ω = λ is called the Liouville vector field;(3) a Weinstein cobordism from ( Y − , ξ − ) to ( Y + , ξ + ) if moreover the Liouville vector field is gradient likefor some Morse function with Y ± as the regular level sets of maximum/minimum. We say a cobordism (
X, ω ) from ( Y − , α − ) to ( Y, α + ) is strict iff λ ± | Y ± = α ± . It is clear from definitionthat we can glue strict cobordisms to get a strict cobordism. In general, given two exact cobordisms W , W from Y , Y to Y , Y respectively, the composition W ◦ W from Y to Y is not uniquely defined, but up tohomotopies of Liouville structures [20, § Definition 3.2.
The exact cobordism category of contact manifolds
Con is defined to be the category whoseobjects are contact manifolds and morphisms are exact cobordisms up to homotopy. The composition is givenby gluing cobordisms. We will use
Con k − to denote the subcategory of k − dimensional contact manifolds. LANDSCAPE OF CONTACT MANIFOLDS VIA RATIONAL SFT 19
Similarly, we use
Con W to denote the Weinstein cobordism category and Con S to denote the strong cobordismcategory. All of the categories above have monoidal structures given by the disjoint union. It is clear that we havenatural functors
Con W → Con → Con S , which are identities on the object level. Remark 3.3.
There is a forgetful functor from
Con to the cobordism category of almost contact manifolds,where the cobordisms are almost symplectic cobordisms. In the case of
Con W , there is a forgetful map to thealmost Weinstein cobordism category of almost contact manifolds. These are purely topological objects, thelatter was studied thoroughly in [17, 18] . Roughly speaking, the principle in the symplectic cobordism category is that the complexity of contactgeometry increases in the direction of cobordism. In view of this, we can introduce the following category,which only remembers if there exists a cobordism.
Definition 3.4.
We define
Con ≤ to be the category of contact manifolds, such that there is at most onearrow between two contact manifolds and the arrow exists iff there is an exact cobordism. Similarly, we candefine Con ≤ ,W and Con ≤ ,S . It is a natural question to ask whether
Con ≤ is a poset. It clear that we only need to prove that Y ≤ Y , Y ≤ Y implies that Y = Y . Unfortunately, this is not the case, as we may take Y , Y astwo different 3-dimensional overtwisted contact manifolds [32] or suitable flexible fillable contact manifolds.One extreme case is that Y can be different from Y even if the cobordisms are inverse to each other [26].However, we can modulo out this ambiguity to get a poset. It is clear that the existence of (exact) cobordismsbetween Y , Y in both directions define an equivalence relation. Definition 3.5.
We define
Con ≤ to be the poset, such that the object is an equivalence class of contactmanifolds and there is a morphism [ Y ] ≤ [ Y ] iff there is an exact cobordism from Y to Y . Similarly, wecan define the posets Con ≤ ,W and Con ≤ ,S . Under this condition, all overtwisted contact structure becomes the same minimal object in
Con ≤ [32]. Inthe higher dimensions, overtwisted contact manifolds are minimal objects up to topological constrains [30].It is clear that we have functors Con → Con ≤ → Con ≤ . The theme of this paper is constructing functors from Con to some totally ordered set. Since it always descends to
Con ≤ , results in this paper can be understoodas some structures on the poset Con ≤ . It is also an interesting question on whether Con ≤ is a totally orderedset , see § Con ≤ , Con ≤ ,W , Con ≤ ,S .An exact/Weinstein/strong cobordism from ∅ to Y is called an exact/Weinstein/strong filling of Y . Wealso introduce a category Con ∗ as the over category over the empty set. Definition 3.6.
The objects of
Con ∗ are pairs ( Y, W ) , where W is an exact filling of Y up to homotopy. Amorphism from ( Y , W ) to ( Y , W ) is an exact cobordism X from Y to Y such that X ◦ W = W up tohomotopy, or equivalently an exact embedding of W into W up to homotopy. Example 3.7.
The symplectic cohomology is a functor from
Con ∗ to the category of BV algebras, where thefunctoriality follows from the Viterbo transfer map. The S -equivariant symplectic cochain complex is alsoa functor Con ∗ to the homotopy category of S -cochain complexes. The order of dilation and the order ofsemi-dilation in [78] are functors from Con ∗ to N ∪ {∞} . For the total order, we will not consider ∅ as an object in Con ≤ , as overtwisted contact manifolds and ∅ are obviously notcomparable in Con ≤ . Geometric setups for holomorphic curves.
As usual, an almost complex structure J on the sym-plectization ( R s × Y, d( e s α )) is said to be admissible iff(1) J is invariant under the s -translation and restricts to a tame almost complex structure on ( ξ =ker α, d α ),(2) J sends ∂ s to the Reeb vector R α .Let ( W, λ ) be an exact filling and (
X, λ ) an exact cobordism. An almost complex structure J on completions( (cid:99) W , (cid:98) λ ) or ( (cid:98) X, (cid:98) λ ) is admissible iff(1) J is tame for d (cid:98) λ ,(2) J is admissible on cylindrical ends.Occasionally, we will also consider strong fillings ( W, ω ), where the definition of admissible almost complexstructure on (cid:99) W is similar. For each Reeb orbit γ , we can fix a basepoint b γ on the image. Now fixan admissible J , and consider two collections of Reeb orbits γ +1 , . . . , γ + s + and γ − , . . . , γ − s − , possibly withduplicates. A pseudoholomorphic map in the symplectization R × Y or completions (cid:99) W , (cid:98) X with positiveasymptotics γ +1 , . . . , γ + s + and negative asymptotics γ − , . . . , γ − s − consists of:(1) a sphere Σ, with a complex structure denoted by j ,(2) a collection of pairwise distinct points z +1 , . . . , z + s + , z − , . . . , z − s − ∈ Σ, each equipped with an asymptoticmarker, i.e. a direction in the tangent sphere bundle S z ± i Σ,(3) a map ˙Σ → R × Y, (cid:99) W , (cid:98) X satisfying d u ◦ j = J ◦ d u , where ˙Σ denotes the punctured Riemann surfaceΣ \{ z +1 , . . . , z + s + , z − , . . . , z − s − } ,(4) for each z + i with corresponding polar coordinates ( r, θ ) around z + i such that the asymptotic markercorresponding to θ = 0, we have(3.1) lim r → ( π R ◦ u )( re iθ ) = + ∞ , u z + i ( θ ) := lim r → ( π Y ◦ u )( re iθ ) = γ + i ( 12 π T + i θ )where T + i is the period of the parameterized orbit γ + i and γ + i (0) = b γ + i ,(5) for each z − i with corresponding polar coordinates ( r, θ ) compatible with asymptotic marker, we have(3.2) lim r → ( π R ◦ u )( re iθ ) = −∞ , u z − i ( θ ) := lim r → ( π Y ◦ u )( re − iθ ) = γ − i ( − π T − i θ )where T − i is the period of the parameterized orbit γ − i and γ − i (0) = b γ + i .A holomorphic curve is an equivalence of holomorphic maps modulo biholomorphisms of Σ commutingwith all the data. Throughout this paper, we will work with Z -grading unless specified otherwise. LetΓ + = { γ +1 , . . . , γ + s + } , Γ − = { γ − , . . . , γ − s − } be two ordered sets of Reeb orbits possible with duplicates.Choosing trivializations of ξ over orbits in Γ + , Γ − , we can assign the Conley-Zehnder index µ CZ ( γ ± i ) to eachorbit. With such trivialization, we have a relative first Chern class c : H ( Y, Γ + ∪ Γ − ; Z ) → Z , similarly for W and X . Let A be a relative homology representing the curve u , we have the Fredholm indexof the Cauchy Riemann operator at u minus the dimension of automorphism group (i.e. biholomorphism ofΣ commuting with all the data) is the following,ind( u ) = ( n − − s + − s − ) + s + (cid:88) i =1 µ CZ ( γ + i ) − s − (cid:88) i =1 µ CZ ( γ − i ) + 2 c ( A ) . LANDSCAPE OF CONTACT MANIFOLDS VIA RATIONAL SFT 21
In this paper, we will consider the following moduli spaces.(1) M Y,A (Γ + , Γ − ) is the moduli space of rational holomorphic curves in the symplectization, moduloautomorphism and the R translation. The expected dimension is ind( u ) − M W,A (Γ + , ∅ ) and M X,A (Γ + , Γ − ) are the moduli spaces of rational holomorphic curves in the filling,respectively cobordism, modulo automorphism. The expected dimension is ind( u ).(3) M Y,A,o (Γ + , Γ − ) is the moduli space of rational holomorphic with one interior marked point in thesymplectization modulo automorphism. And the marked point is required to be mapped to (0 , o ) ∈ R × Y for a point o ∈ Y . The expected dimension is ind( u ) + 2 − n .(4) M W,A,o (Γ + , ∅ ) is the moduli space of rational holomorphic with one interior marked point in thefilling modulo automorphism. And the marked point is required to be mapped to o ∈ W . Theexpected dimension is ind( u ) + 2 − n .(5) M X,A,γ (Γ + , Γ − ) is the moduli space of rational curves with one interior marked point in the exactcobordism X modulo automorphism. The marked point is required to go through a path (cid:98) γ , whichis the completion of a path γ from a point in Y + to a point in Y − , i.e. extension by constant mapsin each slice in the cylindrical ends. The expected dimension is ind( u ) + 1 − n .Another fact that is important for our later proof is that(3.3) (cid:90) Γ + α − (cid:90) Γ − α ≥ , whenever M Y,A (Γ + , Γ − ) or M Y,A,o (Γ + , Γ − ) are not empty. All of the moduli spaces above have a SFTbuilding compactification by [9] denoted by M . The orientation convention follows [10], and we need torequire that all asymptotic Reeb orbits are good [76, Definition 11.6]. One property of this convention isthat if we switch two orbits γ , γ that are next to each other in Γ + or Γ − , the induced orientation is changedby ( − | µ CZ ( γ )+ n − |·| µ γ + n − | [76, § . Next we need transversality, where the count is a honest countof orbifold points, or a virtual machinery, where the count is a count of weighted orbifold points in perturbedmoduli spaces [33, 47] or an algebraic count after fixing some auxiliary data [66]. For simplicity, we will justuse M to denote the count.3.3. Contact homology algebra.
We will first recall the definition of the contact homology algebra. Thereare two possible choices of coefficient, the rational number field Q and the Novikov field Λ = { (cid:80) a i T bi | a i ∈ Q , lim b i ∈ R → ∞} . Since the only non-exact symplectic manifold we will consider is strong fillings,therefore the only place we really need to use the Novikov field is augmentations from strong fillings. In thefollowing, we will state the formula using Λ, but we can always set T = 1 to go back to Q whenever theunderlying symplectic manifold is exact.Let V α denote the free Λ-module generated by formal variable q γ for each good orbit γ of ( Y, α ). q γ isgraded as µ CZ ( γ ) + n −
3, which should be understood as a well-defined Z grading in general. The contacthomology algebra CHA( Y ) is the free symmetric algebra SV α . The differential is defined as follows.(3.4) ∂ l ( q γ ) = (cid:88) | [Γ] | = l M Y,A ( { γ } , Γ) T (cid:82) A d α µ Γ κ Γ q Γ . Alternatively, the count is evaluated in the fixed space of an orientation line with a group action, the appearance of a badorbit is exactly when the group action is not trivial, see [66] for details.
The sum is over all multiset [Γ], i.e. sets with duplicates, of size l . And Γ is an ordered representationof [Γ], e.g. Γ = { η , . . . , η (cid:124) (cid:123)(cid:122) (cid:125) i , . . . , η m , . . . , η m (cid:124) (cid:123)(cid:122) (cid:125) i m } is an ordered set of good orbits with η i (cid:54) = η j for i (cid:54) = j and (cid:80) i j = l . We write µ Γ = i ! . . . i m ! and κ Γ = κ i η . . . κ i m η m is the product of multiplicities, and q Γ = q η . . . q η m .We modulo out µ Γ as we should count holomorphic curves with unordered punctures and modulo out κ Γ tocompensate that we have κ γ different ways to glue when we have a breaking at γ . The orientation property of M Y,A ( { γ } , Γ) implies that (3.4) is independent of the representative Γ. The differential on a single generatoris defined by ∂ ( q γ ) = ∞ (cid:88) l =0 ∂ l ( q γ ) , which is always a finite sum by (3.3). Then the differential on CHA( Y ) is defined by the Leibniz rule ∂ ( q γ . . . q γ l ) = l (cid:88) j =1 ( − | q γ | + ... + | q γj − | q γ . . . q γ j − ∂ ( q γ j ) q γ j +1 . . . q γ l . The relation ∂ = 0 follows from the boundary configuration of M Y,A ( { γ } , Γ) with virtual dimension 1.Given an exact cobordism (
X, λ ) from Y − to Y + , then we have an algebra map φ from CHA( Y + ) toCHA( Y − ), which on generator is defined by φ ( q γ ) = ∞ (cid:88) l =0 (cid:88) | [Γ] | = l M X,A ( { γ } , Γ) T (cid:82) A ω µ Γ κ Γ q Γ , where Γ is a collection of good orbits of Y − . The boundary configuration of M X,A ( { γ } , Γ) with virtual dimen-sion 1 gives the relation ∂ ◦ φ = φ ◦ ∂ . Then we have a functor from Con to the category of supercommutativedifferential graded algebras.
Theorem 3.8 ([66]) . The homology H ∗ (CHA( Y )) above realized in VFC gives a monoidal functor from Con to the category of (super)commutative algebras.
Rational SFT as BL ∞ algebras. To assign strict contact manifolds with BL ∞ algebras, we needto consider moduli spaces with multiple inputs and multiple outputs. We use M Y,A (Γ + , Γ − ) to denotethe compactified moduli space of rational curves in class A with positive asymptotics Γ + and negativeasymptotics Γ − in the symplectization R × Y . Then we can define p k,l by(3.5) p k,l ( q Γ + ) = (cid:88) | [Γ − ] | = l M Y,A (Γ + , Γ − ) T (cid:82) A d α µ Γ + µ Γ − κ Γ − q Γ − . Here [Γ − ] is a multiset with Γ − an ordered set representative and | Γ + | = k . In particular, the orientationproperty of M Y,A (Γ + , Γ − ) implies that p k,l is map from S k V α to S l V α . Then the boundary of the 1-dimensional moduli spaces M Y,A (Γ + , Γ − ) would yield that { p k,l } is a BL ∞ algebra RSFT( Y ), see Theorem3.9 for details. LANDSCAPE OF CONTACT MANIFOLDS VIA RATIONAL SFT 23
T T p , p , Figure 6. (cid:98) p ◦ (cid:98) p = 0, where T stands for a trivial cylinder.Similarly for a strict exact cobordism X from Y − to Y + , by considering the moduli spaces M X,A (Γ + , Γ − )of rational curves in X , we can define a BL ∞ morphism from RSFT( Y + ) to RSFT( Y − ) by following,(3.6) φ k,l ( q Γ + ) = (cid:88) | [Γ − ] | = l M X,A (Γ + , Γ − ) T (cid:82) A ω µ Γ + µ Γ − κ Γ − q Γ − , where | Γ + | = k . Then the boundary of the 1-dimensional moduli spaces M X,A (Γ + , Γ − ) would yield that { φ k,l } is a BL ∞ morphism RSFT( Y + ) → RSFT( Y − ). In the following figure we indicate the cobordism levelwith ‘C’. T φ , φ , p , C φ , T p , φ , C Figure 7. (cid:98) φ ◦ (cid:98) p = (cid:98) p ◦ (cid:98) φ If we fix a point o in Y , by consider moduli spaces M Y,A,o (Γ + , Γ − ), we can define a pointed morphism p • by(3.7) p k,l • ( q Γ + ) = (cid:88) | [Γ − ] | = l M Y,A,o (Γ + , Γ − ) T (cid:82) A d α µ Γ + µ Γ − κ Γ − q Γ − . Then the boundary of the 1-dimensional moduli spaces M Y,A,o (Γ + , Γ − ) would yield that { p k,l • } is a pointedmorphism of degree 0. Note that M Y,A,o (Γ + , Γ − ) counts holomorphic curves with a point constraint in thesymplectization with a s -independent J , therefore in the level containing M Y,A,o (Γ + , Γ − ) in a rigid breaking,there is only one nontrivial component.T T p , • p , T T p , p , • Figure 8. (cid:98) p • ◦ (cid:98) p = (cid:98) p ◦ (cid:98) p • For a strict exact cobordism X from Y − to Y + , if we choose a path γ from o − ∈ Y − to o + ∈ Y + . then wecan complete the path γ to a proper (cid:98) γ path in (cid:98) X by constants in the cylindrical ends. Then we claim thatthe pointed morphisms p • , + , p • , − determined by o − , o + and BL ∞ morphism φ from X are compatible, with φ • given by(3.8) φ k,l • ( q Γ + ) = (cid:88) | [Γ − ] | = l M X,A,γ (Γ + , Γ − ) T (cid:82) A ω µ Γ + µ Γ − κ Γ − q Γ − . LANDSCAPE OF CONTACT MANIFOLDS VIA RATIONAL SFT 25 T φ , φ , p , • C φ , T p , • φ , CT φ , φ , • p , C φ , • T p , φ , C Figure 9. (cid:98) q • ◦ (cid:98) φ − (cid:98) φ ◦ (cid:98) p • = (cid:98) φ • ◦ (cid:98) p + (cid:98) q ◦ (cid:98) φ • The main theorem of this section is that after fixing auxiliary choices depending on the choice of virtualmachinery, we almost have a functor from strict contact cobordism category (with auxiliary choice) to thecategory of BL ∞ algebras . Theorem 3.9.
Let ( Y, α ) be a strict contact manifold with a non-degenerate contact form, then we have thefollowing.(1) There exists a non-empty set of auxiliary data Θ , such that for each θ ∈ Θ we have a BL ∞ algebra p θ on V α .(2) For any point o ∈ Y , there exists a set of auxiliary data Θ o with a surjective map Θ o → Θ , such thatfor any θ o ∈ Θ o , we have a pointed map p • ,θ o for p θ , where θ is the image of θ o in Θ o → Θ .(3) When there is a strict exact cobordism X from ( Y (cid:48) , α (cid:48) ) to ( Y, α ) . Let Θ , Θ (cid:48) be the sets of auxiliarydata for α, α (cid:48) , there exist a set of auxiliary data Ξ with a surjective map Ξ → Θ × Θ (cid:48) , such that for ξ ∈ Ξ , there is a BL ∞ morphism φ ξ from ( V α , p θ ) to ( V α (cid:48) , p θ (cid:48) ) , where ( θ, θ (cid:48) ) is the image of ξ under Ξ → Θ × Θ (cid:48) .(4) Assume in addition, we fix a point o (cid:48) ∈ Y (cid:48) that is in the same component of o in X , then for anycompatible auxiliary data θ, θ (cid:48) , θ o , θ o (cid:48) , ξ , we have p • ,θ o , p • ,θ o (cid:48) , φ ξ are compatible. The composition is not discussed, nor is needed for our application (5) For compatible auxiliary data θ, θ o , there exists compatible auxiliary data kθ, kθ o for ( Y, kα ) for k ∈ R + , such that p kθ , p • ,kθ o are identified with p θ , p • ,θ o by the canonical identification between V α and V kα . To make sense of M we need to fix a choice of virtual machinery, the meaning of auxiliary data alsodepends on the choice. If one adopts the perturbative scheme in [33, 47], Theorem 3.9 is a special case oftheir main constructions. On the other hand, since we only consider rational curves, the combinatorics isnot essentially different from the construction of differentials and morphisms in [66]. In particular, Pardon’sVFC works in a verbatim account. We will explain the VFC construction to prove Theorem 3.9 and discussother virtual techniques in § Augmentations and linearized theories.
We start this section with following definition.
Definition 3.10.
For a strict contact manifold ( Y, α ) , we fix an auxiliary choice θ ∈ Θ , then we definealgebraic planar torsion APT(
Y, α, θ ) to be the torsion of the BL ∞ algebra ( V α , p θ ) over Q . As a consequence of Proposition 2.13 and Theorem 3.9, we have APT(
Y, α, θ ) is an invariant for Y in thefollowing sense. Proposition 3.11.
APT(
Y, α, θ ) is independent of α, θ , hence can be abbreviated as APT( Y ) . Moreover, APT :
Con → N ∪ {∞} is monoidal functor, where the monoidal structure on N ∪ {∞} is given by a ⊗ b =min { a, b } .Proof. By (5) of Theorem 3.9, we have ( V α , p θ ) = ( V kα , p kθ ) for any k ∈ R + . Let α (cid:48) be another contact form, θ (cid:48) is an auxiliary data. Then there exists k , k , such that there are strict cobordisms from ( Y, k α ) , ( Y, α (cid:48) )to (
Y, α (cid:48) ) , ( Y, k α ) respectively. Then by (2) of Theorem 3.9 and Proposition 2.13, we have APT( Y, α, θ ) =APT(
Y, α (cid:48) , θ (cid:48) ). For ( Y , α , θ ) , ( Y , α , θ ), the BL ∞ algebra for the disjoint union ( Y (cid:116) Y , α (cid:116) α , θ × θ )is given by ( V α ⊕ V α , { p k,lθ ⊕ p k,lθ } ), i.e. there are no mixed structure maps. Then it is clear that APT( V α ⊕ V α , { p k,lθ ⊕ p k,lθ } ) = min { APT( V α , p θ ) , P T ( V α , p θ ) } . That APT is a functor follows from (2) of Theorem3.9. (cid:3) When APT( Y ) = 0, it is equivalent to that H ∗ CHA( Y ) = 0, which is also known as algebraicallyovertwisted [11] or 0-algebraic torsion [48], and is implied by overtwistedness [15, 77]. Similarly, we candefine APT Λ ( Y ) to be the order of torsion for RSFT( Y ) over the Novikov field Λ. Proposition 3.12.
APT Λ ( Y ) = APT( Y ) .Proof. To differentiate the two vector spaces, we use V α to denote the Q -space and V Λ α to denote the Λ space.For an element q γ ∈ V Λ α , we define the weight w ( q γ ) := (cid:82) γ ∗ α and w ( T A ) := A . Then we define the weight(uniquely) on SV Λ α and EV Λ α by the following properties w ( xy ) = w ( x ) + w ( y ) , w ( x (cid:12) y ) = w ( x ) + w ( y ) and w ( x + y ) ≤ max { w ( x ) , w ( y ) } . Then by (3.5), (cid:98) p preserves the weight on EV Λ α . Now assume APT( Y ) = k < ∞ ,we have 1 = (cid:98) p ( x ) for x ∈ EV α . Then x induces an element x in EV Λ α by sending each q γ to T − (cid:82) γ ∗ α q γ .Therefore x has pure weight 0, i.e. x is a sum of monomials with weight 0. We know that (cid:98) p ( x ) also has pureweight 0. Since (cid:98) p ( x ) | T =1 = (cid:98) p ( x ) = 1, we must have (cid:98) p ( x ) = 1. This proves that APT Λ ( Y ) ≤ k , in particular,APT Λ ( Y ) ≤ APT( Y ). On the other hand, assume APT Λ ( Y ) = k < ∞ , i.e. 1 = (cid:98) p ( x ) for x ∈ EV Λ α . Then wecan assume x has pure weight 0. Since x is in SSV Λ α , x is written as finite linear combination of terms inthe form of w (cid:12) . . . (cid:12) w k with w i = q γ . . . q γ j . Because x has pure weight, we must have the coefficient ofeach those terms is a monomial T A . In particular, it makes sense to define x ∈ EV α by x | T =1 and (cid:98) p ( x ) = 1.Therefore APT( Y ) ≤ APT Λ ( Y ), which finishes the proof. (cid:3) LANDSCAPE OF CONTACT MANIFOLDS VIA RATIONAL SFT 27
Since finite order of torsion is an obstruction to augmentations, finite algebraic planar torsion is anobstruction to symplectic fillings in view of the following.
Proposition 3.13.
Let ( Y, α ) be a strict contact manifold with an auxiliary data θ .(1) If ( W, d λ ) is a strict exact filling, then there is a BL ∞ augmentation to ( V α , p θ ) over Q .(2) If ( W, ω ) is a strict strong filling, then there is a BL ∞ augmentation to ( V α , p θ ) over Λ . Proof.
The augmentation is defined by (cid:15) k ( q Γ + ) = (cid:88) A M W,A (Γ + , ∅ ) T (cid:82) A ω µ Γ + , where | Γ + | = k . The remaining follows from the same argument in Theorem 3.9. In the exact case, it is aspecial case of the cobordism case of Theorem 3.9. (cid:3) Remark 3.14.
The theory with Λ coefficient considered in this paper is a naive version, and can be trans-ferred to and recovered from the Q coefficient version. The more correct version for Λ coefficient theoryshould be completions with respect to the weight considered in the proof of Proposition 3.12. For strongsymplectic cobordisms with non-empty negative boundary, it is necessary to use the completion to describeMaurer-Cartan elements [22] . If we use EV Λ α to denote the completion of EV Λ α and E k V Λ α to denote thecompletion of E k V Λ α . Then it is no longer true that ∈ H ∗ ( E k V Λ α ) implies that ∈ H ∗ ( E k V α ) .Moreover, ∈ H ∗ ( EV Λ α ) does not imply that there is a k > such that ∈ H ∗ ( E k V Λ α ) . Corollary 3.15. If APT( Y ) < ∞ , then Y has no strong filling.Proof. If APT( Y ) = APT Λ ( Y ) < ∞ , then there is no BL ∞ augmentation over Λ by Proposition 2.11. Thenproposition 3.13 implies the claim. (cid:3) Roughly speaking the algebraic planar torsion looks at rigid curves with multiple positive punctures andno negative puncture. One particular situation, where we can infer information of planar algebraic torsion,is the planar torsion introduced by Wendl [73], which generalizes overtwisted contact structures and theGiroux torsion in dimension 3. The following two results were essentially proven in [48].
Theorem 3.16. If Y is a -dimensional contact manifold with planar torsion of order k , then APT( Y ) ≤ k .Proof. This follows from the same argument of [48, Theorem 6] based on a precise description of low energycurves in [48, Proposition 3.6], see also [73]. In fact, we do not need the genus > (cid:3) Theorem 3.17.
For any k ∈ N , there exists a -dimensional contact manifold Y with APT( Y ) = k .Proof. This follows from the same argument of [48, Theorem 4]. In fact, we only need the genus 0 part of[48, Lemma 4.15] to get a lower bound. (cid:3)
Remark 3.18.
Following from [48, Corollary 1] , there are examples Y i with planar torsion of order k , suchthat there is exact cobordism from Y i to Y i +1 but no exact cobordism from Y i +1 → Y i . On the other hand,there is always a strong cobordism from Y i +1 to Y i by [74, Theorem 1] . We will see similar phenomena inhigher dimensions in § § Note that p θ in Q coefficient and Λ coefficient are different. But we can recover one from the other as the weight is determinedby contact action. Remark 3.19.
It is an interesting question to understand the relations between the algebraic planar torsionand algebraic torsion. First we recall the BV ∞ description of SFT and the definition of algebraic torsionfrom [48] . Let SV α [[ (cid:126) ]] be the algebra of power series in (cid:126) with coefficients in SV α . Then we have a full SFTdifferential defined as follows D SFT q Γ = ∞ (cid:88) g =0 (cid:88) A (cid:88) Γ (cid:48) | Γ | (cid:88) k =1 (cid:126) g + k − n A,g (Γ , Γ (cid:48) ; k ) q Γ (cid:48) where n A,g (Γ , Γ (cid:48) ; k ) is the count of holomorphic curves, which possibly have disconnected components butonly one nontrivial component of k positive punctures, genus g , homology class A , and positive/negative asy-motitics Γ and Γ (cid:48) . Then we say Y admits a k -algebraic torsion iff (cid:126) k is in the homology of ( SV α [[ (cid:126) ]] , D SFT ) .Let’s consider the simplest case with an algebraic planar -torsion, i.e. there are two generators q , q suchthat p , ( q , q ) = 1 , p ,l ( q , q ) = 0 for all l > , and p ,l ( p i ) = 0 for all l ≥ and i = 1 , . That is we know (cid:80) A n A,g (Γ , Γ (cid:48) ; k ) for g = 0 and Γ = { γ , γ } . The natural candidate for algebraic torsion is q q , and wecompute D SFT q q = ∞ (cid:88) g =0 (cid:88) A (cid:88) Γ (cid:48) (cid:88) k =1 (cid:126) g + k − n A,g ( { γ , γ } , Γ (cid:48) ; k ) q Γ (cid:48) = (cid:126) + ∞ (cid:88) g =1 (cid:88) A (cid:88) Γ (cid:48) (cid:88) k =1 (cid:126) g + k − n A,g ( { γ , γ } , Γ (cid:48) ; k ) q Γ (cid:48) Since we have no knowledge of n A,g for g > in RSFT, one should not expect that q q is a primitive of (cid:126) in D SFT . We note here the above consideration is a very special case and in general an algebraic planar k -torsion is not equivalent to that (cid:126) k is the image of the genus term of D SFT . In fact, the algebraic torsionand algebraic planar torsion can be viewed as two “independent” axes in a grid of torsions, see § Remark 3.20.
One can define BL ∞ algebras over group rings as in [48] for stable Hamiltonian fillings andweak symplectic fillings. Then the finiteness of algebraic planar torsion in this setup is an obstruction tostable/weak fillings. One example with finite algebraic planar torsion in the group ring setup is those withthe fully separating planar k -torsions [73, Definition 1.3] , where the finiteness follows from the same proofof [48, Theorem 6 (2)] . Giroux was generalized to higher dimensions in [53], the following theorem is a reformulation of [58,Theorem 1.7].
Theorem 3.21. If Y has Giroux torsion, then APT( Y ) ≤ . Now we assume ( V α , p θ ) does have a BL ∞ augmentation (cid:15) over Q , then APT( Y ) is ∞ . In view of § o ∈ Y and an auxiliary data θ o , which give rise to a pointed morphism p • ,θ o .Hence we can define the order O ( V α , (cid:15), p • ,θ o ). In the following, we use Aug Q ( V α ) to denote the set of BL ∞ augmentations over Q . Definition 3.22.
For a strict contact manifold ( Y, α ) with auxiliary data θ , we define O ( Y, α, θ ) := max (cid:8) O ( V α , (cid:15), p • ,θ o ) (cid:12)(cid:12) ∀ (cid:15) ∈ Aug Q ( V α ) , o ∈ Y, θ o ∈ Θ o (cid:9) , where the maximum of an empty set is defined to be zero. LANDSCAPE OF CONTACT MANIFOLDS VIA RATIONAL SFT 29 p , T (cid:15) (cid:15) Figure 10.
A component of (cid:96) (cid:15) Proposition 3.23. O ( Y, α, θ ) is independent of α and θ , hence will be abbreviated as P( Y ) and is calledthe planarity of Y . Moreover, P :
Con → N ∪ {∞} is a monoidal functor, where the monoidal structure on N ∪ {∞} is given by ⊗ a = 0 , ∀ a and a ⊗ b = max { a, b } , ∀ a, b ≥ Proof.
We first show that if there is a strict exact cobordism X from ( Y − , α − ) to ( Y + , α + ), then O ( Y + , α + , θ + ) ≥ O ( Y − , α − , θ − ) for any θ + , θ − . For any o − ∈ Y − , there exists a point o + ∈ Y + , such that there is apath in X connecting o + and o − . Then by (4) of Theorem 3.9 and Proposition 2.22, for any aug-mentation (cid:15) to V α − and auxiliary data θ o − , there exists an auxiliary data ξ ∈ Ξ and θ o + , such that O ( V α + , (cid:15) ◦ φ ξ , p • ,θ o + ) ≥ O ( V α − , (cid:15), p • ,θ o − ). Hence O ( Y, α + , θ + ) ≥ O ( Y, α − , θ − ). Then by the same argu-ment in Proposition 3.11, O ( Y, α, θ ) is independent of α and θ and P is a functor. To prove the monoidalstructure, we first note that ( V ⊕ V , p ⊕ p ) has an augmentation iff V , V both have augmentations, sincethe natural inclusion V → V ⊕ V defines a BL ∞ morphism. This verifies the case for 0 ⊗ a . When both Y and Y has augmentations, it follows from definition that P( Y (cid:116) Y ) = max { P( Y ) , P( Y ) } . (cid:3) Remark 3.24.
Similarly, we can define another invariant (cid:101) P( Y ) using the maximum of ˜ O ( V, (cid:15), p • ) . Thenwe have P( Y ) ≤ ˜P( Y ) by Proposition 2.19. Similarly, we define P Λ ( Y ) using augmentations over Λ, it is not clear to us whether P( Y ) = P Λ ( Y )except the obvious relation P( Y ) ≤ P Λ ( Y ). Since any augmentation over Q will induce an augmentationover Λ, where the degree in T is declared to the contact action of the positive asymptotics. However, aΛ-augmentation may not induce a Q -augmentation. Since finite algebraic planar torsion is an obstruction to BL ∞ augmentation over both Q and Λ, we have that APT( Y ) ≤ ∞ implies that P( Y ) / P Λ ( Y ) = 0. SinceP( Y ) / P Λ ( Y ) = 0 is precisely those without augmentations, the algebraic planar torsion is the inner hierarchyinside P( Y ) / P Λ ( Y ) = 0. However it is still possible (at least on the algebraic level) that P( Y ) / P Λ ( Y ) = 0but APT( Y ) = ∞ , i.e. there is no augmentation nor finite torsion.3.6. Implementation of virtual techniques.
In the following, we will explain how to get the algebraiccount of moduli spaces in Theorem 3.9 using virtual techniques. Any choice of virtual machinery should givea construction of P and APT with the claimed properties, although it is not clear whether different virtualtechniques give rise to the same P and APT. However, the geometric results, examples and applications inthis paper, do not depend on the choice, as we have the following axiom for virtual machinery, which holdsfor any one of the virtual techniques mentioned in this paper.
Axiom 3.25.
A virtual implementation of a holomorphic curve theory has the property that the virtual countof a compactified moduli space equals to the geometric count, when transversality holds for that moduli space.
In the following, we will finish the proof of Theorem 3.9 by implementing Pardon’s implicit atlas and virtualfundamental cycles [66]. The construction is essentially the constructions of contact homology algebra andmorphisms in [66]. As explained in [65, § R modules. We first introduce a category R which will play the same role of S I in [66] to governthe combinatorics of rational holomorphic curves in the symplectization. The objects of R are connectednon-empty directed graphs without cycles, such that each vertex has at least one incoming edge. Edges withmissing source, i.e. input edges, and edges with missing sink, i.e. output edges are allowed. Those edges arecalled external edges and all other edges are called interior edges. The graph T is equipped with decorationsas follows.(1) For each edge e ∈ E ( T ), a Reeb orbit γ e .(2) For each vertices v ∈ V ( T ), a relative homology class β v ∈ H ( Y, { γ e + } e + ∈ E + ( v ) (cid:116) { γ e − } e − ∈ E − ( v ) ),where we denote by E + ( v ) the set of incoming edges at v and E − ( v ) the set of outgoing edges at v ,which can be empty.(3) For each external edge e ∈ E ext ( T ), a basepoint b e ∈ im γ e .A morphism π : T → T (cid:48) in R consists first of a contraction of the underlying graph of T to T (cid:48) by collapsingsome of the interior edges of T . The decorations have the following property.(1) For each non-contracted edge e ∈ E ( T ), we have γ π ( e ) = e .(2) For each vertex v (cid:48) ∈ V ( T (cid:48) ), we have β v (cid:48) = π ( v )= v (cid:48) β v Finally, we specify for each external edge e ∈ E ext ( T ) = E ext ( T (cid:48) ) a path along im γ e between the basepoints b e and b (cid:48) e modulo the relation that identifies such two paths iff their difference lift to γ e . In particular,there are exactly κ γ e different equivalences of paths. Then automorphism group of T with a single vertexis a product of cyclic groups and symmetric groups with cardinality µ Γ + µ Γ − κ Γ + κ Γ − . For T (cid:48) → T , we useAut( T (cid:48) /T ) denote the subgroup of Aut( T (cid:48) ) compatible with T (cid:48) → T .A concatenation in R consists of a finite non-empty collection of objects T i ∈ R along a matching betweensome pairs of output edges and input edges with matching orbit label, such that the resulting gluing is adirected graph without cycles, along with a choice of paths between the basepoints for each pair of matchingedges. Given a concatenation { T i } i in R , there is a resulting object i T i ∈ R . A morphism of concatenations { T i } i → { T (cid:48) i } i means a collection of morphisms T i → T (cid:48) i covering a bijection of index sets. Then a morphism { T i } i → { T (cid:48) i } i induces a morphism i T i → i T (cid:48) i . If { T i } i is a concatenation and T i = j T ij for someconcatenation { T ij } j , then there is a resulting composite concatenation { T ij } ij with natural isomorphisms ij T ij = i j T ij = i T i . We use Aut( { T i } i / i T i ) to represent the group of automorphism of { T i } i actingtrivially on i T i , i.e. the product (cid:81) e Z κ γe over junction edges.The key concept to organize the moduli spaces, implicit atlases, and virtual fundamental cycles is thefollowing R -module. Definition 3.26 ([66, Definition 4.5]) . A R -module X valued in a symmetric monoidal category C ⊗ consistsof the following data. LANDSCAPE OF CONTACT MANIFOLDS VIA RATIONAL SFT 31 (1) A functor X : R → C .(2) For every concatenation { T i } i in R , a morphism ⊗ i X ( T i ) → X ( i T i ) , such that the following diagrams commute: ⊗ i X ( T i ) (cid:47) (cid:47) (cid:15) (cid:15) X ( i T i ) (cid:15) (cid:15) ⊗ i X ( T (cid:48) i ) (cid:47) (cid:47) X ( i T (cid:48) i ) ⊗ i X ( T i ) (cid:38) (cid:38) ⊗ i,j X ( T ij ) (cid:47) (cid:47) (cid:56) (cid:56) X ( ij T ij ) for any any morphism of concatenations and composition of concatenations. Example 3.27.
A holomorphic building of type T ∈ R consists of the following data.(1) For every vertex v , a closed, connected nodal Riemann surface of genus zero C v , along with distinctpoints { p v,e ∈ C v } e indexed by the edges incident at v and a J holomorphic map u v : C v \{ p v,e } e → R × Y up to the R translation.(2) u v converges to γ e + near p v,e + in the sense of (3.1) for e + ∈ E + ( v ) and converges to γ e − near p v,e − in the sense of (3.2) . We use ( u v ) p v,e : S → Y to denote the Y -component of the limit map nearpunctures.(3) For every input/output edge e , an asymptotic marker L e ∈ S p v,e C v which is mapped to the basepoint b e by ( u v ) p v,e .(4) For every interior edge v e → v (cid:48) , a matching isomorphism m e : S p v,e C v → S p v (cid:48) ,e C v (cid:48) intertwining ( u v ) p v,e and ( u v (cid:48) ) p v (cid:48) ,e .An isomorphism between two buildings is a collections of isomorphisms between C v commuting with all thedata. Then we define M ( T ) to be the set of isomorphism classes of holomorphic buildings of type T . Notethat Aut( T ) acts on M ( T ) by changing markings. Then we define M ( T ) := (cid:71) T (cid:48) → T M ( T (cid:48) ) / Aut( T (cid:48) /T ) . The union is over the set of isomorphism classes in the over category R /T . Moreover, M ( T ) is endowed withthe Gromov topology and is a compact Hausdorff space [66, § . Note that here for each v ∈ V ( T ) ,we view u v as a curve in its own copy of symplectization. In particular, we have no level structure andthe topology is slightly different from the buildings in [9] by forgetting all trivial cylinders. However thisposes no difference for the compactness. In particular, there is a surjective map from the compactification in [9] to M ( T ) by collapsing the boundary containing levels with multiple disconnected nontrivial curves intocorners. The functor M is a R module in the category of compact Hausdorff spaces with disjoint union as themonoidal structure. The natural map M ( T ) → R /T is a stratification in the sense of [66, Definition 2.14] .We define vdim( T ) as (cid:80) v ∈ V ( T ) (ind( u v ) − and codim( T (cid:48) /T ) is the number of interior edges collapsed in T (cid:48) → T . Then we have codim( T (cid:48) /T ) + vdim( T (cid:48) ) = vdim( T ) . Example 3.28.
For each non-degenerate Reeb orbit γ (good or bad) and a basepoint b ∈ im γ , [66, Definition2.46] constructs a canonical Z graded line o γ,b with grading µ CZ ( γ ) + n − . Any path b → b (cid:48) givesrise to a functorial isomorphism o γ,b → o γ,b (cid:48) , two paths induces the same isomorphism if the difference canbe lift to γ . As a consequence, Z κ γ acts on o γ,b . Then γ is good iff the action is trivial. Let T be a tree,then we have the determinant line o ◦ T of the linearized Cauchy-Riemann operator at the vertices and conical isomorphism from o ◦ T to ⊗ e + ∈ E + ( v ) o γ e + ,b e + ⊗ e − ∈ E − ( v ) o ∨ γ e − ,b e − . Moreover, o ◦ is a R -module [65, Example4.7] . We define o T by o ◦ T ⊗ ( o ∨ R ) V ( T ) . Moreover, for T (cid:48) → T , there is an induced isomorphism o T (cid:48) → o T by [66, (2.61)] . An object T ∈ R is called effective iff M ( T ) (cid:54) = ∅ . Then for any morphism T → T (cid:48) , if T is effective, so is T (cid:48) . For any concatenation { T i } i , every T i is effective iff i T i is effective. In the following, R will mean thefull subcategory spanned by effective objects, which depends on J . Then R has the following properties,which allows one to apply inductive constructions.(1) Every T can be written as a concatenation of maximal elements i T i , an element T i is maximal iffthere is only isomorphism mapping out of T i . That is T i has only one interior vertex.(2) Let T, T (cid:48) ∈ R , we say T (cid:48) (cid:52) T iff there is a morphism i T → T with some T i isomorphic to T (cid:48) . Thenthere is no infinite strictly decreasing sequences. This is a consequence of compactness or positivityof contact energy (3.3) in the exact cobordism setting of this paper.As a consequence, a lot of the constructions can be built inductively from the minimal elements in ( R , (cid:52) ). Note that maximal T is not necessarily maximal in (cid:52) , but minimal elements of ( R , (cid:52) ) are necessarilymaximal. Example 3.29.
Thickening datum defined in [66, Definition 3.9] works verbatim for our purpose. Then wehave the set of thickening datums A ( T ) . Then A ( T ) := (cid:71) T (cid:48) ⊆ T A ( T (cid:48) ) , where the disjoint union is over all connected subgraphs that are in R . Then clearly A is an R op -module tothe category of sets. Proposition 3.30. M ( T ) is equipped with an implicit atlas A ( T ) with oriented cell-like stratification.Proof. First of all, we have the space of gluing parameters G T/ that associates to each interior edge a numberin (0 , ∞ ]. Since there is no cycles in T , there is no relations among those gluing parameters. In particular G T/ has a cell-like stratification like ( G I ) T/ in [66, Lemma 3.5]. Then the claim follows from the same proofof [66, Theorem 3.23]. The analogues of [66, Theorem 3.31, 3.32] hold for our setup since we only glue onepuncture at a time, hence the gluing analysis in [66, §
5] applies in a verbatim way. (cid:3)
With the existence of implicit atlas with cell-like stratification, the machinery of virtual fundamentalcycles induces a pushforward map(3.9) C ∗ +vdim( T )vir ( M ( T ) rel ∂ ; A ( T )) → C −∗ ( E ; A ( T )) , where E is part of the datum in A ( T ). The remaining of the construction is hinged on purely combinatorialproperties.Following the same procedure of [66, Definition 4.19], there is a canonical construction of R module C ∗ +vdimvir ( M rel ∂ ) by homotopy colimit in the category of cochain complexes such that C ∗ +vdim( T ) vir ( M ( T ) rel ∂ )is quasi-isomorphic to C ∗ +vdimvir ( M ( T ) rel ∂ ; A ( T )). Similarly, by the homotopy colimit as in [66, Definition4.20], there is a R module C ∗ ( E ) and (3.9) leads to a canonical map of R -modules C ∗ +vdimvir ( M rel ∂ ) → C −∗ ( E ). Similar to [66, Definition 4.14], there is R module Q [ R ] governing the boundary information, Q [ R ]( T ) := Q [ R /T ] = (cid:77) T (cid:48) → T o T (cid:48) [vdim( T (cid:48) )] LANDSCAPE OF CONTACT MANIFOLDS VIA RATIONAL SFT 33 with the differential given by the sum of all codimension one maps T (cid:48)(cid:48) → T (cid:48) in R /T of boundary map o T (cid:48) → o T (cid:48)(cid:48) in Example 3.28. Then Q [ R ] is again cofibrant in the sense of [66, Definition 4.24] by thesame argument of [66, Lemma 4.26]. There is a cofibrant replacement C cof ∗ ( E ) with a quasi-isomorphism C cof ∗ ( E ) ∼ → C ∗ ( E ), which is surjective on maximal T by induction on the partial order (cid:52) by [66, Definition4.28]. Definition 3.31.
Given α, J , an element of Θ( α, J ) consists a commuting diagram of R -modules (3.10) Q [ R ] ˜ w ∗ (cid:47) (cid:47) w ∗ (cid:15) (cid:15) C cof −∗ ( E ) ∼ (cid:15) (cid:15) p ∗ (cid:47) (cid:47) Q C ∗ +vdimvir ( M rel ∂ ) (cid:47) (cid:47) C −∗ ( E ) satisfying the following properties.(1) p ∗ induces the canonical isomorphism H cof ∗ ( E ) = H ∗ ( E ) = Q .(2) w ∗ satisfies the property that for any T ∈ R , w ∗ ∈ Hom R /T ( Q [ R ] , C ∗ +vdimvir ( M rel ∂ )) on cohomologylevel represents the constant function ∈ ˇ H ( M ( T )) under the identification in [66, Lemma 4.23] .Proof of (1) of Theorem 3.9. In the context of VFC, Θ( α ) = (cid:116) J Θ( J, Θ( α, J )). Moreover Θ( α, J ) is notempty. The existence of p ∗ follows from [66, Lemma 4.30], the existence of w ∗ follows from [66, Lemma4.31], the existence of lifting ˜ w ∗ follows from [66, Proposition 4.34], as the induction on (cid:52) can be applied.Given a diagram (3.10), we have a R module map p ∗ ◦ ˜ w ∗ : Q [ R ] → Q , which is assigning each T withvdim( T ) = 0 an element M ( T ) vir ∈ ( o ∨ T ) Aut( T ) , which after fixing a trivialization of o γ,b for every Reeborbits is a rational number. If an exterior edge of T is labeled by a bad orbit, then being Aut( T ) invariantimplies that M ( T ) vir = 0. Finally, being a R module implies that0 = (cid:88) codim( T (cid:48) /T )=1 | Aut( T (cid:48) /T ) | M ( T (cid:48) ) vir (3.11) M ( i T i ) vir = 1 | Aut( { T i } i / T i ) | (cid:89) i M ( T i ) vir (3.12)Let T be a tree with one interior vertex, k input edges labeled by Γ + , and l output edges labeled byΓ − . If vdim( T ) = 0, then we define q Γ − coefficient of p k,l ( q Γ + ), i.e. (cid:104) p k,l ( q Γ + ) , q Γ − (cid:105) , by | κ Γ+ || Aut( T ) | M ( T ) vir .In view of Proposition 2.6, we need to prove that (cid:104) p k,l ( q Γ + ) , q Γ − (cid:105) is zero for any multisets Γ + , Γ − with | Γ + | = k, | Γ − | = l . We claim that (cid:104) p k,l ( q Γ + ) , q Γ − (cid:105) = 1 µ Γ + µ Γ − κ Γ − (cid:88) codim( T (cid:48) /T )=1 M ( T (cid:48) ) vir = 0for maximal T with vdim( T ) = 1. AssumeΓ + := { γ , . . . , γ (cid:124) (cid:123)(cid:122) (cid:125) k , . . . , γ m , . . . , γ m (cid:124) (cid:123)(cid:122) (cid:125) k m } , Γ − := { η , . . . , η (cid:124) (cid:123)(cid:122) (cid:125) l , . . . , η s , . . . , η s (cid:124) (cid:123)(cid:122) (cid:125) l s } for (cid:80) mi =1 k i = k , (cid:80) si =1 l i = l and γ i (cid:54) = γ j , η i (cid:54) = η j for i (cid:54) = j . For T (cid:48) with codim( T (cid:48) /T ) = 1, then T (cid:48) has two vertices with one connecting interior edge. We can cut out the interior edge to obtain T (cid:48) , T (cid:48) with the interior edge turning into an output edge for T (cid:48) , i.e. T (cid:48) = T (cid:48) T (cid:48) . Assume the input edges of T (cid:48) is labeled by ˜Γ + := { γ , . . . , γ (cid:124) (cid:123)(cid:122) (cid:125) k (cid:48) , . . . , γ m , . . . , γ m (cid:124) (cid:123)(cid:122) (cid:125) k (cid:48) m } for k (cid:48) i ≥ T (cid:48) is labeled by˜Γ − := { η , . . . , η (cid:124) (cid:123)(cid:122) (cid:125) l (cid:48) , . . . , η s , . . . , η s (cid:124) (cid:123)(cid:122) (cid:125) l (cid:48) s } for l (cid:48) i ≥
0, Then there are (cid:81) mi =1 (cid:0) k i k (cid:48) i (cid:1) (cid:81) si =1 (cid:0) l i l (cid:48) i (cid:1) many codimension one T (cid:48) with such labels. Since Aut( T (cid:48) /T ) = 1, by (3.12), the contribution to (3.11) from T (cid:48) with such labelsmultiplying µ Γ+ µ Γ − κ Γ − is(3.13) (cid:88) γ d − γ · d + γ · (cid:104) p | ˜Γ + | , | (˜Γ − ) c | +1 ( q ˜Γ + ) , q (˜Γ − ) ∪{ γ } (cid:105) · (cid:104) p | ˜Γ + | , | (˜Γ − ) c | +1 ( q (˜Γ + ) c ∪{ γ } ) , q ˜Γ (cid:105) where γ is the label on the interior edge of T (cid:48) with d − γ is the number of γ in (˜Γ − ) c ∪ { γ } and d + γ is thenumber of γ in (˜Γ + ) c ∪ { γ } . Then (3.13) is the part of (cid:104) p k,l ( q Γ + ) , q Γ − (cid:105) , as we have d − γ · d + γ many ways ofgluing that arise in p k,l . Then the sum of all T (cid:48) yields that (cid:104) p k,l ( q Γ + ) , q Γ − (cid:105) = 0. Hence p k,l gives a BL ∞ structure. (cid:3) The next proposition follows from [66, Proposition 4.33]. It is Axiom 3.25 in the context of VFC.
Proposition 3.32. If M ( T ) is cut out transversely with vdim( T ) = 0 , then M ( T ) vir = M ( T ) = M ( T ) for any θ ∈ Θ( α, J ) . R II , R • and R • II modules. In the following, we introduce R II , R • and R • II to govern moduli spacesas well virtual fundamental cycles for BL ∞ morphisms, pointed maps and the homotopy in Definition 2.21.(1) The category R II is the analogue of S II in [66]. The objects of R II are graphs without cycles asbefore, but now each edge e ∈ E ( T ) is labeled with a symbol ∗ ( e ) ∈ { , } such that all inputedges are labeled with 0 and all output edges are labeled with 1. For each vertex v ∈ V ( T ), weassociate it with a pair of symbols ∗ ± ( e ) ∈ { , } such that ∗ + ( v ) ≤ ∗ − ( v ) and ∗ ( e ± ( v )) = ∗ ± ( v ).If ∗ + ( v ) = ∗ − ( v ), then v is called a symplectization vertex and if ∗ + ( v ) < ∗ − ( v ), then v is calleda cobordism vertex. Given an exact cobordism X from Y − to Y + , for every T ∈ R II , we cansimilarly define the moduli space M II ( T ), where the curve attached to a symplectization vertex v with ∗ ± ( v ) = 0 is a holomorphic curve in R × Y + modulo R -translation, the curve attached to asymplectization vertex v with ∗ ± ( v ) = 1 is a holomorphic curve in R × Y + modulo R -translation, thecurve attached to a cobordism vertex v is a holomorphic curve in (cid:98) X . Then we have the analogouscompactification M II ( T ) using the over category over T , which is a R II module.(2) The category R • is similar to R but with exactly one vertex labeled by • . The morphism in R • isagain contractions of graphs such that the • vertex is mapped to the • vertex. For every T ∈ R • ,we can associate a moduli space M • ( T ), which is defined similar to M ( T ) but the map associatedto • vertex is holomorphic curve with a marked point mapped to the fixed point (0 , o ) ∈ R × Y . Wecan similarly define the compactified moduli spaces M • ( T ), which is R • module.(3) The category R • II is the combination of R II and R • , i.e. the objects are the same as R II with oneof the vertex is marked with • . In the definition of M • II ( T ), the curve attached to the • vertex is acurve in the symplectization with a point constraint if the vertex is a symplectization vertex and isa curve in the cobordism with the path constraint if the vertex is a cobordism vertex. Proof of the rest of Theorem 3.9.
We need to argue that M II ( T ) , M • ( T ) , M • II ( T ) are equipped with im-plicit atlases with oriented cell-like stratification. For this, we only need to argue that the gluing parameter LANDSCAPE OF CONTACT MANIFOLDS VIA RATIONAL SFT 35 spaces are cell-like like, the remaining of the argument is the same as [66, Theorem 3.23]. The gluing pa-rameter space ( G • ) T/ for R • T/ is same as the G T/ , i.e. (0 , ∞ ] E int ( T ) , since there is no relations among gluingparameters. The gluing parameter space ( G II ) T/ for ( R II ) T/ is defined as a subset of (cid:110) ( { g e } e , { g v } v ) ∈ (0 , ∞ ] E int, ( T ) × [ −∞ , E int, ( T ) × (0 , ∞ ] V ( T ) × [ −∞ , V ( T ) (cid:111) , subject to the constraints g v = g e + g v (cid:48) , for v e → v (cid:48) with ∗ ( e ) = 0 , g v (cid:48) = g e + g v , for v e → v (cid:48) with ∗ ( e ) = 0 . where g v is interpreted as 0 if v ∈ V ( T ), V ij ( T ) is the set of vertices with ∗ + ( v ) = i, ∗ − ( V ) = j and E int,i ( T ) is the set of interior edges e such that ∗ ( e ) = i . Then g v can be viewed as the height of the vertex v for v ∈ V ij ( T ), where the heights of all cobordism vertices are 0, as all of them are placed in the same level.Following the argument of [66, Lemma 3.6], it is sufficient to prove ( G II ) T/ is a topological manifold withboundary. We can perform the the same change of coordinates h = e − g ∈ [0 ,
1) for v ∈ V ( T ) , e ∈ E int, ( T ),and h = e g ∈ [0 ,
1) for v ∈ V ( V ) , e ∈ E int, ( T ). We allow h ∈ [0 , ∞ ) for convenience. Then the relationbecomes h v = h e h v (cid:48) for ∗ ( e ) = 0 and h v (cid:48) = h e h v for ∗ ( e ) = 1. Now the difference with [66, Lemme 3.6]is that we do not have v max , which in contact homology corresponds to the vertex with the input edge. Inour case, the subgraph generated V ( T ) is a disjoint union of graphs { T i } i ∈ I and the subgraph generated V ( T ) is a disjoint union of graphs { T i } i ∈ I . We pick a vertex v i , v i in T i , T i respectively. Since T i hasno cycles and we can view v i as a root, we can parameterize the gluing parameters associated to T i by h v i ∈ [0 , ∞ ) , q e = h e − h v (cid:48) ∈ R if e is in the same direction with the direction pointed away from the root v i ,and h e ∈ [0 , ∞ ) if e is the opposite direction with the tree direction. In this case, h v ∈ [0 , ∞ ) , q e = h e − h (cid:48) v ∈ R determine h e , h v (cid:48) ∈ [0 , ∞ ) as in [66, Lemma 3.6]. It is clear that such change of coordinate parameterize thegluing parameters by [0 , ∞ ) × R |{ e | in same direction }| × [0 , ∞ ) |{ e | in opposite direction }| . Similarly we parameterizethe gluing parameters on T i by h v i ∈ [0 , ∞ ) , h e ∈ [0 , ∞ ) if e is in the same direction with the tree direction,and q e = h e − h v ∈ R if e is the opposite direction with the tree direction. As a consequence ( G II ) T/ is atopological manifold with boundary, and the top stratum corresponds to the interior. The gluing parameterspace ( G • II ) T/ is same as ( G II ) T/ . Therefore M II ( T ) , M • ( T ) , M • II ( T ) are equipped with implicit atlaseswith oriented cell-like stratification.The virtual fundamental cycles for BL ∞ morphisms, pointed maps and homotopies are module morphisms Q [ R II ] → Q , Q [ R • ] → Q and Q [ R • II ] → Q respectively that are derived from diagrams like (3.10). Thenon-emptiness of such diagrams and surjectivity of the projections of admissible auxiliary data follows from[66, Proposition 4.34]. That module morphisms Q [ R II ] → Q , Q [ R • ] → Q and Q [ R • II ] → Q give rise countsto BL ∞ morphisms, pointed maps and homotopies follows from the same proof of (1) of Theorem 3.9. For(5) of Theorem 3.9, it is clear the whole construction for α can be identified with the construction for kα aslong as we use the same admissible almost complex structure J for k > (cid:3) Polyfold approach.
The polyfold construction of SFT [33], which is described in [34], will imply The-orem 3.9 as well. However, we can not use the “tree-like” compactification as in M ( T ) because the gluingparameter space is only topological manifold with boundary. For the analytic requirement in polyfold, it isimportant to use the building compactification in [9] so that all gluing parameters are independent and forma smooth manifold with boundary and corner. To implement the polyfold construction for our purpose, it issufficient to build polyfold strong bundles with sc-Fredholm sections for the SFT building compactification,which is sketched in [34]. Since we will not need to discuss more subtle cases like neck-stretching and homotopies, the abstracttheory of polyfold developed in [44] suffices to provide transverse perturbations by the similar induction on( R , (cid:52) ) starting from minimal elements in (cid:52) , which are polyfolds without boundaries. The non-empty setΘ in Theorem 3.9 now consists pairs ( J, σ ), where J is an admissible almost complex structure and σ is afamily of compatible sc + -multisections in general position. Then (3.11) and (3.12) follows from the Stokes’theorem in [44], where the coefficients can be explained to be the discrepancies of isotropy among polyfoldswith their boundary polyfolds and boundary polyfolds with product polyfolds.To verify Axiom 3.25, we first note that classical transversality implies polyfold transversality by definition.If M ( T ) is cut out transversely for vdim( T ) = 0, we may still need to perturb the sc-Fredholm section onthe associated polyfold, because we construct perturbations by induction. Even though we know that thesection is transverse on the boundary polyfolds, but the section can be non-transverse on some factor of theboundary, which will be perturbed before we construct perturbations for M ( T ). However, we can choose ourperturbations small enough to get the local invariance of M ( T ) when vdim( T ) = 0. In other words, Axiom3.25 holds if we choose sufficiently small perturbations. This matches with Proposition 3.32, as Θ( α, J ) inVFC can be understood as “infinitesimal” perturbations. Remark 3.33 (Kuranishi approach) . The Kuranishi approach of SFT [47] would also imply Theorem 3.9.Axiom 3.25 should follow from the same argument above for small enough perturbation in a reasonablemeasurement.
Generalized constraints.
There are few directions where one can generalize the constructions above.In the following, we will briefly describe one of such generalizations by considering more general constraints.3.7.1.
General constraints from Y . We can pick a closed submanifold of Y or more generally a closed singularchain C of Y to construct a pointed map p C by considering rational holomorphic curves in R × Y with onemarked point mapped to { } × C . Then given a BL ∞ augmentation (cid:15) , we have an induced chain map (cid:98) (cid:96) C,(cid:15) : ( B k V α , (cid:98) (cid:96) (cid:15) ) → Q defined using p k, C,(cid:15) . When X is an exact cobordism from Y − to Y + , let C − , C + be twoclosed singular chains in Y − , Y + , which define two pointed map p C − , p C + . If C − and C + are homologous in X , then one can show that p C − , p C + and the BL ∞ morphisms induced from X are compatible in the sense ofDefinition 2.21 by counting holomorphic curves in (cid:98) X with a point passing though the singular chain whoseboundary is − C − (cid:116) C + . As a consequence of Proposition 2.22, (cid:98) (cid:96) C,(cid:15) up to homotopy only depends on thehomology class [ C ]. Therefore for any k ≥
1, we have a linear map δ ∨ (cid:15) : H ∗ ( B k V α , (cid:98) (cid:96) (cid:15) ) ⊗ H ∗ ( Y ) → Q , ( x, [ C ]) (cid:55)→ (cid:98) (cid:96) C,(cid:15) ( x ) . Or equivalently, we can write it as δ (cid:15) : H ∗ ( B k V α , (cid:98) (cid:96) (cid:15) ) → H ∗ ( Y, Q ). Combining the method in [12] and theargument in §
5, one can show that H ∗ ( B V α , (cid:98) (cid:96) (cid:15) ) → H ∗ ( Y, Q ) is isomorphic to the map SH ∗ + ,S ( W ; Q ) → H ∗ S ( W ; Q ) := H ∗ +1 ( W ; Q ) ⊗ Q Q [ u, u − ] / [ u ] → H ∗ +1 ( Y ; Q ), where W is an exact filling and (cid:15) is theaugmentation induced from the filling W . This fact was used in [83, 78] (but not phrased in SFT) todefine obstructions to Weinstein fillings. In principle, H ∗ ( B k V α , (cid:98) (cid:96) (cid:15) ) → H ∗ ( Y, Q ) can be used to obstructedWeinstein fillings if the image contains an element of degree > dim Y +12 for all possible augmentations thatcould be from a Weinstein filling, e.g. Z -graded augmentations if c ( Y ) = 0 and dim Y ≥ δ (cid:15) without proof. Given an exact cobordism X from Y − to Y + , let C be a closed chain in X . Then by counting rational holomorphic curves in X with Note that symplectic cohomology in [78] is graded by n − µ CZ , this explains the discrepancy of parity. LANDSCAPE OF CONTACT MANIFOLDS VIA RATIONAL SFT 37 a marked point mapped to C , we obtain a family of maps φ k,lC : S k V α + → S k V α − . Along with the BL ∞ morphism φ k,l from X , we can construct a map (cid:98) φ C : EV α + → EV α − from φ k,lC , φ k,l by the same rule of (cid:98) φ • .Now that C is closed, we have (cid:98) φ C ◦ (cid:98) p + = (cid:98) p − ◦ (cid:98) φ C . Then given an augmentation (cid:15) of V α − , we have a linearizedrelation (cid:98) φ C,(cid:15) ◦ (cid:98) p + ,(cid:15) ◦ φ = (cid:98) p − ,(cid:15) ◦ (cid:98) φ C,(cid:15) . In particular, (cid:98) φ k, C,(cid:15) defines a chain map ( B k V α + , (cid:98) (cid:96) (cid:15) ◦ φ ) → Q . By the similarargument as before, such construction yields a map δ ∨ X,(cid:15) : H ∗ ( B k V α + , (cid:98) (cid:96) (cid:15) ◦ φ ) ⊗ H ∗ ( X ; Q ) → Q where the dual version H ∗ ( B k V α + , (cid:98) (cid:96) (cid:15) ◦ φ ) → H ∗ ( X ; Q ) is denoted by δ X,(cid:15) . If C comes from H ∗ ( Y + ), then themap coincides with δ ∨ (cid:15) ◦ φ by the same argument of Proposition 5.14. If C comes from Y − , then δ ∨ X,(cid:15) factorsthrough (cid:98) φ (cid:15) : H ∗ ( B k V α + , (cid:98) (cid:96) (cid:15) ◦ φ ) → H ∗ ( B k V α − , (cid:98) (cid:96) (cid:15) ) by an argument similar to Proposition 5.14. Dualizing thoseproperties, we have the following commutative diagram, H ∗ ( B k V α + , (cid:98) (cid:96) (cid:15) ◦ φ ) δ X,(cid:15) (cid:47) (cid:47) (cid:98) φ (cid:15) (cid:15) (cid:15) δ (cid:15) ◦ φ (cid:40) (cid:40) H ∗ ( X ; Q ) (cid:47) (cid:47) (cid:15) (cid:15) H ∗ ( Y + ; Q ) H ∗ ( B k V α − , (cid:98) (cid:96) (cid:15) ) δ (cid:15) (cid:47) (cid:47) H ∗ ( Y − ; Q )3.7.2. Tangency conditions.
Another type of generalization is considering point constraint with tangencyconditions, i.e we consider curves in the symplectization passing through a fixed point p and tangent to alocal divisor near p with order m . Such holomorphic curves were considered in [23, 70]. Those curves alsogive rise to pointed maps, hence can be used to define a new order. In many cases, if we have a holomorphicwith a point constraint without tangent conditions, then multiple covers of it might have tangent properties.One can show that the order with tangent condition for ( S n − , ξ std ) , n ≥ Multiple point constraints.
It is natural to consider generalizations of pointed maps of BL ∞ algebrasto maps induced from counting curves with multiple constraints. For example, we can consider rationalholomorphic curves with 2 marked points passing through two fixed points in the contact manifold, wherethe curve can be disconnected. More specifically, we have three families of maps from S k V to S l V thatare p k,l •• coming from counting connected holomorphic curves with two marked points, p k,l • , p k,l • coming fromcounting connected holomorphic curves with each one of the point constraints respectively. Then we canassemble them to (cid:98) p •• by the same rule of (cid:98) p • except the middle level consists of one p •• or both p • , p • .Note that the combinatorics behind (cid:98) p •• is slightly different from (cid:98) p • and (cid:98) p . Namely, (cid:98) p •• does not satisfythe component-wise Leibniz rule on each SV . More precisely, the component from p •• satisfies the Leibnizrule, while the component from p • and p • is a second order differential operator. Nevertheless, we have (cid:98) p •• ◦ (cid:98) p = (cid:98) p ◦ (cid:98) p •• and given a BL ∞ augmentation, we can similarly define (cid:98) p •• ,(cid:15) , which is assembled from p •• ,(cid:15) , p • ,(cid:15) and p • ,(cid:15) . Then we can define ˜ O ( V, (cid:15), p •• ) by the same recipe for ˜ O ( V, (cid:15), p • ). By moving the two point constraints towards infinity in the opposite directions similar to the proof of Proposition 5.14,we obtain the relation (cid:98) p • = (cid:98) p •• up to homotopy on EV . Similarly, we have (cid:98) p • ,(cid:15) and (cid:98) p •• ,(cid:15) are homotopic.Therefore it is direct to see that ˜ O ( V, (cid:15), p •• ) ≥ ˜ O ( V, (cid:15), p • ). It is not clear whether the inequality can bestrict. In general, we can consider disconnected rational holomorphic curves with k marked points passingthrough k point constraints, which give rise to an operator (cid:98) p k on EV . (cid:98) p k is homotopic to (cid:98) p k • and we candefine an order ˜ O ( V, (cid:15), p k ) for a BL ∞ augmentation (cid:15) .Given a strict exact filling ( W, λ ) of (
Y, α ), assume (cid:101) O ( V α , (cid:15), p k ) < ∞ for the BL ∞ augmentation (cid:15) comingfrom W . Then we can define the spectral invariant for l ≥ (cid:101) O ( V α , (cid:15), p k ), r ≤ l (cid:104) p, . . . , p (cid:124) (cid:123)(cid:122) (cid:125) k (cid:105) := inf (cid:110) a : T a ∈ im π Λ ≥ ◦ (cid:98) p k,(cid:15) | H ∗ ( B l SV ≥ α , (cid:98) p (cid:15) ) (cid:111) < ∞ , where V ≥ α is the free Λ ≥ := { (cid:80) a i T b i | b i ≥ , lim b i = ∞ , a i ∈ Q } module generated by good Reeb orbitsof α . This is exactly the higher symplectic capacity r ≤ l (cid:104) p, . . . , p (cid:124) (cid:123)(cid:122) (cid:125) k (cid:105) defined by Siegel in [70, § W with k point constraints, viewed them as a chainmap from S CHA( Y ) = EV α to Λ. Then the capacity is defined to be the infimum of a such that there isa closed class in x ∈ B k SV ≥ α such that x is mapped to T a by the chain map. The equivalence of thesetwo definition can be seen from the same argument of Proposition 5.14. Similarly, combining the tangencyconditions and multiple point constraints, we can define the analogous orders, whose spectral invariant isagain equivalent to the higher capacity r ≤ l (cid:104)T m p, . . . , T m k p (cid:105) in [70, § O ( V, (cid:15), p • ) is equivalent to g ≤ l and the spectral invariant for the analogous version of pointed map withtangency conditions of order m is g ≤ lt m in [70]. Example 3.34.
Let W be an irrational ellipsoid with α the contact form on ∂W . Then we have ˜ O ( V α , (cid:15) W , p k ) =1 for all k ≥ . To see this, we let γ denote the shortest Reeb orbits, then for a generic point in W anda generic admissible almost complex structure, there is one holomorphic plane in (cid:99) W asymptotic to γ andpassing through the point. Since µ CA ( γ (cid:48) ) + n − for any Reeb orbit γ (cid:48) , we have q γ ∈ B SV α is a closed class and π Q ◦ (cid:98) p • ,(cid:15) W ( q γ ) = 1 by the holomorphic curve above. In general, we have k q γ ∈ B SV α is closed by degree reasons. One the other hand, π Q ◦ (cid:98) p k,(cid:15) W | S k V α must only use disconnected curves with k components and each component has one positive puncture, for otherwise, genus has to be created. As aconsequence, we have π Q ◦ (cid:98) p k,(cid:15) W ( k q γ ) = 1 by the disconnected k copies of the curve above. Then one can showsuch argument is independent of the augmentation, as the curve above is contained in the symplectizationafter a neck-stretching, see [83, 79] .Although there is nothing interesting happening for the order ˜ O ( V α , (cid:15) W , p k ) . We know that the spectralinvariant r ≤ l (cid:104) p, . . . , p (cid:124) (cid:123)(cid:122) (cid:125) k (cid:105) is defined for all k ≥ and l ≥ O ( V α , (cid:15) W , p k ) . Those numerical invariants arevery sensitive to the shape of W and are powerful tools to study embedding problems, see [70] for details. Finally, we give the analogue of O ( V α , (cid:15), p • ) when we have multiple point constraints. Let v ∈ B k SV α , wedefine the width w ( v ) to be the maximal number k such that v has a component in the form of ( v . . . v k ) (cid:12) . . . .We define w (0) = ∞ . Since p k, (cid:15) = 0 for k ≥
1, we have w ( (cid:98) p ( v )) ≥ w ( v ). For m ≥
1, let π m denote theprojection B k SV α → B k B m V α , then the kernel of π m is exactly those elements with width > m . Therefore More precisely, the algebraic count of such curves is 1.
LANDSCAPE OF CONTACT MANIFOLDS VIA RATIONAL SFT 39 (cid:98) p m(cid:15) := π m ◦ (cid:98) p (cid:15) | B k B m V α squares to zero. Then (cid:98) p m(cid:15) uses the knowledge of p k,l(cid:15) for l ≤ m and (cid:98) p (cid:15) = (cid:98) (cid:96) (cid:15) . Inparticular, we have π m is a chain map. Moreover, we have the following commutative diagram,(3.14) B k B m V α π Q ◦ (cid:98) p m,(cid:15) | BkBmVα (cid:47) (cid:47) Q B k SV α π Q ◦ (cid:98) p m,(cid:15) (cid:47) (cid:47) π m (cid:79) (cid:79) Q = (cid:79) (cid:79) This is because (cid:98) p m,(cid:15) can have at most m nontrivial connected components. Therefore if w ( v ) ≥ m + 1 then w ( (cid:98) p m,(cid:15) ( v )) ≥
1, in particular, π Q ◦ (cid:98) p m,(cid:15) ( v ) = 0. Definition 3.35.
We define O ( V, (cid:15), p m ) := min (cid:110) k (cid:12)(cid:12)(cid:12) ∈ im π Q ◦ (cid:98) p m,(cid:15) | H ∗ ( B k B m V, (cid:98) p m(cid:15) ) (cid:111) Then (3.14) implies the following.
Proposition 3.36. O ( V, (cid:15), p k ) ≤ ˜ O ( V, (cid:15), p k ) . Remark 3.37.
The spectral invariants for O ( V, (cid:15), p k ) provide a new family of higher symplectic capacities. Higher genera.
Another natural direction to generalize is by increasing the genus of holomorphiccurves, i.e. considering the full SFT. Originally, the full SFT was phrased as a differential Weyl algebra witha distinguished odd degree Hamiltonian H such that H (cid:63) H = 0 [28]. There are two closely related ways toview the full SFT as a functor from Con in a more convenient way, namely the BV ∞ formulation [22] and the IBL ∞ formulation [21]. In the following, we will first briefly recall the definitions of them. In view of the BL ∞ formalism in this paper, we will give a slightly different but equivalent definition of IBL ∞ algebras.Then we will make some speculations assuming the analytical foundation of the full SFT is completed.3.8.1. IBL ∞ algebras and BV ∞ algebras. The original definition of
IBL ∞ algebra on a Z graded vectorspace involves taking a suspension as the formalism for L ∞ algebras in § BL ∞ algebras, we will not take the suspension. In particular, if V is an IBL ∞ algebra in Definition 3.38,then V [ −
1] is an
IBL ∞ algebra in the sense of [21]. Let φ : S k V → S l V , we can define (cid:98) φ : SV → SV by (cid:98) φ = 0 on S m V with m < k and (cid:98) φ ( v . . . v k ) = (cid:88) σ ∈ Sh ( k,m − k ) ( − (cid:5) k !( m − k )! φ ( v σ (1) . . . v σ ( k ) ) v σ ( k +1) . . . v σ ( m ) , for m > k . Definition 3.38 ([21, Definition 2.3]) . Let V be a Z graded vector space over k , an IBL ∞ structure on V is a family of operators p k,l,g : S k V → S l V for k, l ≥ and g ≥ , such that (cid:98) p := ∞ (cid:88) k,l =1 ∞ (cid:88) g =0 (cid:98) p k,l,g (cid:126) k + g − τ k + l +2 g − : SV [[ (cid:126) , τ ]] → SV [[ (cid:126) , τ ]] satisfies that (cid:98) p ◦ (cid:98) p = 0 and | (cid:98) p | = 1 . Here | (cid:126) | = 0 and | τ | = 0In the case of V being Z graded, then one can define IBL ∞ structures of degree d by requiring | (cid:126) | = 2 d and | (cid:98) p | = −
1. In the case of SFT, d = n − n − Y, ξ ) with c ( ξ ) = 0.If moreover we know that for any v , . . . , v k ∈ V and g ≥ l ≥
1, such that p k,l,g ( v . . . v k ) (cid:54) = 0. Then (cid:98) p is well-defined on SV [[ h ]] by setting τ = 1. Note that we only consider p k,l,g with l ≥ IBL ∞ algebras. Definition 3.39.
Let V be a Z graded vector space over k , a (curved) IBL ∞ structure on V is a familyof operators p k,l,g : S k V → S l V for k ≥ , l, g ≥ , such that(1) for v , . . . , v k ∈ V, g ≥ , there are finitely many l such that p k,l,g ( v . . . v k ) (cid:54) = 0 ,(2) (cid:98) p := (cid:80) ∞ k =1 (cid:80) ∞ l,g =0 (cid:98) p k,l,g (cid:126) k + g − : SV [[ (cid:126) ]] → SV [[ (cid:126) ]] satisfies (cid:98) p ◦ (cid:98) p = 0 and | (cid:98) p | = 1 , where (cid:98) p (1) isdefined to be . Note that Definition 3.39 gives rise to a BV ∞ algebra introduced in [22], which was used to define algebraictorsions in [48], c.f. Remark 3.19. For the translations between BV ∞ algebras and differential Weyl algebraswith distinguished Hamiltonians, see [22] for details. In the following, we give an alternative description ofDefinition 3.39. Assume we are given a family of operators operators p k,l,g : S k V → S l V for k ≥ , l, g ≥ EV [[ h ]] to denote S ( SV [[ h ]]), where S is the symmetricproduct over k [[ (cid:126) ]]. Then we define (cid:98) p : EV [[ h ]] → EV [[ h ]] by the following graph description. We use acluster with k + 1 vertices with a label g ≥ (cid:126) g v . . . v k ∈ SV [[ (cid:126) ]]. Then we use thegraph with to represent operators p k,l,g as before, but we label the with a number g ≥
0. To define (cid:98) p , we represent a class in EV [[ (cid:126) ]] by a row of clusters with labels, then we glue in one graph representing p k,l,g and dashed vertical lines representing the identity map. Here we allow cycles being created. The rulefor the output is the same as before with the degree of (cid:126) is determined by the sum of the labels g in theconnected component of the glued graph with the number of (independent) cycles in that component. Then (cid:98) p is the sum of all such glued graphs. g g g g Figure 11.
A component of (cid:98) p from (cid:126) g S V (cid:12) (cid:126) g S V (cid:12) (cid:126) g S V to (cid:126) g + g + g +1 S V (cid:12) (cid:126) g S V using p , ,g One can think of the cluster representing a class in SV [[ (cid:126) ]] as a genus g surface with k negative puncturesand p k,l,g is represented by a genus g surface with k positive punctures and l negative punctures. Then aglued graph represented a possibly disconnected surface with only negative punctures, which represents aclass in EV [[ (cid:126) ]] with the degree of (cid:126) in each (cid:12) component is represented by the genus of the connectedcomponent. Definition 3.40. ( V, p k,l,g ) is a (curved) IBL ∞ algebra iff (cid:98) p : EV [[ h ]] → EV [[ h ]] defined above satisfies (cid:98) p ◦ (cid:98) p = 0 and | (cid:98) p | = 1 LANDSCAPE OF CONTACT MANIFOLDS VIA RATIONAL SFT 41
Proposition 3.41.
Definition 3.39 and 3.40 are equivalent.Proof.
Note that (cid:98) p : SV [[ (cid:126) ]] ⊂ EV [[ (cid:126) ]] → SV [[ (cid:126) ]] as we can not increase the number of connected componentsin the graph description of (cid:98) p . However (cid:98) p | SV [[ (cid:126) ]] is exactly the (cid:98) p in Definition 3.39 as the (cid:126) k − is exactly thenumber of cycles in the glued graph. Therefore Definition 3.40 implies Definition 3.39.On the other hand, if (cid:98) p : SV [[ (cid:126) ]] → SV [ (cid:126) ] squares to zero. If we consider the (cid:126) N S m V part of (cid:98) p ◦ (cid:98) p ( v . . . v n ),which should be zero. We need to consider disconnected graphs with n input vertices and m output verticesand two vertices such that if we glue all input vertices together by adding a new vertices, the resultedgraph has N cycles. Then there are two cases, (1) the two vertices are in different components, then all ofsuch graphs will pair up and cancel each other by | (cid:98) p | = 1; (2) the vertices are in the same component withthe remaining components are compositions of dashed lines. Assume the component with has a inputsand b outputs and has genus k . We will call ( a, b, k ) the signature of the glued graph. Then we must have n − a = m − b is the number of components from dashed lines and a + k − N . We use p a,b,k to denotethe map from S a V to S b V defined by two components such that the glued graph with signature ( a, b, k ).Or equivalently, p a,b,k is defined by all possible two-level breaking of a the graph with two interior verticesand signature ( a, b, k ). Then the vanishing of the (cid:126) N S m V part of (cid:98) p ◦ (cid:98) p ( v . . . v n ) implies n (cid:88) a =1 p a,m − n + a,N − a +12 ∗ n − a id = 0 , for all n ≥ , m, N ≥
0. By setting n = 1, we have p ,m,N = 0 for m, N ≥
0. Then by setting n = 2 and p ,m,N = 0, we have p ,m,N = 0. Similarly we have p n,m,N = 0 for all n ≥ , m, N ≥
0. This is exactlydescribing all maps from two-level breaking of a connected graph with signature ( n, m, N ) should sum upto zero. It implies that (cid:98) p = 0 on EV [[ (cid:126) ]] by the same argument in Proposition 2.6. (cid:3) Remark 3.42.
From the proof of Proposition 3.41. Both Definition 3.39 and Definition 3.40 are equivalentto p n,m,N = 0 , which in the SFT world, corresponding to the algebraic counts of all rigid codimension breaking of connected holomorphic curves with n positive punctures and m negative punctures and genus N sum up to zero. Note that SV ⊂ SV [[ h ]] induces an inclusion EV ⊂ EV [[ h ]], and we use π to denote the naturalprojection EV [[ (cid:126) ]] → EV . It is easy to check that if x ∈ ker π , then (cid:98) p ( x ) ∈ ker π . As a consequence, π ◦ (cid:98) p | EV : EV → EV squares to zero. Moreover, π ◦ (cid:98) p | EV is assembled from p k,l, . Then we have thefollowing instant corollary. Corollary 3.43.
Let ( V, p k,l,g ) be a (curved) IBL ∞ algebra, then ( V, p k,l := p k,l, ) is a BL ∞ algebra. A grid of torsions.
Given a family of operators φ k,l,g : S k V → S l V (cid:48) for k ≥ , l.g ≥
0, assume thatfor any g ≥ v , . . . , v k ∈ V , there are at most finitely many l such that φ k,l,g ( v . . . v k ) (cid:54) = 0. Then wecan assemble (cid:98) φ : EV [[ (cid:126) ]] → EV [[ (cid:126) ]] by the same rule for BL ∞ morphisms except that cycles are allowed tobe created and the rule for the order of (cid:126) is same as (cid:98) p . Definition 3.44.
The family of operators is an
IBL ∞ morphism from ( V, p k,l,g ) to ( V (cid:48) , q k,l,g ) iff (cid:98) q ◦ (cid:98) φ = (cid:98) φ ◦ (cid:98) p . Then we have a trivial
IBL ∞ algebra := { } with p k,l,g = 0. is an initial object in the categoryof IBL ∞ algebras with φ k,l,g = 0. Then an IBL ∞ augmentation of V is an IBL ∞ morphism from V to . IBL ∞ augmentation may not always exist. One obstruction is torsion. Unlike the torsions for BL ∞ algebras, there are many more torsions for IBL ∞ algebras. Let E k V [[ (cid:126) ]] := B k SV [[ (cid:126) ]]. Definition 3.45.
For n, m ≥ , we say V has a ( n, m ) torsion if [ (cid:126) n ] = 0 ∈ H ∗ ( E m +1 V [[ (cid:126) ]]) . Then the algebraic torsion in [48] is the ( n,
0) torsion of the
IBL ∞ algebra associated to a contact manifoldby SFT. Proposition 3.46. If V has ( n, m ) torsion, then V has ( n + 1 , m − torsion.Proof. We use (cid:126) − EV [[ (cid:126) ]] to denote EV [[ (cid:126) ]] ⊗ k [[ (cid:126) ]] (cid:126) − k [[ (cid:126) ]], (cid:126) − k [[ (cid:126) ]] is the k [[ (cid:126) ]] module generated by (cid:126) − .Then we have a map c k : (cid:12) k SV [[ (cid:126) ]] → (cid:126) − (cid:12) k − SV [[ (cid:126) ]] for k ≥ w (cid:12) . . . (cid:12) w k (cid:55)→ (cid:88) i IBL ∞ algebra. Let v ∈ SV [[ (cid:126) ]], we define the (cid:126) -width w (cid:126) ( v ) ∈ N ∪ {∞} to be themaximal k , such there exists v (cid:48) ∈ SV [[ (cid:126) ]] with (cid:126) k v (cid:48) = v . In particular, w (cid:126) (0) = ∞ . For v ∈ EV [[ (cid:126) ]], w (cid:126) ( v )is uniquely characterized by w (cid:126) ( v (cid:12) v ) = w (cid:126) ( v ) + w (cid:126) ( v ) and w (cid:126) ( v + v ) ≥ max { w (cid:126) ( v ) , w (cid:126) ( v ) } . Thenwe have w (cid:126) ( (cid:98) p ( v )) ≥ w (cid:126) ( v ) for all v ∈ EV [[ (cid:126) ]]. We define EV [[ (cid:126) ]] m := EV [[ (cid:126) ]] ⊗ k [[ (cid:126) ]] ( k [[ (cid:126) ]] / [ (cid:126) m +1 ]), thenwe consider the projection π m : EV [[ (cid:126) ]] → EV [[ (cid:126) ]] m . Then ker π m is exactly those elements with w (cid:126) > m .We can view Q [[ (cid:126) ]] / [ (cid:126) m +1 ] as polynomials of (cid:126) of degree at most m , we can view EV [[ (cid:126) ]] m ⊂ EV [[ (cid:126) ]]. As aconsequence we have the following commutative diagram,(3.16) EV [[ (cid:126) ]] (cid:98) p (cid:47) (cid:47) π m (cid:15) (cid:15) EV [[ (cid:126) ]] π m (cid:15) (cid:15) EV [[ (cid:126) ]] m (cid:98) p m := π m ◦ (cid:98) p | EV [[ (cid:126) ]] m (cid:47) (cid:47) EV [[ (cid:126) ]] m Then we have (cid:98) p m ◦ (cid:98) p m = 0 with (cid:98) p is the associated BL ∞ structure. If we unwrap the definition, then weknow that (cid:98) p m uses p k,l,g for g ≤ m . Take m = 1 as an example, (cid:98) p uses both p k,l, , p k,l, . However, to get anoutput in EV [[ h ]] , the k inputs of p k,l, must glue to k different clusters while at most 2 of the k inputs of p k,l, can glue to the same cluster. We can similarly define E k V [[ (cid:126) ]] m and we have analogous diagrams. LANDSCAPE OF CONTACT MANIFOLDS VIA RATIONAL SFT 43 Definition 3.47. We say ( V, p k,l,g ) has a ( n, m ) k torsion iff [ (cid:126) n ] = 0 ∈ H ∗ ( E m +1 V [[ (cid:126) ]] k ) . Then by definition, we always have ( n, m ) k torsion for n > k . Moreover, (0 , m ) torsion is the m -torsionfor BL ∞ algebras. The ( n, m ) torsion can be viewed as ( n, m ) ∞ torsion. We summarize the basic propertiesof those torsion in the following. Proposition 3.48. The torsions have the following properties.(1) If V has ( n, m ) k torsion then V has ( n, m ) k − , ( n + 1 , m ) k and ( n, m + 1) k torsions. In particularif V has ( n, m ) torsion, then V has ( n, m ) k torsion for any k ≥ .(2) If V has ( n, m ) k torsion, then V has ( n + 1 , m − k torsion.Proof. (1) follows from (3.16), the filtration E k V [[ (cid:126) ]] m and the linear property with respect to (cid:126) . (2) followsfrom the EV [[ (cid:126) ]] m version of (3.15). (cid:3) As a corollary, the existence of (0 , n ) torsion implies both n -algebraic torsion and n -algebraic planartorsion. Contact manifolds in Theorem 3.16 and 3.17 actually has (0 , k ) torsion by the same argument in[48]. Therefore it implies both algebraic torsion and algebraic planar torsion. Roughly speaking, differenttorsions make different requirements on holomorphic curves. For ( n, 0) torsion, we can have higher genuscurves without negative puncture to contribute to the torsion, while all higher genus curves with the samepositive punctures and non-empty negative punctures must sum up to zero. For (0 , n ) torsion, we can onlyhave rational curves without negative puncture to contribute to the torsion, while all higher genus curveswith the same positive punctures and non-empty negative punctures must sum up to zero. For (0 , n ) torsion, we can only have rational curves without negative puncture to contribute to the torsion, and allrational curves with the same positive punctures and non-empty negative punctures must sum up to zero.Let 2 N denote the category of subsets of N × ( N ∪ {∞} ), the arrow from V to W is an inclusion W ⊂ V .2 N is a monoidal category where the monoidal structure is given by taking union. We use SFT( Y ) to denotethe full SFT as an IBL ∞ algebra in the sense of Definition 3.40 for a contact manifold Y . Theorem 3.49. Let Y be a contact manifold, we define T( V ) := { ( m, n, k ) | SFT( Y ) has ( m, n ) k torison } .Then T : Con → N is a covariant monoidal functor. Remark 3.50. For the proof of Theorem 3.49, we only need to construct SFT( Y ) to the same extent as inTheorem 3.9, which is expected from [33, 47] . It is also expected that one can generalize the constructionin [66] , in particular prove the existence of implicit atlases with cell-like stratification. One of the maindifferences is that gluing parameters are subject to more relations if we allow cycles to be created, and wemay not be able to find a “free basis” to build a topological manifold with boundary like ( G II ) T/ in the proofof Theorem 3.9. Analogues of planarity. Given an IBL ∞ augmentation (cid:15) , then we have similar constructions of lin-earization to construct another IBL ∞ structure p k,l,g(cid:15) such that p k,l,g(cid:15) = 0 whenever l = 0. As a consequence,we arrive at an IBL ∞ structure in the sense of Definition 3.38. Then we can introduce the analogue ofpointed maps and the analogue of orders in the context of IBL ∞ algebras, which in the SFT case considersholomorphic curves passing through a fixed point in the symplectization. Moreover, we will have a grid oforders as the case of torsions above. However, it is a much harder task to find examples with holomorphiccurves with higher genus. In fact, we do not know any nontrivial examples with such structures except forpunctured Riemann surfaces as exact fillings of a disjoint union of contact circles. Semi-dilations In this section, we introduce an inner hierarchy called the order of semi-dilation for the P = 1 case. Notethat if P( Y ) = 1, then RSFT( Y ) admits BL ∞ augmentations and for any BL ∞ augmentation (cid:15) , we havethe order is 1 for any point in Y . Note that ( B V α , (cid:98) (cid:96) (cid:15) ) is the chain complex ( V α , (cid:96) (cid:15) ) for the linearized contacthomology. Since P( Y ) = 1, for any point in Y , we have an class x ∈ H ∗ ( V α , (cid:96) (cid:15) ) such that (cid:96) • ,(cid:15) ( x ) = 1.If the augmentation (cid:15) W is from an exact filling W , then by [12, 14], the linearized contact homologyLCH ∗ ( Y, (cid:15) W ) = LCH ∗ ( W ) := H ∗ ( V α , (cid:96) (cid:15) W ) is isomorphic to the equivariant symplectic (co)homology whentransversality holds, which as a S -equivariant theory carries a H ∗ ( BS ) = Q [ u ]-module structure and fitsinto the following Gysin sequences, . . . (cid:47) (cid:47) SH n − − k + ( W ) (cid:47) (cid:47) (cid:15) (cid:15) LCH k ( W ) u (cid:47) (cid:47) (cid:39) (cid:15) (cid:15) LCH k − ( W ) (cid:47) (cid:47) (cid:39) (cid:15) (cid:15) SH n − − k + ( W ) (cid:15) (cid:15) (cid:47) (cid:47) . . .. . . (cid:47) (cid:47) SH n − − k + ( W ) (cid:47) (cid:47) SH n − − k + ,S ( W ) u (cid:47) (cid:47) SH n − − k + ,S ( W ) (cid:47) (cid:47) SH n − − k + ( W ) (cid:47) (cid:47) . . . As we use homological convention in this paper, u has degree − Z grading for LCH. And u has degree 2 for the S -equivariant symplectic cohomology, which is graded by n − µ CZ . In fact, the map u is defined on the linearized contact homology H ∗ ( V α , (cid:96) (cid:15) ) for any augmentation (cid:15) . And for any element x ∈ H ∗ ( V α , (cid:96) (cid:15) ) there exists k ∈ N + such that u k ( x ) = 0. In the following, we first recall on the definition of u for linearized contact homology.4.1. H ∗ ( V α , (cid:96) (cid:15) ) as a Q [ u ] module. To explain the u -map, we recall the following two moduli spaces from[12, § M Y,A ( γ + , γ − , Γ − ) . Let γ + , γ − be two good Reeb orbits and Γ − be an ordered multiset of goodReeb orbits of cardinality k ≥ 0. Then an element in M Y,A ( γ + , γ − , Γ − ) consists of the following data.(1) A sphere (Σ , j ), with one positive puncture z + and 1 + k negative punctures z − , z − , . . . , z − k . We pickan asymptotic marker on z + , then by choosing a global polar coordinate on Σ \{ z + , z − } , there is aconically induced asymptotic marker on z − by requiring it having the same angle as the asymptoticmarker at z + in the polar coordinate. We also pick free asymptotic markers on z − i for 1 ≤ i ≤ k .(2) A map u : ˙Σ → R × Y such that d u ◦ j = J ◦ d u and [ u ] = A modulo automorphism and the R -translation, where ˙Σ is the 2 + k punctured sphere.(3) u is asymptotic to γ + , γ − , Γ − near z + , z − and { z − i } i Then for an exact cobordism X , we can similarly define M X,A ( γ + , γ − , Γ − ) where we do not modulo the R translation.4.1.2. M Y,A ( γ + , γ, γ − , Γ − , Γ − ) . Let γ + , γ, γ − , be three good Reeb orbits and Γ − , Γ − be two orderedmultisets of good Reeb orbits of cardinality k , k ≥ 0. Then an element in M Y,A ( γ + , γ, γ − , Γ − , Γ − ) consistsof the following data.(1) Two spheres (Σ , j ) , ( ˜Σ , j ), each with one positive puncture z + , ˜ z + , and 1 + k negative punctures z − , z − , . . . , z − k , 1 + k negative punctures, ˜ z − , ˜ z − , . . . , ˜ z − k respectively. Each puncture is equippedwith an asymptotic marker. LANDSCAPE OF CONTACT MANIFOLDS VIA RATIONAL SFT 45 (2) Two holomorphic curves u, ˜ u from ˙Σ , ˙˜Σ to R × Y modulo automorphism and R -translations, suchthat [ u ] γ [˜ u ] = A .(3) u is asymptotic to γ + , γ, Γ − and ˜ u is asymptotic to γ, γ − , Γ − .(4) Let L − and L + be two asymptotic markers on z − and ˜ z + that are induced from the chosen asymptoticmarkers on z + and ˜ z − by global polar coordinates . Then we can define ev L − ( u ) , ev L + (˜ u ) to be thelimit point in the Y component evaluated along the asymptotic markers L − , L + . Then we require( b γ , ev L − ( u ) , ev L + (˜ u )) is the natural order on im γ , where b γ is the marked point on im γ .We can similarly define M , ↑ X,A ( γ + , γ, γ − , Γ − , Γ − ) and M , ↓ X,A ( γ + , γ, γ − , Γ − , Γ − ) for an exact cobordism X .The difference is that the former one has u in (cid:98) X and the latter one has ˜ u in (cid:98) X . We use M and M todenote their compactification. Note that in the case of M , we need to add in the stratum correspondingto the collision of ( b γ , ev L − ( u ) , ev L + (˜ u )) in addition to usual building structures.Given an dga augmentation (cid:15) to CHA( Y ), i.e. a map (cid:15) : V α → Q , which extends to an algebra map (cid:98) (cid:15) : CHA( Y ) → Q such that (cid:98) (cid:15) ◦ (cid:98) p = 0. Then u : V α → V α is defined by u ( q γ + ) = (cid:88) γ − , [Γ − ] κ γ − µ Γ − κ Γ − M Y,A ( γ + , γ − , Γ − ) (cid:89) γ (cid:48) ∈ Γ − (cid:15) ( γ (cid:48) ) q γ − + (cid:88) γ − ,γ, [Γ − ] , [Γ − ] κ γ − κ γ µ Γ − µ Γ − κ Γ − κ Γ − M Y,A ( γ + , γ, γ − , Γ − , Γ − ) (cid:89) γ (cid:48) ∈ Γ − ∪ Γ − (cid:15) ( γ (cid:48) ) q γ − (4.1) Remark 4.1. The M Y,A in [12, § requires modulo an equivalence ( L − , L + ) (cid:39) ( L − + πκ γ , L + + πκ γ ) , i.e.the moduli space should be thought as glued two-level buildings. Here we do not introduce the equivalence,the discrepancy is just the extra κ γ in (4.1) compared to [12, (85)] . The reason that u is a chain map from ( V α , (cid:96) (cid:15) ) to itself follows from the boundary of 1-dimensional M Y,A and M Y,A . More precisely, the codimension 1 boundary of M Y,A consists of (1) a level breaking where thelower level does not contain z − and (2) a level breaking where the lower level contains z − . For case one, suchcontribution is zero when capping Γ − off with (cid:15) by the relation (cid:98) (cid:15) ◦ (cid:98) p = 0. For case two, the contribution willcancel with the codimension 1 boundary part of M Y,A corresponding to the collision of ev L − ( u ) , ev L + (˜ u ).The other parts of codimension 1 boundary of M Y,A consists of (1) a level breaking of u where the lowerlevel does not contain z − , this is again killed by the capping off with (cid:15) ; (2) A level breaking of u where thelower level contains z − , the corresponds to a component of (cid:98) (cid:96) ◦ u , where the u part is contributed by a M Y,A ;(3) Similar level breakings for ˜ u ; (4) The collision of b γ and ev L − ( u ), this corresponds a component of (cid:96) (cid:15) ◦ u ,where the u part is contributed by a M Y,A , similarly, the collision of b γ and ev L + (˜ u ) is the remaining partof u ◦ (cid:96) (cid:15) .Similarly, given an exact cobordism, we can show that chain morphism φ , (cid:15) : V α → V α (cid:48) is commutativewith u up to homotopy, where the homotopy is defined by M X,A , M , ↑ X,A , and M , ↓ X,A by a similar formulato (4.1) with a similar argument. In particular, they may be different from the chosen asymptotic markers on z − and ˜ z + The extra coefficient µ Γ comes from that we consider Γ as an ordered set, and µ Γ is the size of the isotropy coming frompermutation. The following proposition asserts that the counts of moduli spaces above can be defined after appropriatesetup of virtual machinery. In particular, it follows from the same argument of Theorem 3.9. Proposition 4.2. Let ( Y, α ) be a non-degenerate contact manifold and θ be an auxiliary data which is usedin defining a BL ∞ structure p θ .(1) There is an auxiliary data θ u for the definition of u , such that for any BL ∞ augmentation (cid:15) of ( V α , p θ ) , we have a map u θ u : H ∗ ( V α , (cid:96) (cid:15) ) → H ∗− ( V α , (cid:96) (cid:15) ) and for any x ∈ H ∗ ( V α , (cid:96) (cid:15) ) there exists k such that u kθ u ( x ) = 0 .(2) When there is a strict exact cobordism X from ( Y (cid:48) , α (cid:48) ) to ( Y, α ) with admissible auxiliary data θ, θ (cid:48) , θ u , θ (cid:48) u for α, α (cid:48) and their u -maps respectively, then there exists auxiliary data ξ , such that the φ , ξ,(cid:15) : H ∗ ( V α , (cid:96) (cid:15) ◦ φ ξ ) → H ∗ ( V α (cid:48) , (cid:96) (cid:15) ) commutes with the u -maps for any BL ∞ augmentation (cid:15) for ( V α (cid:48) , p θ (cid:48) ) .(3) For any k ∈ R + , there exists kθ u , such that u kθ u is canonical identified with u θ u That u kθ u ( x ) = 0 follows from that u strictly decreases the contact action for non-degenerate α . Definition 4.3. Let Y be a contact manifold with P( Y ) = 1 , then we define the order of semi-dilation SD( Y ) as follows, SD( Y ) := max (cid:110) min (cid:110) k (cid:12)(cid:12)(cid:12) u k +1 ( x ) = 0 , x ∈ H ∗ ( V α , (cid:96) (cid:15) ) , (cid:96) • ,(cid:15) ( x ) = 1 (cid:111) (cid:12)(cid:12) o ∈ Y, (cid:15) ∈ Aug Q ( V α ) (cid:111) SD can be defined on all of Con , by declaring SD( Y ) = ∞ if P( Y ) ≥ Y ) = − Y ) = 0. Proposition 4.4. For those contact manifolds Y with P( Y ) = 1 , the assignment of SD( Y ) is a monoidalfunctor from the full subcategory of Con to N ∪ {∞} , where the monoidal structure on N is defined by a ⊗ b = max { a, b } ,Proof. That SD( Y ) is independent of all choices follows from the same argument of Proposition 3.11. Themonoidal structure follows from H ∗ ( V ⊕ V (cid:48) , (cid:96) (cid:15) ⊕ (cid:96) (cid:15) (cid:48) ) = H ∗ ( V (cid:48) , (cid:96) (cid:15) (cid:48) ) ⊕ H ∗ ( V (cid:48) , (cid:96) (cid:15) (cid:48) ) as Q [ u ]-modules. (cid:3) k -dilation and k -semi-dilation were introduced in [78] as structures on S -equivariant symplectic co-homology, which are generalizations of symplectic dilation of Seidel-Solomon [69]. More precisely, anexact domain W carries a k -dilation, iff there is a class x ∈ SH ∗ + ,S ( W ) such that x is sent to 1 by SH ∗ + ,S ( W ) → H ∗ +1 S ( W ) and u k +1 ( x ) = 0, where H ∗ S ( W ) := H ∗ ( W ) ⊗ Q [ u ] ( Q [ u, u − ] / [ u ]). W carries a k -semi-dilation iff x is sent to 1 in SH ∗ + ,S ( W ) → H ∗ +1 S ( W ) → H ∗ +1 S ( ∂W ) and u k +1 ( x ) = 1. Under theisomorphism SH ∗ + ,S ( W ) = LCH n − −∗ ( W ) [12, 14], the element we are looking for in Definition 4.3 is x ∈ SH ∗ + ,S ( W ) that is sent to one in SH ∗ + ,S ( W ) → H ∗ +1 S ( W ) → H ( W ) with u k +1 ( x ) = 0, where the lastmap is the natural projection. It is natural to expect that examples with nontrivial SD come from exampleswith nontrivial k -dilation found in [78] and we will show in § Remark 4.5. A prori, the semi-dilation used in this paper is weaker than the semi-dilation in [78] forexact fillings. However, we do not know any example justifying that there are differences between those twodefinitions.Proof of Theorem A. It follows from Proposition 3.11, Proposition 3.23, and Proposition 4.4, with themonoidal stricture explained therein. (cid:3) LANDSCAPE OF CONTACT MANIFOLDS VIA RATIONAL SFT 47 Remark 4.6. In fact, ∞ APT , ∞ SD are the only two elements in H that we do not know if it is in the imageof H cx . H cx ( Y ) = ∞ APT corresponds to that RSFT( Y ) has no BL ∞ augmentation while RSFT( Y ) hasinfinite torsion. Note that a BL ∞ augmentation is essentially a solution to a family of algebraic equations(in a infinite dimensional space). Using that Q is not algebraically closed, it is easy to define a (finitedimensional) BL ∞ algebra over Q with no augmentation and infinite algebraic planar torsion. However, itis unclear how to construct a geometric example. It is also an interesting question on obstructions to BL ∞ augmentations beyond torsion besides using that Q is not algebraically closed. Planarity and semi-dilation for fillings. Since fillings of a contact manifold give rise to BL ∞ augmentations, one can define planarity and order of semi-dilation for fillings as follows. Definition 4.7. Let W be an exact domain, we define P( W ) as P( W ) := max { O ( V α , p • , (cid:15) W ) | o ∈ Y, α, (cid:15) W } , the maximal is taken over all non-degenerate contact forms α and all BL ∞ augmentations (cid:15) W from thefilling (i.e. for all choices of auxiliary data). Similarly, we define SD( W ) in the case of P( W ) = 1 , SD( W ) := max (cid:110) min (cid:110) k (cid:12)(cid:12)(cid:12) u k +1 ( x ) = 0 , x ∈ H ∗ ( V α , (cid:96) (cid:15) W ) , (cid:96) • ,(cid:15) W ( x ) = 1 (cid:111) | o ∈ Y, α, (cid:15) W (cid:111) . We define SD( W ) = ∞ if P( W ) > . Remark 4.8. In fact, the choice of α, (cid:15) W is redundant. P( W ) and SD( W ) can be computed using just one α and (cid:15) W . However this requires introducing the notation of homotopy between BL ∞ augmentations forlinearized theories, which will be carried out in the future. The trick in Proposition 3.11,3.23, 4.4 can nothelp dropping the dependence on α, (cid:15) W , since it requires that the composition of the morphism from an exactcobordism X and the augmentation from an exact filling W is (homotopic to) an augmentation from X ◦ W .However this involves neck-stretching and is essentially a BL ∞ homotopy. Planarity P( W ) can also be computed from (5.1) by Proposition 5.14. Claim 4.9. P and SD are functors from Con ∗ to N + ∪ {∞} . SD( W ) ≤ k iff there exists x ∈ SH ∗ + ,S ( W ) that is mapped to in SH ∗ + ,S ( W ) → H ∗ +1 S ( W ) → H ( W ) and u k +1 ( x ) = 0 . In particular, if W carries a k -semi-dilation, then SD( W ) ≤ k . We leave it as a claim, because the proof of functoriality of P and SD requires building the full packageRSFT to discuss linearized theory up to homotopy, and the second claim requires proving the isomorphism SH ∗ + ,S ( W ) = LCH n − −∗ ( W ) for any exact domain W , i.e. implementing virtual machinery for [12, 14].5. Lower bounds for planarity As explained in § 3, the curve responsible for finiteness of planarity is a curve with multiple positivepunctures and a point constraint. Since planarity does not depend on the choice of the point, one shouldexpect that finiteness of planarity implies uniruledness. In this section, we will prove such implication anda lower bound for planarity. We first recall the notion of uniruledness from [55].5.1. Order of uniruledness.Definition 5.1 ([55, § . Let ( W, λ ) be an exact domain. A d λ -compatible almost complex structure J on W is convex iff there is a function φ such that(1) φ attains its maximum on ∂W and ∂W is a regular level set, (2) λ ◦ J = d φ near ∂W . Definition 5.2 ([55, Definition 2.2]) . Let k > be an integer and Λ > a real number. We say thatan exact domain ( W, λ ) is ( k, Λ) uniruled if, for every convex almost complex structure J on W and every p ∈ W ◦ (the interior of W ) where J is integrable near p , there is a proper J -holomorphic map u : S → W ◦ passing through p and the following holds,(1) S is a genus Riemann surface and the rank of H ( S ; Q ) ≤ k − ,(2) (cid:82) S u ∗ d λ ≤ Λ .We say W is k -uniruled if W is ( k, Λ) uniruled for some Λ > . The number Λ depends on the Liouville form λ which is not relevant for our purpose. However the number k only depends on the Liouville structure up to homotopy. Definition 5.3. Let W be an exact domain, we define the order of uniruledness U( W ) := min { k | W is k uniruled. } The following was proven by McLean [55]. Proposition 5.4. U is a functor from Con ∗ to N + ∪ {∞} .Proof. Let V ⊂ W be an exact subdomain, then U( V ) ≤ U( W ) by [55, Proposition 3.1]. It is clear fromdefinition that U( V, λ ) = U( V, tλ ) for t > 0. Since for any Liouville structure θ on V that is homotopic to λ , we have exact embeddings ( V, t − λ ) ⊂ ( V, θ ) ⊂ ( V, tλ ) for t (cid:29) 0, therefore U is a well-defined functor on Con ∗ . (cid:3) Remark 5.5. A worth noting point is that the definition and functorial property of U do not depend on anyFloer theory. However U gives a measurement of “complexity” of exact domains. By [78, Theorem 3.23] ,the exist of k -(semi)-dilation implies that the order of uniruledness is . Hence the order of (semi)-dilationin [78, Corollary D] is a refined hierarchy in U = 1 . For an affine variety V , we define the order of algebraically uniruledness AU( V ) be the minimal number k such that V is algebraically k uniruled, i.e. through every generic p ∈ V there is a polynomial map S → A passing through p with S is a punctured CP with at most k punctures. Proposition 5.6 ([55, Theorem 2.5]) . Let V be an affine variety then U( V ) ≥ AU( V ) . Example 5.7. Let S k be the k -punctured sphere. Then U( S k ) = k . Let Σ g,k be the k -punctured genus g ≥ surface, then U(Σ g,k ) = ∞ . It is clear that S k embeds exactly into S k +1 . However S k +1 can only be embeddedin S k symplectically but not exactly. In general we have the following. Theorem 5.8. We have U(( S k ) n ) = k and U((Σ g,k ) n ) = ∞ for g ≥ . In particular, U is a surjectivefunctor in any dimension ≥ .Proof. Note that S nk has a projective compactification ( CP ) n , using the fact for any compatible almostcomplex structure J there is a holomorphic curve passing any fixed point in the class [ CP × { pt } × . . . × { pt } ]and intersecting each divisor { p i } × ( CP ) n − exactly once for 1 ≤ i ≤ k , where p i is the i th puncture. Wemay assume J is an extension of a convex almost complex structure on S nk . Therefore U(( S k ) n ) ≤ k , byneck-stretching. LANDSCAPE OF CONTACT MANIFOLDS VIA RATIONAL SFT 49 On the other hand, when we view ( S k ) n and (Σ g,k ) n as affine varieties with the product complex structure.We know a rational algebraic curve in ( S k ) n and (Σ g,k ) n projects to each factor a rational algebraic curve.Then every rational curve must have at least k punctures. Therefore AU(( S k ) n ) = k and AU((Σ g,k ) n ) = ∞ ,and the claim follows from Proposition 5.6 (cid:3) As a consequence, we find in each dimension a nested sequence of exact domains V ⊂ V . . . , such that V i can not be embedded into V j exactly if i > j . Sequences with such property in dim ≥ 10 were also obtainedin [50, Corollary 1.5]. Remark 5.9. In § 6, we will show that P( ∂ ( S k ) n ) = k if n ≥ . Therefore not only there is no exactembedding from ( S k +1 ) n to ( S k ) n , but also there is no exact cobordism from ∂ ( S k +1 ) n to ∂ ( S k ) n . Remark 5.10. From U on Con ∗ , we can build a functor U ∂ on Con as follows U ∂ ( Y ) := max { U( W ) | W is an exact filling of Y } . Then Corollary 5.15 below implies that U ∂ ≤ P . The equality does not always hold. For example, U ∂ ( RP n − , ξ std ) =0 for n (cid:54) = 2 k by [79] , but P( RP n − , ξ std ) = 1 when n ≥ by Theorem 7.28. Those discrepancies come fromthe difference between fillings and augmentations. It is possible to generalize the notation of order of unir-uledness U , U ∂ to strong fillings or even weak fillings, but we will not pursue this in this paper. In the following, we introduce an alternative definition of k -uniruledness. Definition 5.11. Let ( W, λ ) be an exact filling with a non-degenerate contact boundary, we say the com-pletion (cid:99) W is k -uniruled if there exists a Λ > , such that for every p ∈ W ◦ and every admissible almostcomplex structure J that is integrable near p , there is a rational holomorphic curve passing through p withat most k positive punctures and contact energy of the curve is at most Λ . Proposition 5.12. An exact filling ( W, λ ) is k -uniruled iff (cid:99) W (cid:15) is k -uniruled, where W (cid:15) is Liouville homotopicto W with a non-degenerate contact boundary.Proof. We first show that ( W, λ ) is k -uniruled implies (cid:99) W (cid:15) is k -uniruled. WLOG, we can take W (cid:15) ⊂ W , sincewe can rescale W (cid:15) . By assumption there is a Λ > p ∈ W ◦ (cid:15) and any J integrable near p and convex near ∂W , there is a J -rational curve u : S → W with (cid:82) S u ∗ d λ < Λ and H ( S ; Q ) ≤ k − 1. Inparticular, we can choose J to be cylindrical convex near ∂W (cid:15) . Then by applying neck-stretching along ∂W (cid:15) ,we must have a rational holomorphic curve u : S (cid:15) → (cid:99) W (cid:15) passing through p with contact energy smaller thanΛ. We know that S (cid:15) is a punctured sphere, as ∂W (cid:15) is non-degenerate. It sufficient to prove rank H ( S (cid:15) ; Z ) =rank H ( S (cid:15) ; Q ) ≤ k − 1. Assume otherwise, we know that H ( S (cid:15) ; Z ) → H ( S ; Z ) is not injective, for otherwise,we have rank H ( S ; Q ) ≥ rank H ( S (cid:15) ; Q ) ≥ k . Therefore we find a class [ γ ] ∈ H ( S (cid:15) ; Z ), such that [ γ ] isrepresented by a disjoint union γ of possibly multiply covered loops around punctures of S (cid:15) and there is animmersed surface A in S \ S (cid:15) . Then in the fully stretched case, u | A corresponds to a holomorphic buildingwith only negative punctures, which is impossible for energy reasons.Now we assume (cid:99) W (cid:15) is k -uniruled. WLOG, we can assume W ⊂ W (cid:15) . By [83, Proposition 5.3], any convexalmost complex structure on W can be extended to an admissible almost complex structure on (cid:99) W (cid:15) . Byassumption, there is a rational curve u : S → (cid:99) W (cid:15) passing through the chosen point p ∈ W with S an atmost k punctured sphere and the contact energy of u is at most Λ. Let S (cid:48) be the connected component of u − ( W ◦ ) containing the point mapped to p . It clear that the area of u | S (cid:48) is bounded by Λ. We claim that H ( S (cid:48) ; Z ) → H ( S ; Z ) is injective. For otherwise, there is a class A ∈ H ( S, S (cid:48) ; Z ) mapped to a nontrivialelement by H ( S, S (cid:48) ; Z ) → H ( S (cid:48) ; Z ). Then we can find a S (cid:48)(cid:48) ⊂ S (cid:48) such that λ ◦ J = d φ on u | S (cid:48) \ S (cid:48)(cid:48) , where φ is the function in Definition 5.1. Then by excision, we have A represented by an immersed surface in S \ S (cid:48)(cid:48) not contained completely in S (cid:48) \ S (cid:48)(cid:48) with boundary in S (cid:48) \ S (cid:48)(cid:48) . Let (cid:98) φ be the extension of φ on (cid:99) W (cid:15) by[83, Proposition 5.3]. In particular, the maximum principle holds for (cid:98) φ . Then we reach at a contradiction,since (cid:98) φ ( u ) | ∂A < (cid:98) φ ( u ) | A . Since Q is flat, we know that H ( S (cid:48) ; Q ) → H ( S ; Q ) is also injective, hencerank H ( S (cid:48) ; Q ) ≤ rank H ( S ; Q ) ≤ k − (cid:3) Uniruledness and planarity. The main theorem of this section is following. Theorem 5.13. If P( Y ) = k , then any exact filling of Y is k -uniruled. Since an exact filling W gives rise to a BL ∞ augmentation (cid:15) W over Q . As a consequence we have a chainmorphism (cid:98) (cid:96) • ,(cid:15) W : BV → Q after fixing a point o in Y and an auxiliary data. We can define a different map η W : BV → Q by(5.1) η W ( q Γ + ) = (cid:88) A µ Γ + M W,A,p (Γ + , ∅ )for a fixed point p ∈ W such that p, o are in the same connected component of W , and | Γ + | = k . Proposition 5.14. η W is a chain morphism and is homotopic to (cid:98) (cid:96) • ,(cid:15) W with appropriate choices of auxiliarydata, where (cid:15) W is the augmentation from W . Moreover η W is compatible with the word length filtration, i.e. η W is a chain map from B k V for any k ≥ , and is homotopic to (cid:98) (cid:96) • ,(cid:15) W on B k V .Proof. Let γ be a path in W connecting p to o and we use (cid:98) γ to denote the completion of γ in (cid:99) W . Then thehomotopy H : BV → Q is defined by H ( q Γ + ) = (cid:88) A µ Γ + M W,A,γ (Γ + , ∅ ) , The realization of those operators using virtual techniques is similar to φ • in (4) of Theorem 3.9 in § M W,A,γ (Γ + , ∅ ). It is clear that both η W and H are compatible with the word length. (cid:3) Proof of Theorem 5.13. The theory of BL ∞ algebra considered for contact manifold is equipped with afiltration by the contact action, where the action A ( q γ ) of a generator is (cid:82) γ ∗ α . Then the action canbe extended to EV and SV uniquely by the following two property A ( x · / (cid:12) y ) = A ( x ) + A ( y ) and A ( x + y ) ≤ max {A ( x ) , A ( y ) } . Then all of the operators for contact manifolds and exact cobordisms willdecrease the action as explained in § Y ) = k is bounded for all BL ∞ augmentations. But for the augmentation (cid:15) from an exact filling W , we have thespectral invariant is bounded, i.e. there is a Λ > x ∈ B k V with A ( x ) ≤ Λ, (cid:98) (cid:96) (cid:15) ( x ) = 0,and (cid:98) (cid:96) • ,(cid:15) ( x ) = 1. Then by Proposition 5.14, we have η W ( x ) = 1. By Axiom 3.25, we must have the geometric M W,A,p (Γ + , ∅ ) is not empty for some Γ + with | Γ + | ≤ k and (cid:80) γ ∈ Γ + (cid:82) γ ∗ α ≤ Λ. This shows that (cid:99) W is ( k, Λ)uniruled, by Proposition 5.12, W is k -uniruled. (cid:3) Theorem 5.13 provides a lower bound for P. Corollary 5.15. Let W be an exact filling of Y , then P( Y ) ≥ U( W ) . If W is an affine variety, then P( Y ) ≥ AU( W ) . LANDSCAPE OF CONTACT MANIFOLDS VIA RATIONAL SFT 51 Upper bounds for planarity The strategy for obtaining an upper bounds P( Y ) ≤ k on the planarity of a contact manifold ( Y, ξ ) is viathe following algebraic-geometric condition: Lemma 6.1. Let ( Y, ξ ) be a contact manifold. Assume the following holds: ( (cid:63) ) k There exists a point o ∈ R × Y , a contact form α for ξ , a choice of α -compatible cylindrical almostcomplex structure J on R × Y , and some collection Γ = ( γ , . . . , γ k ) of precisely k distinct, non-degenerate and simply-covered α -Reeb orbits, for which the following holds: (1) k If Γ + ⊆ Γ , and Γ − (cid:54) = ∅ , then M Y,A (Γ + , Γ − ) = ∅ for every homology class A . (2) k The moduli space M Y,A,o (Γ , ∅ ) is transversely cut out for every A with expected dimension . (3) k For some choice of coherent orientations, the algebraic count of the k -punctured spheres in (cid:83) A, vdim=0 M Y,A,o (Γ , ∅ ) is nonzero.Then P( Y ) ≤ k .Proof. By the first property, we have M Y,A,o (Γ + , Γ − ) = ∅ for any Γ + ⊆ Γ, and Γ − (cid:54) = ∅ . Therefore byAxiom 3.25, the second and third condition implies that (cid:98) (cid:96) • ,(cid:15) ( q Γ + ) (cid:54) = 0 for any augmentation (cid:15) (if there isno augmentation, then P( Y ) = 0 by definition). Moreover by the first property, we have q Γ + is closed in( SV, (cid:98) (cid:96) (cid:15) ) for any augmentation (cid:15) . Then the claim follows. (cid:3) Remark 6.2. In some situations, we will need to relax (1) k to the following: (1) (cid:48) k If Γ + ⊆ Γ , and Γ − (cid:54) = ∅ , then M Y,A (Γ + , Γ − ) = ∅ for every homology class A , unless | Γ + | = | Γ − | =1 . In addition, we have M Y,A,o ( { γ + } , { γ − } ) = ∅ for any γ + ∈ Γ + and any γ − . Moreover the M Y,A ( { γ + } , { γ − } ) is cut out transversely for any γ + ∈ Γ + and any γ − and the compactification M Y,A ( { γ + } , { γ − } ) only involves cylinders. Finally M Y,A ( { γ + } , { γ − } ) = 0 when the expecteddimension is .We also need to modify (2) k accordingly. (2) (cid:48) k The moduli space M Y,A,o (Γ , ∅ ) is transversely cut out for every A with expected dimension . Assume { γ +1 , . . . , γ + j } ⊂ Γ , such that there are γ − , . . . , γ − j and A , . . . , A j with M Y,A i ( γ + i , γ − i ) (cid:54) = ∅ for ≤ i ≤ j , then we have M Y,A (cid:48) ,o (Γ (cid:48) , ∅ ) = ∅ , where Γ (cid:48) = (Γ \{ γ +1 , . . . , γ + j } ) ∪ { γ − , . . . , γ − j } and the expecteddimension of M Y, i A i A (cid:48) ,o (Γ , ∅ ) is zero.Along with (3) k , we have P( Y ) ≤ k .Proof. By (1) (cid:48) k , we have M Y,A (Γ + , Γ − ) = ∅ unless | Γ + | = | Γ − | = 1. And when | Γ + | = | Γ − | = 1, we have M Y,A (Γ + , Γ − ) is cut out transversely with algebraic count 0 when the expected dimension is 0. Thereforeby Axiom 3.25, q Γ is a closed class for any augmentation. Since we know that M Y,A,o (Γ + , Γ − ) = ∅ aslong as Γ − (cid:54) = ∅ by (1) (cid:48) k . We know that (cid:98) (cid:96) • ,(cid:15) ( q Γ ) is solely contributed by M Y,A,o (Γ + , ∅ ). Moreover, in thecompactification M Y,A,o (Γ + , ∅ ), we only need to worry about buildings containing upper levels without thepoint constraint. Then by (2) (cid:48) k , either the upper level is empty or the lower level is empty. In particular,we have M Y,A,o (Γ + , ∅ ) = M Y,A,o (Γ − , ∅ ) and it is cut out transversely with total algebraic count (cid:54) = 0. As aconsequence, (cid:98) (cid:96) • ,(cid:15) ( q Γ ) (cid:54) = 0 for any augmentation by Axiom 3.25. Hence P( Y ) ≤ k . (cid:3) In practice, we usually construct a homological foliation on the symplectization R × Y (i.e. a modulispace of curves for which the above algebraic-geometric condition holds for generic points o ), with strong uniqueness properties, although this is stronger than needed. We shall do this for a sufficiently large classof examples, as follows.6.1. Iterated planar open books.Definition 6.3 ([2]) . An iterated planar Lefschetz fibration f : ( W n , ω ) → D on a n -dimensional Wein-stein domain ( W n , ω ) is an exact symplectic Lefschetz fibration satisfying the following properties:(1) There exists a sequence of exact symplectic Lefschetz fibrations f i : ( W i , ω i ) → D for i = 2 , . . . , n with f = f n .(2) The total space ( W i , ω i ) of f i is a regular fiber of f i +1 , for i = 2 , . . . , n − .(3) f : ( W , ω ) → D is a planar Lefschetz fibration, i.e. the regular fiber of f is a genus zero surfacewith nonempty boundary, which we denote by W . Definition 6.4 ([2]) . An iterated planar open book decomposition of a contact manifold ( Y n +1 , ξ ) is anopen book decomposition for Y whose page W admits an iterated planar Lefschetz fibration, which supportsthe contact structure ξ in the sense of Giroux. We say that ( Y, ξ ) is iterated planar (IP). If the number of boundary components of W in the above definition is k , we say that ( Y, ξ ) is k -iteratedplanar or k -IP. We remark that the collection of IP contact manifolds is already a large class of examples,as e.g. the fundamental group is not an obstruction in any fixed dimension at least 5 [4, Theorem 1.4]. Theorem 6.5. Let ( Y, ξ ) be a k -IP contact manifold. Then P( Y ) ≤ k .Proof. We proceed by induction on dimension.If dim Y = 3, then an IP contact 3-manifold is simply a planar contact 3-manifold. Fix a choice ofplanar open book supporting the contact structure, with page a sphere with k -disks removed, and so withbinding consisting of k circles. One then constructs an adapted Giroux form, so that each component of thebinding is a non-degenerate and simply-covered orbit, and a holomorphic open book as e.g. in [71]. Thisprovides a Fredholm-regular foliation of R × Y whose leaves are either trivial cylinders over the binding, orholomorphic Fredholm-regular k -punctured spheres projecting to pages and asymptotic to the binding. Onecan prove via standard 4-dimensional arguments coming from Siefring intersection theory (the same as inhigher-dimensions, as used below), that any curve whose positive asymptotics are a subset of the binding,is a leaf of this foliation. While this a priori holds for an almost complex structure which is compatible witha SHS deforming the contact form, one may perturb this SHS to nearby contact data without changing theisotopy class of the contact form. After perturbing the almost complex structure to make it compatible withthis nearby contact data and generic, the curves in the foliation survive by Fredholm regularity, and theuniqueness statement still holds if the perturbation is small enough (as follows easily from a SFT compactnessargument). In particular, (1) k and (2) k in Lemma 6.1 hold, for the perturbed J . In this case the geometric(and hence the algebraic) count of these curves with a point constrain is 1 for any generic point o , and so(3) k is also satisfied. We fix such a o for which we have this uniqueness property.If dim Y ≥ 5, we fix an IP open book π : Y \ B → S supporting ξ , with binding B ⊂ Y , a codimension-2contact submanifold. Since B is also k -IP if Y is, we may assume by induction that ( (cid:63) ) k holds for B . We maythen extend the Giroux contact form on B for which ( (cid:63) ) k holds to a Giroux contact form on Y , in such a waythat all k Reeb orbits Γ = ( γ , . . . , γ k ) from the induction step are still non-degenerate orbits in Y . On theother hand, the holomorphic open book construction can also be done in arbitrary dimensions (again, afterdeforming the Giroux form away from B to a stable Hamiltonian structure, cf. [19, Appendix A], [58, 59]).The choice of almost complex structure can be taken to agree with the one from the inductive step along LANDSCAPE OF CONTACT MANIFOLDS VIA RATIONAL SFT 53 H B := R × B , which is then a holomorphic submanifold. The leaves of the resulting codimension-2 foliationare now either H B , or a codimension-2 holomorphic submanifold, a copy of the Liouville completion of thepage, which is asymptotic to H B at infinity in the sense of [60]. We let F denote this codimension-2 foliationon R × Y . The moduli space M B of k -punctured spheres defined on R × B in the inductive step extends to amoduli space M Y on R × Y , consisting of curves having the same positive asymptotics as curves in M B . Anapplication of Siefring intersection theory as in [59, 60] shows that any holomorphic curve u whose positiveasymptotics are a subset of Γ either completely lies in H B , or its image lies completely in a non-cylindricalleaf H of F . In the first case, u cannot have negative ends by (1) k applied to B . In the second case, since H has no negative ends, u also, and so (1) k holds on R × Y . This also shows that M Y,A,o (Γ , ∅ ) = M B,A,o (Γ , ∅ )for every o ∈ H B . If moreover we take o ∈ H B to be the point given by the inductive step, for which (by thebase case) we may assume that we have the uniqueness property that curves in M B,A,o (Γ; ∅ ) are necessarilyelements in M B (for every A ), and in particular transversely cut out inside H B . The same analysis ascarried out in [58, Lemma 4.13] (by splitting the normal linearized CR-operator into tangent and normalcomponents with respect to H B , and using automatic transversality on the normal summand) shows thatcurves in M B are transversely cut out in R × Y . Then (2) k holds, and (3) k also (and in fact the geometriccount is 1 for our particular choice of o ). Note that all of these conditions still hold after perturbing tonearby contact data, via the same argument as above.An appeal to Lemma 6.1 finishes the proof. (cid:3) It is clear from the definition that the Weinstein conjecture holds for contact manifolds with finite planarity.In the case of iterated planar open books, the Weinstein conjectures was proven for dimension 3 in [1] andhigher dimensions in [2, 5]. Theorem 6.5 proves the Weinstein conjecture for a slightly larger class of contactmanifolds than iterated planar. In view of the proof of Weinstein conjecture, Theorem 6.5 is of the samespirit as the proofs in [1, 2, 5]. However, more importantly, Theorem 6.5 endows the holomorphic curve withSFT meaning. We further remark that the proof of the above theorem in the 5-dimensional case actuallyprovides a foliation, as opposed to a homological one, as shown in [59].Theorem 6.5 can be viewed as a special case of the following conjecture. Conjecture 6.6. Let Y be an open book whose page is W , then P( Y ) ≤ P( W ) and SD( Y ) ≤ SD( W ) . In the context of semi-dilations in symplectic cohomology, such claim was proven in [78] for Lefschetzfibrations. The geometric intuition behind the conjecture is clear and was used in Theorem 6.5, the difficultylies in making the virtual machinery compatible with the geometry for general Y and W .6.2. Trivial planar SOBDs. We now consider a related example as to the ones considered above. Fix( S k , dλ ) a sphere with k -disks removed together with a Liouville form λ , and let ( M, dα ) be any Liouvilledomain. Define ( V := S k × M, ω = d ( λ + α )), endowed with the product Liouville domain structure. Let( Y = ∂V, ξ = ker( α + λ )), the contact manifold filled by V . Theorem 6.7. We have P( Y ) ≤ k .Proof. We first note that Y admits a supporting (trivial) SOBD, as considered in [52, 58]. In other words,we have a decomposition Y = Y S ∪ Y P , where Y S = ∂S k × M is the spine and Y P = S k × ∂M is the paper , and we have trivial fibrations π S : Y S → M,π P : Y P → ∂M. We view the first one as a contact fibration over a Liouville domain, and the second, as a Liouville fibrationover a contact manifold (its fibers are called the pages ).We then construct a holomorphic foliation of R × Y , i.e. we make our SOBD holomorphic (this is theanalogous construction for trivial SOBDs, to the one considered in the proof of Theorem 6.5 for the caseof open books; see [58] for full details), as follows. By choosing a Morse function H on the vertebrae M (the base of the spine) which vanishes near Y P and has a unique maximum and no minimum, we mayperturb the contact form along Y S to e (cid:15)H ( α + dθ ) , where θ ∈ S parametrizes each connected componentof ∂S k . As explained in [58], each critical point p ∈ M of H corresponds to a Reeb orbit of the form γ p = S × { p } (one for each of the k components of Y S ; they are non-degenerate by non-degeneracy of H ).One then deforms the contact form to a stable Hamiltonian structure which coincides with H = ( α, d ( α + λ ))on Y P , and so its kernel there is T S k ⊕ ker α , tangent to S k . After this, one can construct a compatiblecylindrical almost complex structure, for which there exist a foliation of R × Y by Fredholm regular curveswhose leaves come in three types: trivial cylinders R × γ p over critical points, holomorphic pages (which areLiouville completions of S k , project to the pages, and are asymptotic to the trivial cylinders at infinity),and holomorphic flow-line cylinders (which project to M as Morse flow lines of H , and are also asymptoticto trivial cylinders). A generic point p on Y S lies in a flow-line which enters Y S from Y P and reaches themaximum. The corresponding flow-line cylinder is an asymptotic of a generic page, which has Fredholmindex dim M (and so has index zero after including a point constraint). By the corresponding version of theuniqueness theorem [58, Theorem 3.9], which proves that any holomorphic curve with positive asymptoticsa subset of the γ p must be a curve in the foliation, we see that ( (cid:63) ) (cid:48) k is satisfied for generic choice of o ∈ Y .This uniqueness statement also survives sufficiently small perturbations to nearby contact data and adaptedgeneric J . The result follows. (cid:3) Remark 6.8. The same exact proof works for the SOBDs considered in [58] . The difference for thoseexamples is that the paper has two connected components Y ± P , Y − P having genus zero pages, Y + P havingpositive genus ones (i.e. the SOBD is not symmetric). While ( (cid:63) ) (cid:48) k is satisfied for generic points o ∈ Y − P , thisis not true for o ∈ Y + P . This is explained as follows: these examples have finite algebraic planar torsion bythe proof of [58, Theorem 1.4] , in particular, there is no BL ∞ augmentation, hence the planarity is . Notethat the proof of Lemma 6.1 and Remark 6.2 shows that the ruling curve has homological meaning only ifthere are augmentations. Corollary 6.9. Let V be an affine variety, such that AU( V ) ≥ k . Then P( ∂ ( S k × V )) = k .Proof. By Theorem 6.7, we have P( ∂ ( S k × V )) ≤ k . One the other hand it is easy to see that AU( S k × V ) = k ,as every algebraic curve in V × S k projects two algebraic curve in both V and S k . Then the claim followsfrom Corollary 5.15. (cid:3) IP Bourgeois examples. In [8], given a contact manifold ( Y, ξ ) together with a supporting openbook decomposition, Bourgeois constructs an associated contact structure ξ BO on Y × T . These contactmanifolds were studied more systematically in [51, 19]. Amongst these examples, the natural candidates forwhich we may estimate the planarity is precisely those for which the inital open book is iterated planar.We will focus on the 5-dimensional case, i.e. when Y is a 3-manifold and the open book is planar, sincecontrolling holomorphic curves becomes much more approachable, due to dimensional reasons having to dowith intersection theory. We say that the 5-dimensional Bourgeois manifold ( Y × T , ξ BO ) is k -planar if thestarting open book has genus zero pages with k -boundary components, k ≥ LANDSCAPE OF CONTACT MANIFOLDS VIA RATIONAL SFT 55 Theorem 6.10. If ( Y × T , ξ BO ) is k -planar, then P ( Y × T ) ≤ k .Proof. If ( Y, ξ ) = OBD (Σ , φ ) is supported by an open book with page Σ and monodromy φ , we consider theassociated SOBD ( Y × T , ξ BO ) = Y S (cid:83) Y P supporting the Bourgeois contact structure, where Y S = B × D ∗ T ,with B = ∂ Σ the binding in Y , and Y P = Σ φ × T , with Σ φ the mapping torus of φ (see [19, § F of [19,Appendix A, § B = B × T × { } (a T Morse-Bott family of Reeb orbits), or Liouville completions of Σ which are asymptotic to orbits in B , onein each of the k components of B . If we introduce a Morse perturbation by choosing a Morse function on T , critical points of this function corresponds to non-degenerate orbits lying in B , and we further obtainflow-line holomorphic cylinders.We claim that, before or after a Morse perturbation, any holomorphic curve u in the symplectization R × Y × T , whose positive asymptotics are simply covered, and each one lie in a different connectedcomponent of B , is necessarily a leaf in F . This claim implies that ( (cid:63) ) (cid:48) k in Lemma 6.1 is satisfied (and thisstill holds after perturbing to nearby contact data) and so the theorem follows. This is again proved by anadaptation of [58, Theorem 3.9], as in [19, Lemma 7.4] (note, however, that [19, Lemma 7.4] assumes thestronger assumption that the curve has precisely k asymptotics, which is not enough for the conditions ofLemma 6.1; this is not the case for [58, Theorem 3.9], which is more general).We give a guide to the argument for convenience of the reader. First, one can arrange that the negativeends of u also asymptote orbits in B , and their number is bounded above by the number of positive asymp-totics of u [19, Lemma 7.2]. Then one separates two cases: either u lies completely in Y S (case A), or it doesnot (case B). In case A, u has only one positive end by assumption, and since orbits are non-contractiblein Y S , u has precisely one negative end. The Morse-Bott case is then easily dealt with using energy ( u isnecessarily a trivial cylinder); the Morse case is obtained from gluing analysis for holomorphic cascades asin [7] (see [58, Theorem 3.9, case A]). The proof for Case B is then almost word by word as that in [19,Lemma 7.2]; see also [58, Theorem 3.9, case B]. (cid:3) Remark 6.11. For the higher-dimensional case, it suffices to show that a holomorphic curve u as in theproof above lies in a leaf of F , and appeal to what we proved above for the case of IP-contact manifolds.While we expect this to hold, its proof needs rely on a different argument, since the foliation is now not -dimensional (nor -codimensional), and so the intersection theory is not so useful a priori. Examples and applications In this section, we will discuss two more classes of examples, where we can compute the hierarchy functors.The first case is smooth affine varieties with a CP n compactification, or more generally a Fano hypersurfacecompactification. The second case is links of singularities, including links of Brieskorn singularities andquotient singularities by the diagonal action of cyclic groups. In particular, we will finish the proof ofTheorem B.7.1. Affine varieties. Let V be a smooth affine variety, then V is naturally a Weinstein manifold by viewing V ⊂ C N and the function | x − x | on C N restricted to V is a Morse function with finitely many criticalpoints for a generic x ∈ C N [57, § V with a large enough ball. We will use ∂V to denote the contact boundary.An alternative way of associating a Weinstein structure to V is by using a smooth projective compactifi-cation V with an ample line bundle L with a holomorphic section s such that s − (0) is normal crossing and V = V \ s − (0). We choose a metric on L , such that the curvature is a K¨ahler form ω on V . Then by [68, Lemma 4.3], h = − log | s | and − d c h defines a Weinstein structure (possibly after perturbation) on V . Theequivalence of these two definitions can be found in [55].We first give a description on the embedding relations of affine varieties with the same projective com-pactification. Lemma 7.1. Let X be a smooth projective variety with a very ample line L . For s ∈ H ( L ) , we use V s todenote the Liouville domain associated to the affine variety X \ s − (0) . Then for s (cid:54) = 0 ∈ H ( L ) , there exists (cid:15) > , such that for all t ∈ H ( L ) with | s − t | < (cid:15) , we have V s embeds exactly into V t .Proof. With the very ample line bundle L , X can be embedded in P H ( L ) such that every s ∈ H ( L )corresponds to a hyperplane H s ⊂ P H ( L ) and s − (0) = X ∩ H s . Since we can view the Liouville domain V s as the intersection of X with a large ball in the identification of C N with P H ( L ) \ H s . Then for t sufficientlyclose to s , i.e. H t sufficiently closed to H s , the Liouville form of V t restricts V s ∩ S R is a contact form, where S R the radius R (cid:29) C N . The Gray stability theorem implies that all of them induced the samecontact structure on ∂V s for t sufficiently close to s , hence V s embeds exactly into V t . (cid:3) Remark 7.2. In principle, the exact embedding from V s to V t should be built from a Weinstein cobordism.Hence one expects a more precise description of the Weinstein cobordism, which depends the deformationfrom s to t . Some results in this direction can be found in [3, 61] . Roughly speaking, we should have a stratification on P H ( L ) indexed by the singularity type of s − (0).The index set forms a category by declaring a morphism from stratum A to stratum B if the closure of B contains A . Then Lemma 7.1 implies that we have a functor from the index set (which should be a poset)to Con ∗ . Making such description precise is not easy, as we do not have a classification of singularities of s − (0). However, we can describe some subcategory of the index set. The following lemma is also very usefulin understanding the embedding relations of affine varieties arose from different line bundles. Lemma 7.3 ([68, Lemma 4.4]) . Assume the smooth affine variety V has a smooth projective compactification V . Assume there are two ample line bundles L i with sections s i , such that s − (0) = s − (0) = V \ V isnormal crossing, but possibly with different multiplicities. Then the Liouville structures on V defined by s i are homotopic. Example 7.4. CP n minus k generic hyperplanes can be viewed as the complement of ( s ⊗ . . . ⊗ s k ) − (0) for generic sections s i of O (1) . On the other hand, CP n minus k − generic hyperplanes can be viewedas the complement of ( s ⊗ s ⊗ s ⊗ . . . ⊗ s k ) − (0) by Lemma 7.3. As a consequence of Lemma 7.1, wehave an exact embedding of CP n minus k − generic hyperplanes to CP n minus k generic hyperplanes. Asa simple example, CP minus a line is C , CP minus two generic lines is C × T ∗ S and CP minus threegeneric lines is T ∗ T . It is clear that we have the embedding relations. Moreover some of the relations cannot be reversed, e.g. T ∗ T can not be embedded exactly into C or C × T ∗ S . But C × T ∗ S can be embeddedback into C by adding an handle corresponding to the positive Dehn twist in the trivial open book for ∂ ( C × T ∗ S ) . More generally, CP n minus k generic hyperplanes is C n +1 − k × T ∗ T k − for k ≤ n , and theycan be embedded into each other exactly. Example 7.5. CP minus hyperplanes passing through the same point is C × S , where S is the thricepunctured sphere. Since CP minus hyperplanes can still be viewed as a further degeneration, we have C × T ∗ S embeds to C × S , which is obviously true. On the other hand, CP minus generic hyperplanes,i.e. T ∗ T contains C × S as an exact domain. Moreover, CP minus a smooth degree curve is T ∗ RP ,which is obtained from attaching a -handle to C × T ∗ S , i.e. CP minus generic lines. CP minus a LANDSCAPE OF CONTACT MANIFOLDS VIA RATIONAL SFT 57 smooth degree curve can be described as attaching three -handles to T ∗ T , see [3] for details. It is notclear if the complement of a smooth degree curve embeds exactly into the complement of a smooth degree curve. However, the former embeds exactly into the complement of a smooth degree curve by Lemma 7.1and Lemma 7.3. Let D be a divisor, we use D c to denote the complement affine variety. Our main theorem in this sectionis the following. Theorem 7.6. Let D be k generic hyperplanes in CP n for n ≥ , then we have the following.(1) P( ∂D c ) ≥ k + 1 − n for k > n + 1 .(2) P( ∂D c ) = k + 1 − n for n + 1 < k < n − and n odd.(3) P( ∂D c ) = 2 for k = n + 1 .(4) H cx ( ∂D c ) = 0 SD for k ≤ n . The strategy to obtain Theorem 7.6 is first prove P( ∂D c ) ≥ max { , k + 1 − n } by index computation, thenwe obtain that the planarity of the affine variety D c is at most max { , k + 1 − n } by looking at the affinevariety D c s , where D s is the smoothing of D , i.e. a smooth degree k hypersurface. Finally, we use indexcomputation to show that the relevant portion of computation is independent of the BL ∞ augmentation forRSFT( ∂D c ) when n + 1 < k < n − . The n being odd condition is to obtain automatic closedness for achain in SV ∂D c s for any augmentations and is expected to be irrelevant. However, to drop this constraint,we need to use stronger transversality properties supplied by [82], see Remark 7.20 for more discussion. The n +1 < k < n − condition is likely not optimal and it is not clear whether it is necessary. It is a difficult taskto compute planarity for all augmentations. There are many affine varieties with a CP n compactificationsuch that the contact boundary has infinite planarity, while the planarity of the affine domain, i.e. usingthe augmentation for the affine variety is finite, see Theorem 7.12. In particular, different augmentations domake a difference. Therefore it is a subtle question to determine which affine variety has finite planarity. Ingeneral, we need to develop a computation method of RSFT from (log/relative) Gromov-Witten invariantslike the symplectic (co)homology computation in [27].7.1.1. Reeb dynamics on the divisor complement. In this part, we describe the Reeb dynamics on the bound-ary of a tubular neighborhood of a simple normal crossing divisor. The general description was obtained in[54], see also [35, 36, 56]. For our purpose, we are only interested in the following two special cases. A smooth degree k hypersurfaces in CP n for n ≥ . Let D denote a smooth degree k hypersurface in CP n , then the contact boundary of the concave boundary of the O ( k ) line bundle over D carries a naturalMorse-Bott contact form. We can pick a C -small Morse function f on D , such that Reeb orbits (up to anarbitrarily high action threshold) have the following properties.(1) There is a simple Reeb orbit γ p over every critical point p of f and these are all of the simple Reeborbits. We use γ mp to denote the m -th cover of γ p . All of the Reeb orbits are good and non-degenerate.(2) [ γ p ] ∈ H ( ∂D c ) ∈ H ( D c ) is a generator of order k .(3) Using the obvious disk bounded by γ mq in O ( k ) | D , which induces an trivialization of det C ξ , we havethat the Conley-Zehnder index has the following property, n − − µ CZ ( γ mp ) = 2 m − p ) , where ind( p ) is the Morse index of p . k generic hyperplanes in CP n for k ≥ n + 1 . Let D , . . . , D k denote the k hyperplanes. Let I ⊂ { , . . . , k } be a set of cardinality at most n . We define D I to be the intersection ∩ i ∈ I D i which is a copy of CP n −| I | . We define ˇ D I by D I \ ∪ i/ ∈ I D i . We pick exhausting a Morse function f I on each ˇ D I . The Reeb dynamics hasthe following properties.(1) For each critical point p of f I and a function s : { , . . . , k } → N k with supp s = I , we have a T | I |− Morse-Bott family of Reeb orbits γ sp .(2) H ( D c ) = H ( ∂D c ) is generated by the simple circles [ β i ] wrapping around D i once subject to therelation (cid:80) ki =1 [ β i ] = 0. The homology class [ γ sp ] over ˇ D I is (cid:80) i ∈ I s ( i )[ β i ].(3) The generalized Conley-Zehnder index using the obvious disk whose the intersection number with D i is s ( i ) for i ∈ I , is given by n − − µ CZ ( γ sp ) = 2 (cid:88) s ( i ) − p ) + | I | − . After the perturbation, the T | I |− family of Reeb orbits degenerate to 2 | I |− many non-degenerateorbits, the Conley-Zehnder indices span the following region, n − − µ CZ ∈ [2 (cid:88) s ( i ) − p ) , (cid:88) s ( i ) − p ) + | I | − . We use ˇ γ sp to denote the orbit with n − − µ CZ (ˇ γ sp ) = 2 (cid:80) s ( i ) − p ) and ˆ γ sp to denote theorbit with n − − µ CZ (ˆ γ sp ) = 2 (cid:80) s ( i ) − p ) + | I | − Proposition 7.7. Let D = D ∪ . . . ∪ D k denote the k > n + 1 generic hyperplanes in CP n for n ≥ .(1) For any Reeb orbits set Γ := { γ , . . . , γ r } for r < k + 1 − n with (cid:80) [ γ i ] = 0 ∈ H ( D c ) , the virtualdimension of the moduli space M D c ,A,o (Γ , ∅ ) is less than for any A .(2) For any Reeb orbits set Γ := { γ , . . . , γ k +1 − n } with (cid:80) [ γ i ] = 0 ∈ H ( D c ) and the virtual dimension ofthe moduli space M D c ,A,o (Γ , ∅ ) ≥ , then there is a partition of { , . . . , k } into I , . . . , I k +1 − n , suchthat Γ = { ˇ γ σ Ii p i, min } i , where p i, min is the minimum on ˇ D I i and σ I i is the indication function supportedon I i .Proof. Note that c ( D c ) = 0, we have the virtual dimension does not depend on A , hence we will abbreviateit in the following discussion (the same applies to everywhere in this subsection). Given a curve u in thesame homotopy class of a curve in M D c ,o (Γ , ∅ ), we use u i to denote the natural disk cap of γ i , then we haveind( u ) + r (cid:88) i =1 ( n − − µ CZ ( γ i )) = 2 c ( u ri =1 u i ) − . We assume γ i is from γ s i p i after perturbation, then they are subject to the condition (cid:80) ri =1 s i = ( N, . . . , N )and c ( u ri =1 u i ) = N ( n + 1). Then we haveind( u ) ≤ N ( n + 1) − − r (cid:88) i =1 (2 (cid:88) s i − p i ))(7.1) ≤ N ( n + 1) − − r (cid:88) i =1 (cid:88) s i + 2 r (7.2) = 2 N ( n + 1 − k ) − r = 2( N − n + 1 − k ) + 2( r + n − k − < LANDSCAPE OF CONTACT MANIFOLDS VIA RATIONAL SFT 59 when r < k + 1 − n. If r = k + 1 − n , to have ind( u ) ≥ 0, we must have N = 1. In this case, both inequalities(7.1) and (7.2) must be equality. In particular, ind( p i ) = 0 and γ i must be a check orbit, i.e. the claimholds. (cid:3) Then the by Proposition 5.14, we have the following. Corollary 7.8. If s is a perturbation of the k generic hyperplanes for k > n + 1 and n ≥ , then P( ∂V s ) ≥ k + 1 − n .Proof. Let D be k generic hyperplanes, then η D c on B r V ∂D c in Propitiation 5.14 is zero for r < k + 1 − n bydimension reasons. Therefore P( ∂D c ) ≥ k + 1 − n . The remaining of the claim follows from Lemma 7.1. (cid:3) A neck-stretching argument.Proposition 7.9. Let D s be a smooth degree k hypersurfaces in CP n , then the following holds.(1) If k ≤ n , for a point o ∈ D c , there is a Reeb orbit γ kp with ind( p ) = 2( n − k ) and an admissiblecomplex structure, such that M D c s ,o ( { γ kp } , ∅ ) is cut out transversely and M D c s ,o ( { γ kp } , ∅ ) (cid:54) = 0 .(2) If k ≥ n + 1 , for a point o ∈ D c s , there are two Reeb orbits γ np min , γ p min with p min is the minimum on D s and an admissible almost complex structure, such that M D c s ,o ( { γ np min , γ p min , . . . , γ p min (cid:124) (cid:123)(cid:122) (cid:125) k − n } , ∅ ) is cutout transversely with nontrivial algebraic count.Proof. This follows from applying neck-stretching to CP n along ∂D c s . We denote the relative Gromov-Witten invariant that counts genus 0 holomorphic curves in class A with k marked point going through C , . . . , C k ∈ H ∗ ( CP n ) and l marked point going through E , . . . , E l ∈ H ∗ ( D s ) and intersect D s with mul-tiplicity at least s , . . . , s l respectively by GW CP n ,D s ,k, ( s ,...,s l ) ,A ( C , . . . , C k , E , . . . , E l ) [46]. The source of holo-morphic curves is from the non-vanishing relative Gromov-Witten invariants GW CP n ,D s , , ( k ) ,A ([ pt ] , [ D s ] ∩ n − k [ H ])and GW CP n ,D s , , ( n, ,..., ,A ([ pt ] , [ D s ] , . . . , [ D s ]) respectively from [38], where H ∈ H n − ( CP n ) is the hyperplaneclass and A is the generator of H ( CP n ). Since the curve is necessarily somewhere injective and not con-tained in D s because we can choose the [ pt ] class in D c s , one can assume transversality in the process ofneck-stretching.In the fully stretched picture, the bottom curve has at most max { , k + 1 − n } positive punctures forotherwise genus has to be created. If the bottom curve has 0 < r < max { , k + 1 − n } positive punctures,in particular, k ≥ n + 1, we assume the positive asymptotics are Γ + = { γ d i p i } i . Then by homology reasonswe have (cid:80) d i = km . Then the expected dimension of M D c ,o (Γ + , ∅ ) is given byind( u ) = 2 m ( n + 1) − − r (cid:88) i =1 ( n + 3 − µ CZ ( γ s i p i )) ≤ m ( n + 1) − − r (cid:88) i =1 d i + 2 r = 2 m ( n + 1 − k ) + 2 r − m − n + 1 − k ) + 2( r + n − k − < { , k + 1 − n } positive punctures. Moreover, from the abovecomputation, we separate the proof into three cases. (1) k > n + 1. To have ind( u ) ≥ 0, we must have that p i is the minimum p min and m = 1. That isthe positive asymptotics of the bottom curve are { γ d i p min } ≤ i ≤ k +1 − n with (cid:80) d i = k . Then other levels mustbe either a cylinder in the symplectization or a disk in the symplectic cap. Note that the disk v in thesymplectic cap that intersect D s with order n must be asymptotic to γ nq for a critical point q . Assumeotherwise that the multiplicity is n + km for m ≥ 1, then the the relative homology class of v is the sameas the sum of the natural disk of γ n + kmq and − mA for the positive generator A ∈ H ( D s ) (cid:16) H ( CP n )when n ≥ 3. Then the symplectic action of such disk is negative for m ≥ n = 2, since π ( D s ) = 0, it isnecessarily to have m = 0 as γ n + kmq and γ nq are not homotopic in the neighborhood of D . Therefore thenegative asymptotics of the disks in the symplectic cap must be γ nq and γ q (cid:48) . Then by computing the expecteddimension of the cylinders in the symplectization, the only possibility (i.e. those with non-negative dimension)is q, q (cid:48) are the minimum p min and there is no nontrivial curve in the symplectization. The bottom curvemoduli space M D c s ,o ( { γ np min , γ p min , . . . , γ p min (cid:124) (cid:123)(cid:122) (cid:125) k − n } , ∅ ) consists of somewhere injective curves, for otherwise, assume u ∈ M D c s ,o ( { γ np min , γ p min , . . . , γ p min (cid:124) (cid:123)(cid:122) (cid:125) k − n } , ∅ ) is a branched cover over u (cid:48) , then we can cap off u (cid:48) with natural disksto obtain a homology class A in H ( CP ) with A ∩ D s < k , which is a contradiction. It is direct to check thatthe holomorphic disks in the symplectic cap (i.e. disk fibers) are cut out transversely, hence transversalityholds for the fully stretched situation. Therefore we have M D c s ,o ( { γ np min , γ p min , . . . , γ p min (cid:124) (cid:123)(cid:122) (cid:125) k − n } , ∅ ) (cid:54) = 0.(2) k = n + 1. Then to have ind( u ) ≥ 0, we must have that p i is the minimum of p min but m ≥ 1. By thesame area argument for the cap, we have the negative asymptotics of the disks must be γ nq and γ q (cid:48) . Thenwe must have m = 1, for otherwise the total contact action of the symplectization levels is negative. Thenthe remaining of the argument is the same as before.(3) k ≤ n . Since the bottom level has one positive puncture, that is asymptotic to γ kmq . By the samearea and action argument, we have m = 1. Assume the negative asymptotic of the disk is γ kp , then wemust have ind( p ) ≥ n − k ) to have non-negative expected dimension for the disk. On the other hand, forthe bottom curve, we must have ind( q ) ≤ n − k ) to have non-negative expected dimension. Therefor if p (cid:54) = q , the expected dimension of the cylinders in the symplectization is negative. Hence we have p = q withind( p ) = 2( n − k ). Then we know that there is at least one critical point p with ind( p ) = 2( n − k ) such that M D c s ,o ( { γ kp } , ∅ ) (cid:54) = 0 and the unstable manifold of p represents multiples of [ D s ] ∩ n − k [ H ]. (cid:3) Corollary 7.10. Let D s be the smooth degree k hypersurface in CP n for k > n + 1 and n ≥ . Then η D c s ( q γ np min q k − nγ p min ) (cid:54) = 0 and q γ np min q k − nγ p min is closed in ( B k +1 − n V ∂D c , (cid:98) (cid:96) (cid:15) D c s ) .Proof. We may assume the Morse function on D s is perfect, this follows from direct check for n = 2, [43] for n = 3, and the Lefschetz hyperplane theorem and the h -cobordism theorem for n ≥ 4. Then Proposition 7.9implies that η D c s ( q γ np min q k − nγ p min ) (cid:54) = 0. It suffices to prove that q γ np min q k − nγ p min is closed. Since we have the parity ofthe SFT grading is the same as the Morse index, and q γ np min q k − nγ p min has even grading, we only need to consider (cid:104) (cid:98) (cid:96) (cid:15) D c s ( q γ np min q k − nγ p min ) , q γ mp (cid:105) with ind( p ) = n − n even. As a consequence, we need consider M ∂D c s (Γ + , Γ − )for Γ + ⊂ { γ np min , γ p min , . . . , γ p min } and Γ − = { γ mp , γ d p , . . . γ d s p s } , then we close off { γ d i p i } i by the augmentationfrom D c s . On the other hand, by homology reason, we know the sum of multiplicities of Γ − equals to the sumof multiplicities of Γ + which is no larger than k . As a consequence, there is no subset of { γ d i p i } i whose sum LANDSCAPE OF CONTACT MANIFOLDS VIA RATIONAL SFT 61 represents a null-homologous class in D c s . In particular, to consider (cid:104) (cid:98) (cid:96) (cid:15) D c s ( q γ np min q k − nγ p min ) , q γ mp (cid:105) , we only need toconsider M ∂D c s (Γ + , { γ mp } ) where m is the sum of the multiplicities of Γ + . It is direct to check the expecteddimension of this moduli space is ( n − 2) + 2 | Γ + | − 2, which is strictly positive whenever n ≥ 3. When n = 2,it is direct check that only case with expected dimension 0 is M ∂D c s ( { γ p min } , { γ p } ) and M ∂D c s ( { γ p min } , { γ p } ),each of them corresponds to moduli space of gradient trajectories from minimum p min to the index 1 criticalpoint p , whose algebraic count is zero, as our Morse function is perfect [27, 35]. Therefore q γ np min q k − nγ p min isclosed in ( B k +1 − n V ∂D c s , (cid:98) (cid:96) (cid:15) D c s ). (cid:3) Remark 7.11. In the proof of Corollary 7.10, we use the topology of the filling D c to get some restrictionson the augmentation, in particular the augmentation respects the homology classes of orbits. However, wecan not run such argument for general augmentations to obtain Theorem F and we use n being odd to getautomatic closedness. Roughly speaking, we proved that P( D c s ) ≤ k + 1 − n for a smooth degree k > n + 1 divisor D s . If thefunctoriality of P for exact domains was proven in Claim 4.9, then we can conclude that P( D c ) ≤ k +1 − n for D the k generic hyperplanes. We still need to argue that the computation is independent of augmentation.In the following, we first show that the computation of P( ∂D c s ) does depend on augmentations in some cases Theorem 7.12. Let D s be a smooth degree k > n − hypersurface in CP n for n ≥ odd, then P( ∂D c s ) = ∞ .Proof. As assume as before the Morse function on D s is perfect. Since n ≥ (cid:15) k : SV ∂D c s → Q is a BL ∞ augmentation. We claim if we pick (cid:15) k = 0 for k ≥ 1, then the order of planarity is ∞ . To obtain this, we need to prove that M ∂D c s ,o (Γ + , ∅ ) = 0.For this we use a cascades model (but only the compactness), i.e. we consider the Boothby-Wang contact formon ∂D c s . Following the compactness argument in [13], if we degenerate the contact form on D s (as perturbedby the Morse function) to the Boothby-Wang contact form, the M ∂D c s ,A,o (Γ + , ∅ ) degenerates to cascades.But since | Γ + | (cid:54) = ∅ , there is one level containing nontrivial holomorphic curves in the symplectization ofthe Boothby-Wang contact form, which projects to a holomorphic sphere in D s . However since k > n − D s . As a consequence M ∂D c s ,A,o (Γ + , ∅ ) = ∅ and the claims follows. (cid:3) Proposition 7.13. Let D s be a smooth degree k ≥ n + 1 hypersurface in CP n , assume Γ + is a proper subsetof { γ np min , γ p min , . . . , γ p min (cid:124) (cid:123)(cid:122) (cid:125) k − n } . Then for Γ − (cid:54) = ∅ , M ∂D c s ,o (Γ + , Γ − ) = 0 unless Γ + = { γ np min } , Γ − = { γ np max } or Γ + = { γ p min } , Γ − = { γ p max } .Proof. We can assume Γ − = { γ d p , . . . , γ d r p r } with (cid:80) d i = d by homology and action reasons. Then we can runthe Morse-Bott compactness argument as in Theorem 7.12 for M ∂D c s ,o (Γ + , Γ − ), in the limit cascades modulispace, we necessarily have the holomorphic curve part have zero energy and hence a constant. Thereforedue to the generic point constraint o , we must have p = . . . = p r = p max . Then the expected dimension ofsuch moduli space is 2 d − | Γ + | + vdim M ∂D c s ,A,o (Γ + , Γ − ) − | Γ − | − d = − . Hence vdim M ∂D c s ,A,o (Γ + , Γ − ) = 2 | Γ + | + 2 | Γ + | − 4, which is zero iff | Γ + | = | Γ − | = 1. Hence the claimfollows. (cid:3) Remark 7.14. In the case considered in Proposition 7.13, the only non-empty moduli spaces contributingto the pointed map are M ∂D c s ,o ( γ np min , γ np max ) and M ∂D c s ,o ( γ p min , γ p max ) . Moreover, the algebraic count is not zero as the gradient trajectories from p min to p max traverse the whole manifold. This follows from a cascadesconstruction with gluing as in [13] . Let D be k generic hyperplanes and D s the degree k smooth hypersurface. We use X to denote the exactcobordism from ∂D c to ∂D c s . Our strategy for Theorem 7.6 is showing P( ∂D c s ) is independent of (cid:15) ◦ φ for anyaugmentation (cid:15) of RSFT( ∂D c ) and φ is the BL ∞ morphism induced from X . Then we use the functorialityin Proposition 2.22 and argue that the computation we did with the filling D c s in Corollary 7.10 is in theform (cid:15) D c ◦ φ , where (cid:15) D c is the augmentation of RSFT( ∂D c ) from D c . In principle, this involves a homotopyargument by neck-stretching. To avoid the overhead of introducing homotopies of BL ∞ morphisms, we showthat the formula can be identified on the nose, due to the fact that when transversality in neck-streakingholds, we can identify a fully-stretched moduli space with a sufficiently stretched moduli space by classicalgluing. In the following, we first prove a property explaining the role of k < n − . Proposition 7.15. Let X be the cobordism from ∂D c to ∂D c s as above. Assume Γ + = { γ k p max , γ k p min , . . . , γ k s p min } for (cid:80) si =1 k s ≤ k , we have vdim M X (Γ + , Γ − ) < if Γ − (cid:54) = ∅ and s < n +12 .Proof. Assume Γ − = { γ − r } r ∈ R , which are perturbations from { γ s r p r } r ∈ R , with (cid:80) (cid:80) s r = (cid:80) si =1 k i by homologyand action reasons. Then the expected dimension of M X,A (Γ + , Γ − ), i.e. ind( u ) for u ∈ M X,A (Γ + , Γ − ),satisfies 2 n − s (cid:88) i =1 k i − 1) + ind( u ) + R (cid:88) r =1 ( µ CZ ( γ − r ) + n − 3) = 2 n − . Since ( µ CZ ( γ − s ) + n − ≥ n − − (cid:80) s r − ind( p r ) − | supp s r | and ˇ D supp s r is Weinstein by k ≥ n + 1, wehave ind( p r ) ≤ n − | supp s r | and ( µ CZ ( γ − s ) + n − ≥ ( n − − (cid:80) s r . As a consequence, we haveind( u ) ≤ − − | R | ( n − 3) + 2 s. In particular, ind( u ) < | R | (cid:54) = 0 and s < n +12 . (cid:3) Proposition 7.16. Let φ denote the BL ∞ morphism from the cobordisms X and (cid:15) D c s , (cid:15) D c augmentationsfrom D c s , D c respectively, then we have (cid:98) (cid:96) • ,(cid:15) D c s ( q γ np min q k − nγ p min ) = (cid:98) (cid:96) • ,(cid:15) D c ◦ (cid:98) φ (cid:15) D c ( q γ np min q k − nγ p min ) (cid:54) = 0 , where (cid:98) φ (cid:15) D c is defined in Proposition 2.22, i.e. the map on the bar complex for the linearized L ∞ morphismfrom ( V ∂D c s , { (cid:96) k(cid:15) D c ◦ φ } k ≥ ) to ( V ∂D c , { (cid:96) k(cid:15) } k ≥ ) .Proof. We will apply a neck-stretching for M D c s ,o ( { γ np min , γ p min , . . . , γ p min } , ∅ ) in (2) of Proposition 7.9 along ∂D c for o ∈ D c . Every curve in M D c s ,o ( { γ np min , γ p min , . . . , γ p min } , ∅ ) is somewhere injective. For otherwise,assume u ∈ M D c s ,o ( { γ np min , γ p min , . . . , γ p min } , ∅ ) is a branched cover over u (cid:48) , then we can cap off u (cid:48) with naturaldisks to obtain a homology class A in H ( CP ) with A ∩ D s < k , which is a contradiction. Therefore it issafe to assume M D c s ,o ( { γ np min , γ p min , . . . , γ p min } , ∅ ) is cut out transversely for the stretching J t . In the fullystretched picture, the bottom level containing the marked point o must have k + 1 − n positive punctures.This is because we must have the number of positive punctures no larger than k + 1 − n for otherwise genushas to be created. If there are fewer punctures, then by Proposition 7.7, the curve can not exist. By thesame capping argument, we know that the bottom curve is necessarily somewhere injective. Then by thedimension computation in Proposition 7.7, the only possible bottom level is described in Proposition 7.7.As a consequence, all the levels above the bottom level must be unions of cylinders because of the number LANDSCAPE OF CONTACT MANIFOLDS VIA RATIONAL SFT 63 of positive punctures. Then by considering homology of the cobordism X , we must have the positiveasymptotics of the bottom level is the form of ˇ γ σ I p I, min ∪ { γ p i, min } i ∈ I c , where I ⊂ { , . . . , k } is a subset ofsize n , p I, min , p i, min are minimums. Then by a dimension argument, it is easy to obtain every level abovethe bottom is also cut out transversely. In fact, we only one more level consists of M X ( { γ np min } , { ˇ γ σ I p I, min } )and k − n copies of M X ( { γ p min } , { γ p i, min } ). The transversality of neck-stretching, implies that this 2-levelbreaking can be identified with M D c s ,o ( { γ np min , γ p min , . . . , γ p min } , ∅ ) for sufficiently stretched J t . By Axiom3.25, we can count them to obtain that η D c s ( q γ np min q k − nγ p min ) = η D c ◦ (cid:98) φ (cid:15) D c ( q γ np min q k − nγ p min ) . Then we can use Proposition 5.14 to relate η back to (cid:98) (cid:96) • ,(cid:15) , since q γ np min q k − nγ p min is closed in ( B k +1 − n V ∂D c s , (cid:98) (cid:96) (cid:15) D c s )by Corollary 7.10. The non-vanishing follows from Corollary 7.10. (cid:3) Proposition 7.17. If k < n − , then (cid:98) (cid:96) • ,(cid:15) ◦ (cid:98) φ (cid:15) ( q γ np min q k − nγ p min ) (cid:54) = 0 is independent of the augmentation (cid:15) of RSFT( ∂D c ) .Proof. When k < n − , we have 1 + k − n < n +12 . Then the independence follows from Proposition 7.13 andProposition 7.15, as a component to (cid:98) (cid:96) • ,(cid:15) ◦ (cid:98) φ (cid:15) ( q γ np min q k − nγ p min ) with influence from (cid:15) is described in the graphbelow, which does not exist by dimension reasons by Proposition 7.15. γ np min γ p min γ p min γ p min p , • γ np max φ , φ , (cid:15) (cid:15) (cid:15) Figure 12. The red part has negative dimensionThe non-vanishing then follows from Proposition 7.16. (cid:3) Proof of Theorem 7.6. . If k ≤ n , then D c = T ∗ T k − × C n − k +1 , then H cx ( ∂D c ) = 0 SD by Theorem 7.31. If k = n + 1, then P( ∂D c ) = 2 by Corollary 6.9. For k > n + 1, the lower bound follows from Corollary 7.8.When k < n − and n odd, we have for any augmentation (cid:15) of RSFT( ∂D c ), q γ np min q k − nγ p min ) represents a closedclass in ( B k +1 − n V ∂D c s , (cid:98) (cid:96) (cid:15) ◦ φ ), as the SFT grading of RSFT( ∂D c s ) is even for all generators. In particular, (cid:98) φ (cid:15) ( q γ np min q k − nγ p min ) is closed in ( B k +1 − n V ∂D c , (cid:98) (cid:96) (cid:15) ) for any (cid:15) . Then by Proposition 7.17, (cid:98) (cid:96) • ,(cid:15) ◦ (cid:98) φ (cid:15) ( q γ np min q k − nγ p min ) (cid:54) = 0for any (cid:15) , and we conclude that P( ∂D c ) = k + 1 − n if n + 1 < k < n − . (cid:3) Remark 7.18. Our computation method above can be summarized as finding a curve contributing to the pla-narity by relative Gromov-Witten invariants and then argue the independence of augmentation by dimensioncomputation. The trick we use is arguing closedness in the smooth divisor, where generators are simpler,and prove the upper bounds using the functoriality and arguing everything interesting about the functorial-ity happens purely in the X (i.e. not dependent on augmentation for RSFT( ∂D c ) ). A more scientific wayof computing planarity is deriving a formula for the BL ∞ algebra as well as the augmentation from theaffine variety using log/relative Gromov-Witten invariants. In the context of symplectic (co)homology, suchformula was obtained in [27] . Theorem 7.19. Assume D s is a smooth degree ≤ k < n +12 hypersurface in CP n for n ≥ , then P( ∂D c s ) = 1 and H cx ( ∂D c s ) ≤ ( k − SD . When n is odd, then same holds for ≤ k < n , and moreover we have H cx ( ∂D c s ) ≥ ( k − SD .Proof. Let p be the critical point in (1) of Proposition 7.9, Then we have η D c s ( q γ kp ) (cid:54) = 0 by the same argumentof Corollary 7.10. We can pick the Morse function on D s similar to [79, Proposition 3.1], such that theperturbed contact form has the following property,(7.3) (cid:90) α ∗ γ dp − j (cid:88) i =1 (cid:90) α ∗ γ d i p i < , for d ≤ k, (cid:80) d i = d and one of p i has the property that ind( p i ) < ind( p ) . This energy constraint will helpus exclude certain configurations. Claim. q γ kp is closed in ( B V ∂D c s , (cid:96) (cid:15) ) for any augmentation (cid:15) for ≤ k < n +12 or n odd with ≤ k < n .Proof. Since the parity of the SFT grading of γ dq is the same as the parity of ind( q ). As a consequence,we only need to consider (cid:104) (cid:96) (cid:15) ( q γ kp ) , q γ dq (cid:105) for ind( q ) = n − n is even. By k < n +12 , we have ind( p ) =2 n − k > ind( q ) = n − 1. Then by (7.3), all relevant moduli spaces are empty by action reasons. Hence theclaim follows. (cid:3) Claim. (cid:96) • ,(cid:15) ( q γ kp ) is independent of (cid:15) .Proof. It is sufficient to show that M ∂D c s ,o ( { γ kp } , Γ − ) is empty for Γ − (cid:54) = ∅ . If Γ − (cid:54) = ∅ , then Γ − = { γ d i p i } ≤ i ≤ r with (cid:80) d i = k . Then claim follows from the same argument of Proposition 7.13 and ind( p ) > (cid:3) Claim. We have H cx ( ∂D c s ) ≤ ( k − SD .Proof. We need to show that u k ( q γ kp ) = 0 for any augmentation. By homology reason and (7.3), for d ≤ k , u ( q γ dq ) can only have nontrivial coefficient for q γ d (cid:48) q (cid:48) for d (cid:48) < d and ind( q (cid:48) ) ≥ ind( q ) and ind( q (cid:48) ) = ind( q )mod 2, and for q γ dq for ind( q (cid:48) ) > ind( q ) and ind( q (cid:48) ) = ind( q ) mod 2. Therefore, we have u k ( q γ kp ) = 0 forany augmentation. (cid:3) Note that in our setup here higher ind( p ) means smaller contact action, since we apply the perturbation in the cap of thepositive prequantization bundle instead of the filling of the negative prequantization bundle, see [56]. In particular, the order isreversed compared to [79] and the proof of Theorem 7.28 below. LANDSCAPE OF CONTACT MANIFOLDS VIA RATIONAL SFT 65 Claim. When n is odd, we have H cx ( ∂D c s ) ≥ ( k − SD .Proof. The linearized contact homology has an action filtration, such the filtered theory around period (cid:82) ( γ kp ) ∗ α = k (cid:82) γ ∗ p α , is generated by the k th covered orbits. By the argument in [12], as transversality holdsin this case, the u map on this filtered theory is the same as the u map on the filtered S -equivariantsymplectic cohomology represented by multiplying c ( O ( k ) | D ) to the cochain represented by the criticalpoint, i.e. the Poincar´e dual of the unstable manifold. We have u k − ( q γ kp ) = k k − q γ kp max +terms lowermultiplicities for the maximum p max with ind( p max ) = 2 n − 2. When n is odd, all generators have even SFTdegree, hence u k − ( q γ kp ) (cid:54) = 0 in homology. (cid:3)(cid:3) Remark 7.20. The n being odd condition in Theorem 7.6, 7.12, 7.19, as well as 7.22 below is not necessary,as one can show q γ np min q k − nγ p min is always closed. This is because a differential from q γ np min q k − nγ p min involves counting M Y (Γ + , Γ − ) with Γ + is a subset of { γ np min , γ p min , . . . , γ p min } and Γ − = { γ d i p i } ≤ i ≤ r for (cid:80) d i = k . Therefore ifwe use the Morse-Bott contact form and cascades construction, the relevant holomorphic curve must be coversof trivial cylinders. Then the moduli space M Y (Γ + , Γ − ) is the fiber product of D × M with unstable/stablemanifolds of p min , p i , where M is the space of meromorphic functions on CP with a pole of order n and k − n simple poles and a zero with order p i for all ≤ i ≤ r modulo the R rescaling on meromorphic functionsand the automorphism of the punctured Riemann surface. Then by the nontrivial S action on meromorphicfunctions, we have M Y (Γ + , Γ − ) = 0 unless | Γ + | = | Γ − | = 1 . If | Γ + | = | Γ − | = 1 , M Y (Γ + , Γ − ) is identifiedwith Morse trajectories (here the S action on meromorphic function is identical with the S action inthe automorphism group of surface, hence is trivial on the quotient), whose algebraic count is zero, as weassume the Morse function is perfect. To make this precise, one can follow a Morse perturbation of thecontact form as before. And we use a J that is S -invariant under the rotation in the fiber direction, thenapply the S -equivariant transversality for quotients from [82] , we can argue that M Y (Γ + , Γ − ) = 0 unless | Γ + | = | Γ − | = 1 similar to Floer’s proof of the isomorphism between Hamiltonian Floer cohomology andMorse cohomology. This argument requires building our functors using polyfolds as in [33] . Nevertheless we have the following result that is independent of the parity of n or size of k . Theorem 7.21. Let D be k generic hyperplanes in CP n for n ≥ , then U( D c ) = max { , k + 1 − n } .Proof. The n = 1 case is obvious. For n ≥ 2, we can use Proposition 7.7 to claim that M D c ,o (Γ + , ∅ ) = ∅ forgeneric J as along as | Γ + | < max { , k + 1 − n } . This is because we can obtain the classical transversality of M D c ,o (Γ + , ∅ ) = ∅ , as every curve is a branched cover of a somewhere injective curve with negative expecteddimension. Therefore, we have U( D c ) ≥ max { , k + 1 − n } by Proposition 5.12. On the other hand, thenontrivial relative Gromov-Witten invariant used in Proposition 7.9 implies that U( D c ) ≥ max { , k + 1 − n } by neck-stretching. (cid:3) We also have the following generalization of Theorem 7.6 by the same argument. Theorem 7.22. Let X be a smooth degree m hypersurface in CP n +1 for ≤ m ≤ n and D be k ≥ n generichyperplanes, i.e. D = ( H ∪ . . . ∪ H k ) ∩ X for H i is a hyperplane in CP n +1 in generic position with eachother and X , then P( ∂D c ) = k + m − n for n odd and k + m < n +12 .Proof. We separate the proof into several steps. The Reeb dynamics on ∂D c has the same property withthe CP n case, with the only difference that the minimal Chern number of X is n + 2 − m , which will enterinto the computation of virtual dimensions. Claim. For any Reeb orbits set Γ := { γ , . . . , γ r } for r < k + m − n with (cid:80) [ γ i ] = 0 ∈ H ( D c ) , the virtualdimension of the moduli space M D c ,A,o (Γ , ∅ ) is negative for any A .Proof. This follows from the same argument in Proposition 7.7, with the difference that c ( u ri =1 u i ) =2 N ( n + 2 − m ). Therefore we haveind( u ) ≤ N ( n + 2 − m ) − − r (cid:88) i =1 (2 (cid:88) s i − p i )) ≤ N ( n + 2 − m ) − − kN − r = 2( N − n + 2 − m − k ) + 2( r + n − m − k ) < , since r < m + k − n and k ≥ n, m ≥ 2. This computation also implies the following claim. This also showsthe lower bound. (cid:3) Claim. Assume D s in the generic intersection of a degree k hypersurface in CP n +1 with X . Then we have η D c s ( q γ n +1 − mp min q k + m − n − γ p min ) (cid:54) = 0 , and q γ n +1 − mp min q k + m − n − γ p min is closed in ( B k + m − n V ∂D c s , (cid:98) (cid:96) (cid:15) D c s ) .Proof. That η D c s ( q k + m − n − γ p q γ np ) (cid:54) = 0 follows from the non-vanishing of GW X,D , , ( n +1 − m, ,..., ,A ([ pt ] , [ D s ] , . . . , [ D s ] (cid:124) (cid:123)(cid:122) (cid:125) k + m − n )from [38] for A is the positive generator of H ( X ) and the same argument in Proposition 7.9. The remainingof the argument is exactly same as Corollary 7.10. (cid:3) Then by the same neck-stretching argument in Proposition 7.16, we have (cid:98) (cid:96) • ,(cid:15) D c s ( q γ n +1 − mp min q k + m − n − γ p min ) = (cid:98) (cid:96) • ,(cid:15) D c ◦ (cid:98) φ (cid:15) D c ( q γ n +1 − mp min q k + m − n − γ p min ) (cid:54) = 0 . Next Proposition 7.13 and Proposition 7.15 also holds, as the dimension computation there is essentially fortrivial homology class, which does not depend on m . It is important to note that in the proof of Proposition7.15, we use that ˇ D I is Weinstein is obtain an upper bound of Morse indices. Such property also holds hereas we assume k ≥ n . Then the remaining of the proof is the same as Theorem 7.6. (cid:3) From the proof above, the source of holomorphic curves is supplied by the degree 1 holomorphic curvesin X for m ≤ n . For m = n + 1, the degree 1 curve does not unirule X anymore, but a degree 2 curveunirules X . In the proof Theorem 7.6 and Theorem 7.22, being degree 1 is used in several places to obtainsomewhere injectivity (the capping argument). Indeed, for m = n + 1, the situation is different, we willprove P( ∂D c ) ≥ D is a generic intersection of X with a hyperplane in CP n +1 . For m ≥ n + 2, then X is not uniruled, which implies D c is not k uniruled for any k by [55], therefore P( ∂D c ) = ∞ by Corollary5.15.In view of Theorem 5.13, Theorem 7.6, Theorem 7.21, and Theorem 7.22, we make the following conjecture. Conjecture 7.23. V is a k -uniruled affine variety then P( V ) < ∞ and P( V ) = U( V ) = AU( V ) . On the other hand, by Theorem 7.12, it is not true that any uniruled affine variety has a contact boundarywith finite planarity. It is subtle question to determine which affine variety with a CP n compactification hasa finite planarity boundary. Question 7.24. Let D be k generic hyperplanes in CP n , is P( ∂D c ) always finite? LANDSCAPE OF CONTACT MANIFOLDS VIA RATIONAL SFT 67 Theorem 7.6 and Theorem 7.22 along with Lemma 7.1 and Lemma 7.3 imply that there are many sequencesof contact manifolds where exact cobordisms only exist in one direction. On the other hand, exact embeddingproblems in the flavor of Theorem 7.21 are studied in [37]. It is an interesting question to determine whetherthose embedding obstructions can lift to cobordism obstructions.7.3. Links of singularities. Another natural source of contact manifolds is links of isolated singularities.In the following, we will consider the Brieskorn singularities and quotient singularities from cyclic actionson C n .7.3.1. Brieskorn singularities. A Brieskorn singularity is of the following form x a + . . . + x a n n = 0 , for 2 ≤ a ≤ . . . ≤ a n . We use a to denote the sequence, the link LB ( a ) is defined to be the intersection LB ( a ) := { ( x , . . . , x n ) ∈ C n +1 | x a + . . . + x a n n = 0 } ∩ S n +1 , which is (2 n − LB ( a ) is exactly fillable by the smooth affine variety x a + . . . + x a n n = 1, which is called theBrieskorn variety. Moreover, we have the following fact about embedding relations for Brieskorn varieties. Proposition 7.25. We say a ≤ b iff a i ≤ b i for all i . Then if a ≤ b , the Brieskorn variety of a embedsexactly into the Brieskorn variety of b . In particular LB ( a ) ≤ LB ( b ) in Con ≤ . Brieskorn varieties were showed in [78, Theorem A] to support k -dilations. In the following, we willcompute the hierarchy functor H cx for LB ( a ). In particular, we will have either a computation or anestimate of H cx ( LB ( a )) for any a from the theorem below and Proposition 7.25. Theorem 7.26. We use LB ( k, n ) to denote the contact link of the Brieskorn singularity x k + . . . + x kn = 0 ,then H cx ( LB ( k, n )) is(1) ( k − SD if k < n ;(2) ≥ ( k − SD if k = n and > P if k = n + 1 ;(3) ∞ P , if k > n + 1 .Proof. The associated Brieskorn variety V ( k, n ) carries a k − k ≤ n .Moreover, the dilation is provided by a simple Reeb orbit γ p with ind( p ) = (2 n − k ). Since we can considerthe filtered theory generated by all simple Reeb orbits, transversality conditions in [12] hold, and the orderof semi-dilation of LB ( k, n ) for augmentation from V ( k, n ) is k − 1. In particular,H cx ( LB ( k, n )) ≥ ( k − SD for k ≤ n . Next we need to argue that this semi-dilation is independent of the augmentation when k < n .To see that, we first claim that q γ p contribute to P = 1 is independent of augmentation. If not, we havea non-empty moduli space M LB ( k,n ) ,o ( { γ p } , { γ q } ), whose expected dimension is ind( q ) − ind( p ) + 2 − n =ind( q ) + 2 k + 2 − n < k < n since ind( q ) ≤ n − 2. Therefore planarity of is always 1 if k < n .Moreover, u i ( q γ p ) is independent of augmentation as we are at the minimal period, there is no room for u todepend on augmentations. To see the case k = n + 1, since the log-Kodaira dimension of the correspondingBrieskorn variety V is 0, we know that V is not algebraically 1-uniruled. Hence the planarity is greater than1 by Corollary 5.15. When k > n + 1, the Brieskorn variety admits a compactification that is not uniruled,hence the planarity is infinity by Corollary 5.15. (cid:3) Remark 7.27. If Conjecture 6.6 was proven, one can get better estimate for H cx ( LB ( a )) by writing LB ( a ) as an open book with a Brieskorn variety page. In the context of symplectic cohomology, computation insuch spirit can be found in [78, Proposition 3.27] . Proof of Theorem B. This theorem is a combination of Theorem 3.16, Theorem 3.17, Theorem 3.21, Corol-lary 5.15, Theorem 6.5, Corollary 6.9 and Theorem 7.26. (cid:3) Quotient singularities by cyclic groups. Let Z k acts on C n on the diagonal action by multiplying e πik ,then the link of the quotient singularity C n / Z k is the quotient contact manifold ( S n − / Z k , ξ std ). Suchcontact manifolds provide many examples of strongly fillable but not exactly fillable contact manifolds [79].In fact, the symplectic part of [79] is a computation of the hierarchy functor H cx in the context of symplecticcohomology, which will be rephrased as follows. Theorem 7.28. Let Y be the quotient ( S n − / Z k , ξ std ) by the diagonal action by e πik for n ≥ .(1) If n > k , we have H cx ( Y ) = 0 SD .(2) If n ≤ k , we have H cx ( Y ) ≤ ( n − SD .Proof. We follow the same setup as in [79, Proposition 3.1]. We have a non-degenerate contact form on ξ std by perturbing with a C -small perfect Morse function f on CP n − , such that Reeb orbits are the following.(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 with ind( q i ) = 2 i . (2) The period of γ j is 1 + (cid:15) j .(3) (cid:15) j < (cid:15) j +1 k , (cid:15) j (cid:28) γ ji with the natural disk in O ( − k ) satisfies µ CZ ( γ ji )+ n − i +2 j − Claim. We have P( Y ) = 1 for n ≥ , ∀ k .Proof. By the same argument as [79, Step 3 of Proposition 3.1], we have M Y,o ( { γ k } , ∅ ) = k for n ≥ − (cid:54) = ∅ ,we have M Y,o ( { γ k } , Γ − ) = ∅ by action reasons, unless Γ − = { γ d i } ≤ i ≤ r for (cid:80) d i = k . In this case, a curvein M Y,o ( { γ k } , Γ − ) is necessarily a branched cover over a trivial cylinder. In particular, M Y,o ( { γ k } , Γ − ) = ∅ for generic o . Since all Reeb orbits have even SFT degree, we have q γ k is closed in any linearized contacthomology, and the planarity is 1 for any augmentation (which exists) by q γ k . (cid:3) Claim. If k < n , then we have H cx ( Y ) = 0 SD .Proof. By action reasons, u ( q γ k ) can only have nontrivial coefficients in q γ d for d < k . Note that the filteredlinearized contact homology with action supported around d is generated by q γ dr . Since the transversalityfor all moduli spaces for this filtered homology holds, the argument in [12] implies that it is isomorphic tothe filtered S symplectic cohomology with action centered around d , i.e. the homology is H ∗ ( CP n − ) withthe u map is the multiplication by c ( O ( k )). As a consequence, we have u ( q γ dr ) = kq γ dr − + (cid:80) d − i =1 (cid:80) rj =0 a ij q γ ij by action reasons. Therefore for any augmentation, there exist c ij such that u ( q γ k + (cid:80) k − i =1 (cid:80) k − ij =1 c ij q γ ij ) bythe same argument as [79, (3.2)]. In order to finish the proof, it is sufficient to prove M Y,o ( γ ij , Γ − ) = ∅ for i + j ≤ k and j > 0. This follows from the same dimension computation in [79, Step 7 of Proposition 3.1]and is the place where n > k is essential. Then q γ k + (cid:80) k − i =1 (cid:80) k − ij =1 c ij q γ ij also contributes P = 1 and is killedby u . In particular, H cx ( Y ) = 0 SD . (cid:3) It is important to note that now we perturb f is the prequantization filling in [79], therefore higher Morse index meanslager period, which is reverse to Theorem 7.6. LANDSCAPE OF CONTACT MANIFOLDS VIA RATIONAL SFT 69 Claim. If k ≥ n , then we have SD ≤ H cx ( Y ) ≤ ( n − SD .Proof. First note that all generators have even degree, hence any maps { (cid:15) k } k ≥ form an augmentation. Bythe argument in Proposition 3.13 and Remark 7.14, we have that M Y,o ( { γ dn − } , { γ d } ) = 1 for 1 ≤ d ≤ k − (cid:104) ∂ (ˇ p k + ) , ˆ p k − (cid:105) in Theorem 9.1, Lemma 9.4], weknow that (cid:104) u ( q γ k + i ) , q γ k − i (cid:105) = ( k + − k − ) (cid:15) ( q γ k + − k − ) for augmentation { (cid:15) k } k ≥ . If for every 1 ≤ i ≤ k − n , wehave (cid:15) ( q γ i ) = 0. Then we have u n ( q γ k ) = 0, since we have u ( q γ k ) = (cid:80) k − j =1 ( k − j ) (cid:15) ( q γ k − j ) q γ j . Otherwise, weassume i is the minimum among { , . . . , k − n } such that (cid:15) ( q γ i ) (cid:54) = 0. As a consequence, we have planarity 1contributed by q γ i n − by M Y,o ( { γ in − } , { γ i } ) = 1. Since i the minimal one with nontrivial augmentation,we know that u n ( q γ in − ) = 0. Hence we have H cx ( Y ) ≤ ( n − SD . (cid:3) Claim. If k = n , then we have H cx ( Y ) ≥ SD Proof. It suffices to find one augmentation such that the order of semi-dilation is 1. We choose our augmen-tation to be (cid:15) ( q γ ) = − n and (cid:15) k = 0 in all other cases. We first list the following expected dimensions ofvarious moduli spaces.(1) vdim M Y,o ( { γ nmp i } , ∅ ) = 2 i + 2 n ( m − ≥ 0, which is positive, unless i = 1 , m = 1, where we know M Y,o ( { γ n } , ∅ ) = n .(2) For l > 0, vdim M Y,o ( { γ mm + lp i } , { γ p , . . . , γ p (cid:124) (cid:123)(cid:122) (cid:125) l } ) = 2 i + 2 l + 2 n ( m − i = n − , l = 1 , m = 0, M Y,o ( { γ n − } , { γ } ) = 1; (2) i < n − , l = n − i > , m = 0, then M Y,o ( { γ lp n − l } , { γ p , . . . , γ p (cid:124) (cid:123)(cid:122) (cid:125) l } ) = ∅ by the argument of Proposition 7.13.(3) For l > 0, vdim M Y ( γ nm + lp i , γ lp j , ∅ ) = vdim M Y ( γ nm + lp i , γ ∗∗ , γ lp j , ∅ , ∅ ) = 2 mn + 2 i − j − 2. Then it iszero iff (1) m = 0 , i = j + 1, which corresponding to (cid:104) u ( q γ li +1 ) , q γ li (cid:105) = n ; (2) m = 1 , i = 0 , j = n − (cid:104) u ( q γ n + l ) , q γ ln − (cid:105) = a l .(4) For l, s > 0, vdim M Y ( γ nm + l + sp i , γ lp j , { γ p , . . . , γ p (cid:124) (cid:123)(cid:122) (cid:125) s } ) = 2 mn + +2 s + 2 i − j − 2. Then it is zero iff m = 0 , j = s + i − 1. In this case we have M as well as the corresponding M are non-empty iff s = 1, for the otherwise, we have s + i − > i . We can use the compactness argument in Proposition7.13 and that the stable manifold of q i does not intersect with unstable manifold of q s + i − . When m = 0 , s = 1 , j = i , this corresponds to (cid:104) u ( q γ l +1 i ) , q γ li (cid:105) = (cid:15) ( q γ ) = − n .By (1) and (2), to supply for planarity 1, we must have aq γ n + bq γ n − with a − b (cid:54) = 0. Note that u ( aq γ n + bq γ n − ) = − anq γ n − + bnq γ n − (cid:54) = 0. Therefore if the order of semi-dilation of this augmentation is smaller than1, then there exists A generated by generators other than q γ n , q γ n − , such that u ( A ) = anq γ n − − bnq γ n − .The only way to eliminate − bnq γ n − is have bq γ n − in A , which adds bnq γ n − to u ( A ). The only way tocompensate such term is add a bq γ n − to A . We can keep the argument going, and claim that A has b (cid:80) n − i =2 q iγ n − i , then u ( A ) = bnq γ n − − bnq γ n − (cid:54) = anq γ n − − bnq γ n − , since a − b (cid:54) = 0. The claim follows. (cid:3) Although such structure originally appears as part of differential in symplectic cochain complex, it contributes to the u -mapin the S -equivariant symplectic cohomology, see [81, § 5] for discussion. (cid:3) When n ≤ k , there are augmentations with zero order of semi-dilation. For example, one can use theaugmentation from natural prequantization bundle filling, then the order of semi-dilation is 0 since thesymplectic cohomology vanishes [67]. However, we conjecture that H cx ( Y ) ≥ SD whenever n ≤ k . It ispossible that there are BL ∞ augmentations that are not from (even singular) fillings. Note that n > k isthe region where the quotient singularity is terminal. Hence we ask whether there is relation between thisalgebro-geometric property with contact property of the link via the hierarchy functor H cx . Conjecture 7.29. For discrete G ⊂ U ( n ) , if C n /G is an isolated singularity, then H cx ( S n − /G, ξ std ) = 0 SD if the singularity is terminal. Combining with Theorem 7.26, we can also ask the following question. Question 7.30. Is the planarity of an isolated terminal singularity always ? Is it true for hypersurfacesingularities? By a similar argument to Theorem 7.28, we have the following. Theorem 7.31. Let V be an exact domain, then H cx ( ∂ ( V × D )) = 0 SD .Proof. By Corollary 6.9, we have P( ∂ ( V × D )) = 1. Note the planarity is provided the simple Reeb orbitwrapping around V . Then the transversality in [12] holds, we can use the symplectic cohomology descriptionto compute the order of semi-dilation, which is zero for augmentation from V × D by [63]. The relevant u -mapis again independent of augmentations by action reasons similar to Theorem 7.26. The claim follows. (cid:3) The computation in Theorem 7.31 carried out in the form of symplectic cohomology is the symplecticinput used to prove uniqueness results of exact fillings for ∂ ( V × D ) in [80].7.4. An obstruction to IP. In dimension 3, obstructions to planar open book decomposition were studiedfrom many different perspectives in [31, 64]. In higher dimensions, obstructions to supporting an iteratedplanar structure were found in [5]. By Corollary 5.15 and Theorem 6.5, we the following easy to checkobstruction to iterated planar structure. Corollary 7.32. If contact manifold Y admits an exact filling that is not k -uniruled for any k , then Y isnot iterated planar. As an application of this corollary, we have the following. Corollary 7.33. Let Q be a hyperbolic manifold of dimension ≥ , then S ∗ Q is not iterated planar.Proof. The claim follows from a result of Viterbo [28, Theorem 1.7.5] that T ∗ Q is not k -uniruled for any k . (cid:3) For other classes of cosphere bundles, by Theorem 7.26, H cx ( S ∗ S n ) = 1 SD for n ≥ 2. By Corollary6.9, H cx ( S ∗ T n ) = 2 P for n ≥ 2. Assuming Claim 4.9 holds, since SH ∗ ( T ∗ Q ) (cid:54) = 0 for any Q , we knowH cx ( S ∗ Q ) > SD . As a consequence there is no exact cobordism from S ∗ Q to ∂ ( V × D ) for any Liouvilledomain V , which is a generalization of a result of Gromov [42]. By [78, Proposition 5.1], T ∗ Q admits a k -dilation for some k ≥ n -manifold Q , i.e. if H n ( Q ; Q ) → H n ( Bπ ( Q ); Q ) vanishes,then we can update the estimate H cx ( S ∗ Q ) by figuring out k . For Lagrangian Q that is a K ( π, 1) space, wehave H cx ( S ∗ Q ) ≥ P , since T ∗ Q carries no k -semi-dilation for any k . LANDSCAPE OF CONTACT MANIFOLDS VIA RATIONAL SFT 71 Corollary 7.34. For every n ≥ , there exists a tight S n − with the standard almost contact structure thatis not iterated planar.Proof. Note that the contact boundary of the Brieskorn variety x n +20 + . . . + x n +2 n = 1 has planarity order ∞ by Theorem 7.26. Then we can increases the indexes to get another Brieskorn manifold Y which is anexotic sphere. By the monoidal structure and functoriality of H cx , we know H cx ( k Y ) = ∞ P , where k such that k Y is the standard smooth sphere with thestandard almost contact structure. The claim follows. (cid:3) Corollary 7.35. In all dimension ≥ , if ( Y, J ) is an almost contact manifold which has an exactly fillablecontact representation ( Y, ξ ) . Then there is a contact structure ξ (cid:48) in the homotopy class of J , such that ( Y, ξ (cid:48) ) is not iterated planar. In particular, any almost contact simply connected -manifolds admits acontact representation which is not iterated planar.Proof. Let Y (cid:48) be the tight sphere from corollary 7.34, since P( Y ) > Y has an exact filling, thenH cx ( Y Y (cid:48) ) = ∞ P . By Corollary 7.32, Y Y (cid:48) is not iterated planar. The last claim follows from anyalmost contact simply connected 5-manifold is almost Weinstein fillable [39], in particular, there is a contactrepresentation that is Weinstein fillable by [20]. (cid:3) The order on Con . It is a natural question to ask whether the poset Con ≤ is a totally ordered set. Forthis, we need to throw out ∅ , since an overtwisted contact manifold is obviously not comparable with ∅ . Weuse Con (cid:54) = ∅≤ denote the full subcategory without ∅ . We first give a simple argument for Weinstein cobordisms. Proposition 7.36. In any odd dimension ≥ , There exists two contact manifolds, such that there are noWeinstein cobordisms in either direction.Proof. Take Y = ∂ ( V × T ∗ S ) such that V is the 2 n -dimensional Liouville domain that is not Weinsteinconstructed in [53] for n ≥ 5. Then Y is asymptotically dynamically convex by [83, Theorem K] and im ∆ ∂ contains an element of degree 2 n − 1, hence Y is not Weinstein fillable by [83, Corollary 4.19]. Therefore thereis no Weinstein cobordism from ( S n +5 , ξ std ) to Y . On the other hand, since any exact filling of ( S n +5 , ξ std )has vanishing symplectic cohomology [68, Corollary 6.5] but SH ∗ ( V × T ∗ S ) (cid:54) = 0, we know that there is noexact cobordism from Y to ( S n +5 , ξ std ). (cid:3) Remark 7.37. One can replace ( S n +5 , ξ std ) in the proof above by a flexibly fillable representative of themaximal element in the almost Weinstein cobordism category with vanishing first Chern class [18, Theorem1.2] . Then at least one side of the non-existence does not follow from a topological obstruction. On the other hand, when we consider strong cobordisms, there are a lot more morphisms. In particular, ifwe include ∅ into the discussion, the existence of symplectic cap [25, 49] implies that anything with a strongfilling is equivalent to ∅ in Con ≤ ,S . Even if we throw out ∅ and even restrict to the case of connected strongcobordisms, the existence of strong cobordism is much less rigid compared to the the counterparts for exactor Weinstein cobordisms by [74]. Nevertheless, the following question seems to be open. Question 7.38. Is there a pair of contact manifolds without connected strong cobordisms in either direction? We can also consider the analogous question in Con . Question 7.39. Is there a pair of contact manifolds without connected exact cobordisms in either direction? When dimension is 3, an affirmative answer to the above question is explained to us by Chris Wendlbased on the not exactly fillable contact manifold ( Y, η ) found by Ghiggini [40]. As observed in [16], Y admits a Liouville pair ( η , η ) [53, Definition 1], so in particular there is a connected exact filling for( Y, η ) (cid:116) ( − Y, η ). If there is an exact cobordism from ( Y, η ) to ( S , ξ std ), then there is a connected exactfilling of ( S , ξ std ) (cid:116) ( − Y, η ), contradicting that ( S , ξ std ) is not co-fillable [31]. On the other hand, by [40],there is no exact cobordism from ( S , ξ std ) to ( Y, η ). In the following, we will explain a strategy for higherdimensional cases.We adopt the notation of [19], and consider the Bourgeois manifold BO ( D ∗ S n , τ k ) associated to the openbook OBD ( D ∗ S n , τ k ), where n ≥ k ≥ 1, and τ is the Dehn-Seidel twist. It was shown in [19, TheoremG and Remark 1.2] that the setFill( n ) = { k ∈ Z (cid:12)(cid:12)(cid:12) BO ( D ∗ S n , τ k ) is strongly fillable } is a subgroup, and satisfies Fill( n ) = k ( n ) Z , for some k ( n ) > ∈ N . In particular, for every n , there exists arbitrarily large k for which BO ( D ∗ S n , τ k )is not strongly fillable.On the other hand, those contact structures are weakly fillable by [51, Theorem A]. By [53, Proposition 6],a weak filling can always be deformed into a stable Hamiltonian filling near the positive end , and a stableHamiltonian filling for a contact structure has all the essential structures to define SFT, c.f. [24, 48]. Oneplace we need to pay attention to is potential holomorphic caps possibly with multiple negative puncture,which will break the BL ∞ algebra structure (without deformation) in general. Definition 7.40. Let M be a n − oriented dimensional manifold. A SHS is a pair ( λ, ω ) of a one form λ and a two form ω such that the following holds.(1) λ ∧ ω n − > .(2) ker ω ⊂ ker d λ . The Reeb vector field R associated to ( λ, ω ) is determined by λ ( R ) = 1 and ι R ω = 0. To relate SHS withcontact structure, we will also consider the following cobordisms, which is a combination of [48, Definition1.9] and [76, § Definition 7.41. A symplectic manifold ( X, ω ) is a stable cobordism from a SHS ( M − , ( λ − , ω − )) to a (co-oriented) contact ( M + , ξ ) if the following holds.(1) ∂W = M − (cid:116) M + .(2) ω − = ω | M − , ω | ξ is non-degenerate and the induced orientation is the same as the orientation on ξ .(3) The stabilizing vector field V − determined λ − = ι V − ω near M − points inward along M − .(4) There exists a non-degenerate contact form α for ξ , such that the Reeb vector field R α generates thecharacteristic line field on M + . The stabilizing vector field V + determined α = ι V + ω near M + pointsoutward along M + . .(5) ξ admits a complex structure J which is tamed by both d λ | ξ and ω | ξ . We refer readers to [76, § 6] for almost complex structure J on the completion (cid:98) X . One key condition is that J is tamed by both d λ and ω on ξ on the positive cylindrical end. We shall call such almost complex structureadmissible. The importance of the last two conditions of the Definition 7.41 is that the compactification of This is not true for negative end. LANDSCAPE OF CONTACT MANIFOLDS VIA RATIONAL SFT 73 holomorphic curves in the completion (cid:98) X has a upper level from SFT buildings in the symplectization of thecontact manifold ( M + , ξ ). In particular, one expects a version of functorial properties for SFT invariants of( M + , ξ ) just like the usual cobordism case if the breaking combinatorics is the same as before. In the caseof BO ( D ∗ S n , τ k ), we need the following fact from the combination of [51, Theorem A] and [53, Proposition6]. Proposition 7.42. There is stable cobordism X := ([0 , × OBD ( D ∗ S n , τ k ) × T , ω ) from the SHS H − :=( OBD ( D ∗ S n , τ k ) × T , ( α, d α + ω T )) to the contact manifold Y + := BO ( D ∗ S n , τ k ) , where α is any contactform on OBD ( D ∗ S n , τ k ) . Moreover, we have ω = π ∗ ω T ∈ H ( X ) for π : X → T with ω T is a rationalvolume form on T . We note the following. Lemma 7.43. For any admissible J on the cobordism in Proposition 7.42, there exists no non-constantpunctured rational J -holomorphic curve with no positive ends.Proof. Assume that u is a non-constant rational holomorphic curve having only negative ends, if any. Sincethe Reeb vector field of H is R α on OBD ( D ∗ S n , τ k ), the map v = π ◦ u , where π is the projection to T isthe natural projection, extends to a continuous map on the closed sphere. Since π ( T ) = 0, v is necessarilycontractible, and so (cid:82) u ω T = (cid:82) v ω T = 0. If A ≥ α -action of the negative asymptotics of u , we then see that 0 < (cid:90) u ω = − A ≤ , a contradiction. (cid:3) Due to the absence of caps, we can define a BL ∞ algebra for H − and the stable cobordism in Proposition7.42 defines a BL ∞ morphism from RSFT( Y + ) to RSFT( H − ), with one caveat that we need to use a groupring coefficient as in [48, Proposition 2.5] to obtain compactness. More precisely, RSFT( H − ) is definedusing Q [ H ( H − ) / ker ω T ] (the completion of the group ring with respect to the evaluation by ω T ). Itis natural to expect that the hierarchy of the split H − using the group ring coefficient is at least thehierarchy of OBD ( D ∗ S n , τ k ) as we can use a split J for the SHS. The functorial property of the stablecobordism holds for RSFT( Y + ) with coefficient in Q [ H ( Y + ) / ker ω | Y + ]. Since H cx ( OBD ( D ∗ S n , τ k )) ≥ SD by Theorem 7.26, whose argument works for any coefficient for k ≥ 2, then we have the hierarchy of Y + using Q [ H ( Y + ) / ker ω | Y + ] coefficient is at least 1 SD for k ≥ 2. Hence, we ask the following question. Question 7.44. Is H cx ( BO ( D ∗ S n , τ k )) ≥ SD (in Q coefficient) for k ≥ ? In general, is H cx ( BO ( V, φ )) ≥ H cx ( OBD ( V, φ )) ? If the answer to the above question is affirmative, then BO ( D ∗ S n , τ k ) , ( S n +3 , ξ std ) are pair of contactmanifolds without exact cobordisms in either directions for k ≥ / ∈ Fill( n ) by Theorem A, as the hierarchyof ( S n +1 , ξ std ) is 0 SD for any coefficient by the argument of Theorem 7.31. In the group ring context, sincethe group ring is generated by [ T ], the functorial property of BL ∞ morphism for the hypothetical exactcobordism from BO ( D ∗ S n , τ k ) to ( S n +3 , ξ std ) holds for the group ring coefficient if H ( BO ( D ∗ S n , τ k )) → H ( X ) is injective on [ T ], see [48, § Proposition 7.45. There is no exact cobordism X from Y := BO ( D ∗ S n , τ k ) for k ≥ to ( S n +3 , ξ std ) suchthat H ( Y ) → H ( X ) is injective on [ T ] . The rationality of ω T is required in [24, Prop 2.18] to obtain [53, Proposition 6]. The hierarchy functor H cx can only obstruct exact cobordisms in one direction. Another natural way toanswer Question 7.39 is building a functor from Con to a poset, then the preimage of two incomparableelements from the poset will be incomparable in Con . A natural candidate theory is the grid of torsions (orthe analogous grid of planarity) introduced in § Con ≤ in dimension 3 [32], i.e. thereis an exact cobordism from an overtwisted manifold to any contact 3-manifold. It is natural to questionabout the other direction. An element in a poset is called maximal, if there is no element strictly greaterthan it. An element is greatest if every element is smaller than it. Question 7.46. Is there a maximal element in Con ≤ ? Is there is a greatest element in Con ≤ ? In the context of Weinstein cobordism, there are geometric constructions of a contact manifold Y thatis Weinstein cobordant both from Y and Y given that there are Weinstein cobordism from Y (cid:48) to Y , Y indimension ≥ Con ≤ or Con ≤ ,W . On the other hand, in Con ≤ ,S , ∅ is the greatest element. 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