Towards a good definition of algebraically overtwisted
aa r X i v : . [ m a t h . S G ] F e b TOWARDS A GOOD DEFINITION OF ALGEBRAICALLY OVERTWISTED
FR´ED´ERIC BOURGEOIS AND KLAUS NIEDERKR ¨UGER
Abstract.
Symplectic field theory (SFT) is a collection of homology theories that provideinvariants for contact manifolds. We show that vanishing of any one of either contact homology,rational SFT or (full) SFT are equivalent. We call a manifold for which these theories vanish algebraically overtwisted . Introduction
A (coorientable) contact structure ξ on a (2 n + 1)–dimensional manifold M is a hyperplanefield of the tangent bundle that can be written as the kernel of a 1–form α that satisfies theinequality α ∧ dα n = 0. On closed manifolds contact structures are stable under deformations,and their equivalence classes are discrete sets. Much effort has been invested in understanding3–dimensional contact manifolds, and a rich theory has been created. For this, many differenttechniques have been applied ranging from topological ones to different algebraic invariants likeHeegaard Floer theories [OzSz05], or contact homology. One of the first basic properties that werediscovered for 3–manifolds was the distinction between overtwisted and tight contact structures.Being overtwisted is a topological property, but it has many consequences for algebraic invariants.The algebraic invariant under consideration is Symplectic Field Theory (SFT), introduced byEliashberg, Givental and Hofer [EGH00]. This large formalism contains in particular severalversions of contact invariants, such as contact homology. These invariants, described in Section 2,are based on the count of holomorphic curves in the symplectization of a contact manifolds andare defined for contact manifolds of any odd dimension.It has been proved by Eliashberg and Yau [Yau06] that the contact homology of any overtwisted3–manifold is trivial. Given that the classification of such manifolds is purely topological [Eli89],it was to be expected that also the other invariants of SFT do not provide interesting information.In this article we confirm this conjecture by a more general result for contact manifolds of anyodd dimension. In fact, it follows already from purely algebraic properties that vanishing of anyof the different homology theories implies that the other ones also have to be trivial. Theorem 1.
Let ( M, α ) be a (2 n − –dimensional closed contact manifold with a non degeneratecontact form. All of the following statements are equivalent: (i) The contact homology (without marked points) of ( M, α ) vanishes. (ii) The rational SFT (without marked points) of ( M, α ) vanishes. (iii) The SFT (without marked points) of ( M, α ) vanishes. (iv) The contact homology with marked points of ( M, α ) vanishes. (v) The rational SFT with marked points of ( M, α ) vanishes. (vi) The SFT with marked points of ( M, α ) vanishes.Remark . Note that any of these invariants may be defined over different coefficient rings. In thetheorem we assume that the same ring is used for the different homologies.To date no final generalization of overtwisted contact manifolds to higher dimensions has beenfound. This theorem, together with [Yau06], motivates the following definition that can be easilyapplied to any dimension.
Definition.
A contact manifold (
M, α ) is called algebraically overtwisted , if any of the ho-mologies listed in Theorem 1 vanishes.
Several examples of algebraically overtwisted contact structures are known: As stated abovethe contact homology of overtwisted manifolds vanishes [Yau06]. Similarly Otto van Koert andthe first author of this article have extended this result by showing that the contact homology ofnegatively stabilized contact manifolds of any dimension is trivial [BvK08].
Corollary 2.
All of these examples are thus algebraically overtwisted, and have vanishing SFT.
A tentative generalization of the notion of overtwisted to higher dimensions was given in [Nie06],where
P S –overtwisted manifolds were defined. Current work by the authors [BN] will show that
P S –overtwisted manifolds also have vanishing contact homology.
Acknowledgments.
At the time of the creation of this article, we were both working at the
Universit´e Libre de Bruxelles . The second author was being funded by the
Fonds National de laRecherche Scientifique (FNRS).We thank Otto van Koert for fruitful discussions, and Hansj¨org Geiges for valuable comments.2.
Contact homology and variants of SFT
The different contact invariants above are homologies of certain differential graded algebraswith a –element, that means Definition.
A differential graded algebra ( A , ∂ ) is a graded algebra, equipped with a differential ∂ : A ∗ → A ∗− such that ∂ = 0 and which satisfies the graded Leibniz rule ∂ ( a · b ) = ( ∂a ) · b +( − | a | a · ( ∂b ).The vanishing results in this article are all based on the following easy remark. Remark . Let ( A , ∂ ) be a differential graded algebra with , and denote the homology over thatalgebra by H ∗ ( A , ∂ ). The homology vanishes if and only if is an exact element, that means ifthere is an element a ∈ A such that ∂a = . Proof.
The –element is closed, because ∂ = ∂ ( · ) = ( ∂ ) · + ( − | | · ∂ = 2 ∂ , hence itis obvious that has to be exact for the homology to vanish. On the other hand, if there is anelement a ∈ A such that ∂a = , then the whole homology has to vanish, because we can writean arbitrary cycle b ∈ A as b = · b = ( ∂a ) · b = ∂ ( a · b ) − ( − | a | a · ∂b = ∂ ( a · b ). (cid:3) Our proof of the main theorem is based on using algebraic properties and exploiting that thecontact homology algebra embeds naturally into both the rational SFT algebra and the full SFT al-gebra. In particular the –elements all coincide under these inclusions. Before starting to describethe actual proof, we will briefly repeat how the algebras are defined, and what the correspondingboundary operators are (see [EGH00]). Since the algebras of the different versions of SFT are allbuild up by using closed Reeb orbits, and the corresponding differentials all count certain holo-morphic curves, we will first fix some common notation. Readers familiar with symplectic fieldtheory can safely skip the next section, and skim back when needed. Notation: Closed Reeb orbits and holomorphic curves.
Let α be a contact form for the(2 n − M, ξ ). The associated Reeb vector field R α is defined asthe unique solution of the equations α ( R α ) = 1 and i R α dα = 0 . A closed Reeb orbit γ of R α is called non degenerate, if the corresponding Poincar´e return mapdoes not have eigenvalues of size 1. We call α a non degenerate contact form, if all of its Reeborbits are non degenerate. Any contact form can be made non degenerate by a small perturbation,and so we will always assume from now on that α is non degenerate. The associated Reeb vectorfield R α then has only countably many closed orbits, and we can introduce a total order on the setof closed Reeb orbits γ = ( γ , γ , . . . ). Note that multiple orbits are considered to be completelyunrelated to the corresponding simple orbits. Denote the period of an orbit γ by T ( γ ), and itsmultiplicity by κ γ . We fix a parametrization for each closed orbit γ k by choosing a base point on γ k . A convenient short hand notation to handle ordered tuples of closed Reeb orbits is to consider OWARDS A GOOD DEFINITION OF ALGEBRAICALLY OVERTWISTED 3 sequences I = ( i k ) k ∈ N N with only finitely many non zero elements, and to denote by γ I thetuple of orbits ( γ , . . . , γ | {z } i , . . . , γ N , . . . , γ N | {z } i N ) , where N is large enough to capture all non vanishing elements of I . We allow for I also thesequence = (0 , . . . ) giving rise to the empty tuple γ = (). Finally, let | I | be the number of nonzero components in the sequence I , and C ( I ) be the integer C ( I ) = | I | ! i ! · · · i N ! κ i γ · · · κ i N γ N again for N large enough.To compute the Conley-Zehnder index CZ( γ ) of a closed Reeb orbit γ , we have to fix a trivial-ization of the contact structure ξ along γ . To do this in a unified way, choose a basis A , . . . , A s of H ( M, Z ) (for H ( M, Z ) with torsion, we refer to [EGH00, Section 2.9.1]) and for each element A j a closed path ψ j representing A j . Fix a trivialization of ξ along ψ j in an arbitrary way, andchoose for every closed Reeb orbit γ , a surface S γ bounding γ and the corresponding combinationof ψ j ’s that represent [ γ ] ∈ H ( M, Z ), then use S γ to extend the trivialization of ξ from the ψ j ’sto γ .Let (cid:0) R × M, d ( e t α ) (cid:1) be the symplectization of ( M, α ). A complex structure J on a symplecticvector bundle ( E, Ω) is called compatible with Ω, if Ω( J · , J · ) = Ω( · , · ), and if Ω( J · , · ) defines ametric. We choose a compatible R –invariant complex structure J on the symplectic vector bundle( ξ, dα ) and extend it to an almost complex structure on the symplectization by J ∂∂t = R α . Todefine the differentials for the different homologies, we have to enumerate certain holomorphiccurves. Let (Σ g , j ) be a compact Riemann surface of genus g , and let I + = ( i + k ) k , and I − = ( i − l ) l be finite sequences of integers. Associate to every i + k = 0 points x k , . . . , x i + k k ∈ Σ g , to every i − l = 0points x l , . . . , x i − l l ∈ Σ g , together with nonzero tangent vectors v ik ∈ T x ik Σ g and v jl ∈ T x jl Σ g respectively. For reasons that will become clear below, we call x = { x lk } the positive, x = { x lk } the negative punctures, and the attached vectors are called asymptotic markers. Additionally letthere be m marked points y , . . . , y m on the Riemann surface Σ g . All the marked points and thepositive and negative punctures have to be pairwise distinct.A map ˜ u = ( a, u ) : (Σ g \ ( x ∪ x ) , j ) → ( R × M, J )is a ( j, J )–holomorphic map, if J ◦ D ˜ u = D ˜ u ◦ j for every point of Σ g \ ( x ∪ x ). Additionallywe require the following properties at the punctures. Choose for every puncture p ∈ x ∪ x aholomorphic chart D → Σ g such the origin is mapped to p , and such that the asymptotic markerpoints along the positive real axis. In polar coordinates (cid:0) ρe iϑ (cid:1) ∈ D , the following asymptoticconditions have to be satisfied by ˜ u :lim ρ → a (cid:0) ρe iϑ (cid:1) = ( + ∞ if p ∈ x , −∞ if p ∈ x , and lim ρ → u (cid:0) ρe iϑ (cid:1) = ( γ k ( − T k π ϑ ) if p = x ik , γ l ( T l π ϑ ) if p = x jl ,where T k denotes the period of the orbit γ k . When we do not want to fix the complex structureon Σ g , we call such a map a J –holomorphic map.Choose an additional puncture x with asymptotic marker that will be asymptotic to a closedReeb orbit γ . We denote by M Ag,m (cid:0) γ I − ; γ I + , γ (cid:1) the space of J –holomorphic maps as above thathave an additional positive puncture x , and by M Ag,m (cid:0) γ, γ I − ; γ I + (cid:1) the space of J –holomorphicmaps that have an additional negative puncture x . In both cases, we assume that the surfaceobtained by gluing u (Σ g \ ( { x } ∪ x ∪ x )) with suitable surfaces S γ k represents the homology class A ∈ H ( M, Z ).Let (Σ g , j ) and (Σ ′ g , j ′ ) be compact Riemann surfaces equipped with positive and negativepunctures x , x and x ′ , x ′ respectively and with m marked points y , . . . , y m and y ′ , . . . , y ′ m . Wecall a diffeomorphism ϕ : Σ g → Σ ′ g a reparametrization, if it is a biholomorphism that is compatiblewith all special points. This means that ϕ satisfies the equation ϕ ∗ j = j ′ , and ϕ ( y k ) = y ′ k , andcorresponding relations for the ordered punctures. The map also has to respect the asymptotic F. BOURGEOIS AND K. NIEDERKR¨UGER markers at each puncture. Define an equivalence relation ∼ on the space of maps M Ag,m ( . . . ) bysaying that two maps ˜ u = ( a, u ) and ˜ u ′ = ( a ′ , u ′ ) are equivalent, if there is a shift τ ∈ R , and areparametrization ϕ : (Σ g , j ) → (Σ g , j ′ ) such that( a, u ) = ( a ′ ◦ ϕ + τ, u ′ ◦ ϕ ) . The moduli spaces c M Ag,m ( . . . ) = M Ag,m ( . . . ) / ∼ are obtained by dividing out the corresponding space of maps by the equivalence relation ˜ u ∼ ˜ u ′ just defined. Denote the first Chern class of the complex vector bundle ( ξ = ker α, J ) by c ( ξ ). Ifthe elements of c M Ag,m ( . . . ) are not branched coverings, then choosing J generically these modulispaces are smooth orbifolds of dimensiondim c M Ag,m (cid:0) γ I − ; γ I + , γ (cid:1) = ( n − (cid:0) − g − (cid:12)(cid:12) I − (cid:12)(cid:12) − (cid:12)(cid:12) I + (cid:12)(cid:12) − (cid:1) − m + CZ( γ )+ 2 h c ( ξ ) | A i + ∞ X j =1 ( i + j − i − j ) CZ( γ j )dim c M Ag,m (cid:0) γ, γ I − ; γ I + (cid:1) = ( n − (cid:0) − g − − (cid:12)(cid:12) I − (cid:12)(cid:12) − (cid:12)(cid:12) I + (cid:12)(cid:12)(cid:1) − m − CZ( γ )+ 2 h c ( ξ ) | A i + ∞ X j =1 ( i + j − i − j ) CZ( γ j )that are equipped with a smooth evaluation map at the marked pointsev : c M Ag,m ( . . . ) → M m , [ a, u ] (cid:0) u ( y ) , . . . , u ( y m ) (cid:1) . The moduli spaces have compactifications M Ag,m (cid:0) γ I − ; γ I + , γ (cid:1) , and M Ag,m (cid:0) γ, γ I − ; γ I + (cid:1) respec-tively consisting of holomorphic buildings of arbitrary height [BEH+03].In the presence of branched coverings, the new ongoing approach to transversality by Cieliebakand Mohnke (see [CM07] for the symplectic case) or the polyfold theory developed by Hofer,Wysocki and Zehnder [Hof08, HWZ07] give to the moduli space M Ag,m ( . . . ) the structure of abranched manifold (with rational weights) with boundary and corners. The presence of theserational weights is due to the use of multivalued perturbations.In the absence of marked points, and when dim c M Ag, ( . . . ) = 0, this moduli space consists offinitely many elements with rational weights. We denote the sum of these rational weights by n Ag (cid:0) I − ; I + , γ (cid:1) or n Ag (cid:0) γ, I − ; I + (cid:1) . When m = 0, we define a multilinear form n Ag,m ( . . . ) on m –tuplesof closed differential forms Θ , . . . , Θ m on M by the formula h n Ag,m ( . . . ) | (Θ , . . . , Θ m ) i = Z M Ag,m ( ... ) ev ∗ (Θ × · · · × Θ m ) . By convention, we set the multilinear form to 0, if P deg Θ j = dim M Ag,m ( . . . ), and we define a0–multivalued form n Ag, ( . . . ) just by using the sum n Ag ( . . . ) of rational weights defined above.To define the algebras, we have to find a suitable coefficient ring. For this choose a submodule R ≤ (cid:8) A ∈ H ( M, Z ) (cid:12)(cid:12) h c ( ξ ) | A i = 0 (cid:9) to construct the group ring Q (cid:2) H ( M, Z ) / R (cid:3) , whose elements will be written as P kj =1 c j e A j , where c j ∈ Q and A j ∈ H ( M, Z ) / R . Different choices of R may lead to different SFT invariants. Wedefine a grading on Q (cid:2) H ( M, Z ) / R (cid:3) by (cid:12)(cid:12) c e A (cid:12)(cid:12) = − h c ( ξ ) | A i .Associate to every closed Reeb orbit γ the formal variables q γ and p γ with gradings | q γ | = CZ( γ ) + n − | p γ | = − CZ( γ ) + n − I , we denote by q I the monomial q i γ · · · q i N γ N and by p I themonomial p i γ · · · p i N γ N for N large enough.Consider a formal variable ~ with grading | ~ | = 2 ( n − OWARDS A GOOD DEFINITION OF ALGEBRAICALLY OVERTWISTED 5
Contact homology.
The algebra A CH of contact homology consists of polynomials in the q γ ’swith coefficients in Q (cid:2) H ( M, Z ) / R (cid:3) . Every element can be written as a finite sum f = K X k =1 f k , where each term f k is of the form f k = c k e A k q I k with every c k e A k ∈ Q (cid:2) H ( M, Z ) / R (cid:3) and I k = (cid:0) i j,k (cid:1) j is a sequence of the type described above.The grading for each such monomial is given by | f k | = ∞ X j =1 (cid:16) CZ( γ j ) + n − (cid:17) i j,k − h c ( ξ ) | A k i . The sum of monomials f = c e A q i γ · · · q i N γ N , and g = c e B q j γ · · · q j N γ N (we assume N to be largeenough to include all non zero terms of both sequences I = ( i k ) k and J = ( j k ) k ) is formal, andthe multiplication of f and g gives f g = c c e A + B q i γ · · · q i N γ N q j γ · · · q j N γ N , where we still have to permute the q –variables to get a monomial in normal form. For this, weimpose supercommutativity q γ q γ ′ = ( − | q γ | | q γ ′ | q γ ′ q γ . The differential on this algebra A CH isdefined by ∂q γ = X A,I n A , (cid:0) γ I ; γ (cid:1) C ( I ) e A q I , where the sum runs over all integer valued sequences I and all homology classes A . Rememberthat n A , (cid:0) γ I ; γ (cid:1) counts (in the sense defined above) punctured holomorphic spheres with a singlepositive puncture asymptotic to γ , and negative punctures asymptotic to the orbits in γ I . Ef-fectively, the sum in the definition of the differential operator is finite. On one hand, the periodof γ gives an upper bound for P i k T ( γ k ) (see for example [BEH+03, Lemma 5.16]), so that onlyfinitely many sequences I need to be taken into account. On the other hand, the compactnesstheorem for the space ∪ A M Ag,m ( γ I ; γ ) with fixed I shows that there may only be holomorphiccurves for finitely many choices of A .For products extend the differential according to the graded Leibniz rule, i.e. ∂ ( f g ) = ( ∂f ) g +( − | f | f ( ∂g ). Rational SFT.
The algebra A rSF T of rational SFT can be interpreted as a Poisson algebra witha distinguished element h . Since we are just interested in showing that its homology vanishes, wewill only describe it as a differential graded algebra. The elements of A rSF T can be written as f = X I + f I + ( q ) p I + , where the sum runs over all finite sequences I + of integers, and the coefficients f I + ( q ) ∈ A CH are elements in the contact homology algebra that depend on I + . In other words, the elementsof A rSF T are formal power series in p –variables with coefficients in the contact homology algebra A CH . The grading of a monomial is given by (cid:12)(cid:12)(cid:12) c e A q I − p I + (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) c e A q I − (cid:12)(cid:12)(cid:12) + ∞ X j =1 (cid:0) n − − CZ( γ j ) (cid:1) i + j . The product between variables is supercommutative q γ q γ ′ = ( − | q γ || q γ ′ | q γ ′ q γ , q γ p γ ′ = ( − | q γ || p γ ′ | p γ ′ q γ and p γ p γ ′ = ( − | p γ || p γ ′ | p γ ′ p γ . The differential d h is defined on a single q –variable by the formula d h q γ = X A,I − ,I + n A , (cid:0) γ I − ; γ I + , γ (cid:1) C ( I − ) C ( I + ) e A q I − p I + F. BOURGEOIS AND K. NIEDERKR¨UGER and on a p –variable respectively by d h p γ = ( − | p γ | +1 X A,I − ,I + n A , (cid:0) γ, γ I − ; γ I + (cid:1) C ( I − ) C ( I + ) e A q I − p I + . We are summing over all combinations of monomials e A q I − p I + . For the definition of the rationalnumbers n A , ( . . . ), we refer to Section “Notation: Closed Reeb orbits and holomorphic curves”.For arbitrary elements in A rSF T extend the operator d h by using the graded Leibniz rule. Full SFT.
The algebra of symplectic field theory A SF T is composed of formal power series of theform F = ∞ X g =0 X I + f g,I + ( q ) p I + ~ g , as above f g,I + ( q ) is an element in the contact homology algebra, and ~ is a new formal variableof degree 2 ( n − ce A q I − p I + ~ g is given by (cid:12)(cid:12)(cid:12) c e A q I − p I + ~ g (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) e A q I − p I + (cid:12)(cid:12)(cid:12) + 2 g ( n − . Unlike A CH and A rSF T , the algebra A SF T is not supercommutative, but instead has the follow-ing commutator relations. For two q –variables or two p –variables the commutator relations areidentical to the ones of rational SFT, but for mixed terms, we require the supersymmetric relation[ q γ , p γ ′ ] := q γ p γ ′ − ( − | q γ || p γ ′ | p γ ′ q γ = ( κ γ ~ if γ = γ ′ p γ as the derivationoperator κ γ ~ ∂/∂q γ . The ~ –variable commutes with the q – and p –variables.In [EGH00], the differential d H of SFT was given as the commutator with a distinguishedelement H , but here we will just specify the effect of d H on the generators of A SF T d H ~ = 0 ,d H q γ = X g,A,I − ,I + n Ag, (cid:0) γ I − ; γ I + , γ (cid:1) C ( I − ) C ( I + ) e A q I − p I + ~ g , and d H p γ = ( − | p γ | +1 X g,A,I − ,I + n Ag, (cid:0) γ, γ I − ; γ I + (cid:1) C ( I − ) C ( I + ) e A q I − p I + ~ g , and extend it to general elements by the graded Leibniz rule. Marked points.
Choose closed differential forms Θ , . . . , Θ d that represent an integral basis forthe de Rham cohomology ring H ∗ dR ( M ). Take any of the algebras described above, i.e., let A be either the contact homology algebra A CH , the rational symplectic field algebra A rSF T , or theSFT algebra A SF T . Define new formal variables t , . . . , t d with grading | t j | := deg Θ j −
2. Let A ∗ be the algebra of formal power series in the t , . . . , t d with coefficients in A such that the t j supercommute among themselves, and with the q –variables, and possibly also (if they are part of A ) with the p –, and ~ –variables.The differential ∂ ∗ on A ∗ vanishes on ~ , and the t , . . . , t d ∂ ∗ t j = 0 , and ∂ ∗ ~ = 0 . To define ∂ ∗ on the q – and p –variables, introduce first the notation Θ = P dj =1 Θ j t j , and h n Ag,m ( . . . ) | ( Θ , . . . , Θ ) i = X ≤ a ,...,a m ≤ d h n Ag,m ( . . . ) | (Θ a , . . . , Θ a m ) i t a · · · t a m , OWARDS A GOOD DEFINITION OF ALGEBRAICALLY OVERTWISTED 7 and then set ∂ ∗ q γ = X g,m,A,I − ,I + h n Ag,m (cid:0) γ I − ; γ I + , γ (cid:1) | ( Θ , . . . , Θ ) i C ( I − ) C ( I + ) e A q I − p I + ~ g , and ∂ ∗ p γ = ( − | p γ | +1 X g,m,A,I − ,I + h n Ag,m (cid:0) γ, γ I − ; γ I + (cid:1) | ( Θ , . . . , Θ ) i C ( I − ) C ( I + ) e A q I − p I + ~ g , where we are summing over all I − described above, and in case A = A CH , we assume that g = 0,and I + = . If A = A rSF T , we still keep g = 0, but allow any sequence I + , and finally if A = A SF T , any integer g ≥
0, and sequence I + is allowed.3. Proof of Theorem 1
The implications ( ii ) ⇒ ( i ) , ( iii ) ⇒ ( i ) , and ( n + iii ) ⇒ ( n ) . The statements are basedon the following trivial remark.
Remark . Let π : A ′ → A be a chain map between two differential graded algebras ( A , ∂ ) and( A ′ , ∂ ′ ). From π ◦ ∂ ′ = ∂ ◦ π it follows immediately that an exact element f ′ ∈ A ′ is mapped toan exact element f = π ( f ′ ) ∈ A .Corresponding to each of the cases ( ii ) ⇒ ( i ), ( iii ) ⇒ ( i ), and ( n + iii ) ⇒ ( n ), we find anelement g in either A rSF T , A SF T , or A ∗ such that d h g = , d H g = , or ∂ ∗ g = .For the first case, define a projection π : A rSF T → A CH by mapping any monomial f I + ( q ) p I + ( f I + ( q ) if I + = π will be an algebra homomorphism, and to seethat it is a chain map, just compare the definitions of the differentials ∂ and d h . All terms countedby the contact homology differential also appear in the rational SFT differential, and it is clear that ∂ ◦ π = π ◦ d h holds, if d h cannot decrease the number of p –variables in any monomial. Note thatalready by the Leibniz rule, the differential d h can decrease the number of factors in a monomialat most by one, and to decrease the number of p –factors, variables p γ have to exist such that d h p γ contains (non zero) terms without any p –coordinates at all. This would only be possible, ifthere were non empty moduli spaces M A , (cid:0) γ ; γ, γ I − (cid:1) of spheres without positive punctures, butby the maximum principle no such curves exist. We show in Appendix A that a weaker form ofthe maximum principle still holds for solutions of perturbed Cauchy-Riemann equations, so thatthe same conclusion remains true. It follows that π (cid:0) d h f I + ( q ) p I + (cid:1) = 0 for any monomial with I + = , and so π ( g ) ∈ A CH will be a primitive of the unit element .The chain map π : A SF T → A CH for the second case will be defined similarly, dropping anymonomial that contains a positive ~ – or p –power. This map is compatible with the commutatorrelations, and it is again a chain map, because d H cannot decrease the number of p –factors withoutraising the ~ –power and vice versa such that π (cid:0) d H f I + ( q ) p I + ~ g (cid:1) = 0, if either g = 0 or I + = ,and so the contact homology algebra is trivial as we wanted to show.To prove ( n + iii ) ⇒ ( n ), let π : A ∗ → A be the projection that drops any monomial containinga t j –variable. Let ∂ be the differential of A (i.e., depending on A either ∂ , d h or d H ). We need toshow π is a chain map. As before the argument here is that ∂ is the zero order term of ∂ ∗ in the t j ,and this is true because the count n Ag ( . . . ) coincides by definition with h n Ag, ( . . . ) | () i , furthermore ∂ ∗ can never decrease the number of t j –variables of a monomial, so that ∂πf = .3.2. The implications ( i ) ⇒ ( ii ) , ( i ) ⇒ ( iii ) , and ( n ) ⇒ ( n + iii ) . For the implication ( i ) ⇒ ( ii ),assume that f is an element in the contact homology algebra A CH such that ∂f = . This f canonically embeds into the rational SFT algebra A rSF T , where we can compute its differential F. BOURGEOIS AND K. NIEDERKR¨UGER d h f = − g . The element g ∈ A rSF T is always closed, because d h g = d h ( − d h f ) = 0, and allof its terms contain at least one p –variable. The formal inverse of − g is given by( − g ) − := ∞ X k =0 g k , where we set g = . This object is well defined in A rSF T , because all terms in g have at least one p –factor, so that only the powers of g up to k = | I + | contribute to the terms p I + in (1 − g ) − , andin particular ( − g ) − is a formal power series in the p –variables. It is obvious that d h (1 − g ) − = 0.Define an element f ∈ A rSF T by f := f ( − g ) − . It easily follows that d h f = d h (cid:0) f ( − g ) − (cid:1) = ( d h f ) ( − g ) − − f d h (1 − g ) − = ( − g ) ( − g ) − = , just as we wanted to show.We will now prove the implication ( i ) ⇒ ( iii ) in a similar way: Assume that f ∈ A CH is suchthat ∂f = . We can canonically embed the contact homology algebra A CH into the algebra ofsymplectic field theory A SF T , since the commutation relations for only q –variables are identicalin both spaces. As we said above, dropping any term with a p – or ~ –variable the differential d H coincides on A CH with the differential ∂ of contact homology. Thus d H f = ∂f − G = − G , where G only contains monomials with a non zero power of ~ or of p , therefore the formal inverse( − G ) − = P ∞ k =0 G k is a well defined element in A SF T . Moreover it is closed, and so F = f ∞ X k =0 G k is a primitive of , because by using the Leibniz rule d H f ∞ X k =0 G k ! = (cid:0) d H f (cid:1) ∞ X k =0 G k − f ∞ X k =0 d H (cid:0) G k (cid:1) = , follows, and the homology of SFT vanishes.Finally, we compare the invariants with and without marked points, and prove ( n ) ⇒ ( n + iii ):Let thus A be either A CH , A rSF T , or A SF T without marked points, and A ∗ the correspondingalgebra with marked points. Use that ∂ and ∂ ∗ are identical in 0–th order of t j –powers, so thatif ∂f = , we have that ∂ ∗ f = − G , where all terms in G have positive t j –powers. As above,the formal inverse (1 − G ) − = P G k is an element of A ∗ , so that we can define F := f (1 − G ) − which is a primitive of with respect to ∂ ∗ . Appendix A. Maximum principle for perturbed holomorphic curves
Let T > M . Let b G be the positive(but not definite) metric on R × M defined by b G ( · , · ) = dα ( · , J · ). Let G be the positive definitemetric on R × M such that the Reeb field R α and the Liouville field ∂∂t are mutually orthogonal,are orthogonal to ξ , and have unit length. In the next proposition, ν ∈ Λ , (Σ g , ˜ u ∗ T ( R × M )) willdenote a perturbation for the Cauchy-Riemann equation. Proposition 3.
A curve ˜ u : Σ g \ ( x ∪ z ) → R × M that satisfies the perturbed Cauchy-Riemannequation d ˜ u + J ◦ d ˜ u ◦ j = ν with k ν k L ( G ) < √ T has to have top punctures x = ∅ . OWARDS A GOOD DEFINITION OF ALGEBRAICALLY OVERTWISTED 9
Proof.
Assume that there exists a map ˜ u : Σ g \ z → R × M satisfying d ˜ u + J ◦ d ˜ u ◦ j = ν that isasymptotic for t → −∞ to the orbits γ , . . . , γ s . By Stokes theorem, we have Z Σ g \ z ˜ u ∗ dα = − s X i =1 T i < − T , where T i is the period of γ i .On the other hand, let z = x + iy be coordinates of a complex chart on Σ g such that k ∂ x k = k ∂ y k = 1. Then dα (cid:0) ∂ x ˜ u, ∂ y ˜ u (cid:1) = − dα (cid:0) J∂ y ˜ u, ∂ y ˜ u (cid:1) + dα (cid:0) ν ( ∂ x ) , ∂ y ˜ u (cid:1) = b G (cid:0) ∂ y ˜ u, ∂ y ˜ u (cid:1) + b G (cid:0) ν ( ∂ x ) , − J ∂ y ˜ u (cid:1) . After a suitable rotation in the ( x, y )–plane, we obtain, using Cauchy-Schwarz inequality, and (cid:0) k d ˜ u k b G − / k ν k b G (cid:1) ≥ dα (cid:0) ∂ x ˜ u, ∂ y ˜ u (cid:1) = k d ˜ u k b G + b G (cid:0) ν ( ∂ x ) , − J ∂ y ˜ u (cid:1) ≥ k d ˜ u k b G − k ν k b G k d ˜ u k b G ≥ − k ν k b G , where k·k b G is the semi-norm induced by b G .Integrating over Σ g \ z , we obtain Z Σ g \ z ˜ u ∗ dα ≥ − k ν k L ( b G ) . Comparing with the Stokes bound for this integral, we obtain k ν k L ( G ) ≥ k ν k L ( b G ) > √ T , acontradiction. (cid:3) References [BEH+03] F. Bourgeois, Y. Eliashberg, H. Hofer, K. Wysocki, and E. Zehnder,
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D´epartement de Math´ematique CP 218, Universit´e Libre de Bruxelles, Boulevard duTriomphe, 1050 Bruxelles, Belgium
E-mail address , K. Niederkr¨uger: [email protected] (K. Niederkr¨uger)(K. Niederkr¨uger)