Infinitely many universally tight torsion free contact structures with vanishing Ozsváth–Szabó contact invariants

Infinitely many universally tight torsion free contact structures with vanishing Ozsváth-Szabó contact invariants

Abstract

Ozsváth–Szabó contact invariants are a powerful way to prove tightness of contact structures but they are known to vanish in the presence of Giroux torsion. In this paper we construct, on infinitely many manifolds, infinitely many isotopy classes of universally tight torsion free contact structures whose Ozsváth–Szabó invariant vanishes. We also discuss the relation between these invariants and an invariant on and construct other examples of new phenomena in Heegaard–Floer theory. Along the way, we prove two conjectures of K Honda, W Kazez and G Matić about their contact topological quantum field theory. Almost all the proofs in this paper rely on their gluing theorem for sutured contact invariants.

Introduction

Contact topology studies isotopy classes of contact structures1. These classes come in two main flavors: overtwisted and tight, the latter being further divided into universally tight and virtually overtwisted. Up to now, besides homotopical data, there are only two algebraic objects which have been successfully used to classify such isotopy classes on a general 3–manifold. The first one is Giroux torsion introduced in [Gir00], its definition is recalled in Section 1. It is either a non-negative integer or infinite and always infinite for overtwisted classes. It is invariant under isomorphisms, not only isotopies. It shares the monotonicity property of symplectic capacities [HZ94] on one hand and the finiteness property of 3–manifolds complexity [Mat90] on the other hand. Indeed, if then and, for fixed and , there are only finitely many isomorphism classes of contact structures on whose torsion is at most . Another way to put it is to say that finite torsion determines contact structures up to isomorphism and a finite ambiguity. More generally, it plays an important role in the coarse classification of tight contact structures [CGH09]. The second object, based on open book decompositions [Gir02], is Ozsváth–Szabó contact invariants introduced in [OS05] which live in the Heegaard–Floer homology of the ambient manifold. They come in various flavors depending on a choice of coefficients. These invariants are a powerful tool to detect tightness and obstructions to fillability by symplectic or complex manifolds. Its main properties are listed in Theorem 12 below.

It is natural to investigate relations between these two invariants. In [GHV], P Ghiggini, K Honda and J Van Horn Morris proved that, whenever Giroux torsion is non zero, the contact invariant over coefficients vanishes (we give a new proof of this result in Section 5). Here we prove that the converse does not hold.

Main theorem (Section 5).

Every Seifert manifold whose base has genus at least three supports infinitely many (explicit) isotopy classes of universally tight torsion free contact structures whose Ozsváth–Szabó invariant over coefficients vanishes.

In the above theorem, the genus hypothesis cannot be completely dropped because, for instance, on the sphere and the torus , all torsion free contact structures have non vanishing Ozsváth–Szabó invariants. However, it may hold for genus two bases. Note that the class of Seifert manifolds is the only one where isotopy classes of contact structures are pretty well understood. So the theorem says that examples of universally tight torsion free contact structures with vanishing Ozsváth–Szabó invariant exist on all manifolds we understand, provided there is enough topology (the base should have genus at least three). In this statement, isotopy classes cannot be replaced by conjugacy classes because of the finiteness property explained above. Along the way we prove Conjecture 7.13 of [HKM08].

Our examples also provide a corollary in the world of Legendrian knots. Ozsváth–Szabó theory provides invariants for Legendrian or transverse knots in different (related) ways, see [SV09] and references therein. In the standard contact 3–spheres there are still two seemingly distinct ways to define such invariants but, in general contact manifolds, the known invariants all come from the sutured contact invariant of the complement of the knot according to the main theorem proved by V Vértesi and A Stipsicz in [SV09]. In this paper they call strongly non loose those Legendrian knots in overtwisted contact manifolds whose complement is tight and torsion free. Corollary 1.2 of that papers states that a Legendrian knot has vanishing invariant when it is not strongly non loose. We prove that the converse does not hold.

Theorem 1 (see the discussion after Proposition 21).

There exists, in an overtwisted contact manifold, a null-homologous strongly non loose Legendrian knot whose sutured invariant vanishes (the construction is explicit).

After studying the relationship between Ozsváth–Szabó invariants and Giroux torsion, we now turn to a more specific relation between these invariants and an invariant defined only on the 3–torus. E Giroux proved that any two incompressible prelagrangian tori of a tight contact structure on are isotopic. We can then define the Giroux invariant to be the homology class of its prelagrangian incompressible tori. Note that there is a “sign ambiguity” because these tori are not naturally oriented. Translated into this language, Giroux proved that two tight contact structures on are isotopic if and only if they have the same Giroux invariant and the same Giroux torsion, see [Gir00]. This invariant is clearly –equivariant. Since this group acts transitively on primitive elements of , we see that all these elements are attained by . This also proves that all tight contact structures on which have the same torsion are isomorphic. This classification of tight contact structures on and a result by Y Eliashberg shows that torsion free contact structures on are exactly the Stein fillable ones.

Theorem 2 (Section 3).

There is a unique up to sign –equivariant isomorphism between and (on the ordinary cohomology side, sends to zero and to by slant product). Under this isomorphism, the Ozsváth–Szabó invariant of a torsion free contact structure on is sent to the Poincaré dual of its Giroux invariant.

Note that, on , cohomology classes can be represented by constant differential forms and 1–dimensional homology classes by constant vector fields. The slant product of the above theorem is then identified with the interior product of vector fields with 2–forms.

The statement about torsion free contact structures is based on the interaction between the action of the mapping class group and first homology group of on its Ozsváth–Szabó homology and ordinary cohomology. It sheds some light on the sign ambiguity of the contact invariant since the sign ambiguity of the Giroux invariant is very easy to understand.

Corollary 3.

There are infinitely many isomorphic contact structures whose isotopy classes are pairwise distinguished by the Ozsváth–Szabó invariant.

Theorem 2 proves, via gluing, a conjecture of Honda, Kazez and Matić about the sutured invariants of –invariants contact structures on toric annuli. This conjecture is stated in [HKM08][top of page 35] and will be discussed in Section 3 Proposition 20.

Theorem 2 also have some consequence for the hierarchy of coefficients because coefficients can distinguish only finitely many isotopy classes of contact structures (since is always finite).

Corollary 4.

There exists a manifold on which the Ozsváth–Szabó invariant over integer coefficients distinguishes infinitely many more isotopy classes of contact structures than the invariant over coefficients.

In the same spirit, we prove that twisted coefficients are more powerful than coefficients even when the latter give non vanishing invariants.

Proposition 5 (see Propositions 20).

There exist a sutured manifold with two contact structures having the same non vanishing Ozsváth–Szabó invariant over coefficients but which are distinguished by their invariants over twisted coefficients.

In Section 1 we review the work of Giroux on certain contact structures on circle bundles, the easy extension of this work to Seifert manifolds and torsion calculations. In Section 2 we review Ozsváth–Szabó contact invariants. In Section 3 we prove Theorem 2. In Section 4 we review the work of Honda, Kazez and Matić on their contact TQFT and upgrade their groups calculations to twisted coefficients. In Section 5, by far the longest, we prove [HKM08][Conjecture 7.13] and the main theorem above.

1 Partitioned contact structures on Seifert manifolds

This section contains preliminary results in contact topology. We first recall the crucial definition of Giroux torsion. The -torsion of a contact manifold was defined in [Gir00][Definition 1.2] to be the supremum of all integers such that there exist a contact embedding of

into the interior of or zero if no such integer exists. Of course all –torsions can be recovered from the –torsion. However when we don’t specify we mean -torsion. This is due to the fact that only -torsion is known to interact with symplectic fillings and Ozsváth–Szabó theory.

A multi-curve in an orbifold surface is a 1–dimensional submanifold properly embedded in the regular part of . When is closed, we will say that a multicurve is essential in if none of its components bound a disk containing at most one exceptional point.

Since we want to extend results from circle bundles to Seifert manifolds and most surface orbifolds are covered (in the orbifold sense) by smooth surfaces, the following characterization will be useful.

Lemma 6.

Let be a multicurve in a closed orbifold surface whose (orbifold) universal cover is smooth. The following statements are equivalent:

  1. is essential;

  2. lifts to an essential multicurve in all smooth finite covers of .

  3. lifts to an essential multicurve in some smooth finite cover of .

Proof.

We first prove (the contrapositive of) (1) (2). Let be a essential multicurve in and be an orbifold covering map from a smooth surface to . Suppose that a component of the inverse image of bounds an embedded disk in . Its image in is a topological disk and we only need to prove that this disk contains at most one exceptional point. Using multiplicativity of the orbifold Euler characteristic under the orbifold covering map from to , we get . This proves that contains at most one exceptional points because its Euler characteristic is with if it has exceptional points so . So (1) implies (2). Since (2) obviously imply (3), we are left with proving (the contrapositive of) (3) implies (1).

Assume that is not essential and let be a connected component of the complement of in which is a disk with at most one exceptional point. In any finite cover of , this disk lifts to a collection of disks bounded by components of the lift of and containing at most one exceptional point. So is non essential in all finite covers of . ∎

The following is the essential definition of this section.

Definition 7 (obvious extension of [Gir01]).

A contact structure is partitioned by a multi-curve in if it transverse to the fibers over and if the surface is transverse to and its characteristics are fibers.

Example 8 ([Lut77, Kt91], see also [Nie]).

Let be a Seifert manifold and be a non empty multi-curve in whose class in is trivial. There is a –invariant contact structure on which is partitioned by . This contact structure is unique up to isotopy among –invariant contact structures.

The following theorem relies on [Mas08][Theorem A] and on easy extensions or consequences of the fourth part of [Gir01]. Of course it also uses a lot the results of [Gir00]. The two papers by Giroux can also be replaced by the Honda versions [Hon00a, Hon00b]. This theorem could be easily improved to say things about Seifert manifolds with non empty boundary but we won’t need such improvements. Recall that a closed Seifert manifold is small if it has at most three exceptional fibers and its base has genus zero. Otherwise it is called large. In particular the bases of large Seifert manifolds admit essential multi-curves. We denote by the rational Euler number of a Seifert manifold . See [Mas08] for the conventions used here for Seifert invariants and Euler numbers. In the statement we exclude for convenience the (finitely many) Seifert manifolds which are torus bundles over the circles (see for instance [Hat] to get the list).

Theorem 9.

Let be a closed oriented Seifert manifold over a closed oriented orbifold surface.

  1. A contact structure on partitioned by a multi-curve is universally tight if and only if one of the following holds:

    1. is empty

    2. is large and is essential

    3. is a Lens space (including and ), , is connected and each component of its complement contains at most one exceptional point.

  2. Any universally tight contact structure on is isotopic to a partitioned contact structure.

  3. Suppose is not a torus bundle over the circle. Let be a contact structure on partitioned by an essential multi-curve . Let be the greatest integer such that there exist closed components of in the same isotopy class of curves. The Giroux torsion of is zero if is empty and at most otherwise.

  4. Let and be contact structures on partitioned by non empty multi-curves denoted by and respectively. If and are isotopic then and are so. If and are isotopic and universally tight then and are isotopic.

We first comment on some consequences of this theorem which have not much to do with the main stream of the present paper. We can deduce from it and [LM04] (or [Mas08]) the list (given in corollary 10 below) of Seifert manifolds which carry universally tight contact structures. This list did not appear in the literature while the (much subtler) list of Seifert manifolds which carry tight contact structures (maybe virtually overtwisted) was obtained (with much more work) by P Lisca and A Stipsicz in [LS]. In addition, the road taken in that paper to prove existence on large Seifert manifold is much heavier than using the above theorem (but the point of that paper is small manifolds).

Corollary 10.

A closed Seifert manifold admits a universally tight contact structure if and only if one of the following holds:

  1. is large

  2. is a Lens space (including and )

  3. has three exceptional fibers which can be numbered such that its Seifert invariants are with

    for some relatively prime integers .

The above theorem also proves that all universally tight contact structures on Seifert manifolds interact nicely with the Seifert structure.

Corollary 11.

If is a universally tight contact structure on a closed Seifert manifold then there exist a locally free action on such that is either transverse to the orbits or invariant.

Note that the alternative in the above corollary is not exclusive. A contact structure which is both invariant and transverse to the orbits of a locally free action exists exactly when , this was proved by Y Kamishima and T Tsuboi in [KT91]. There is only one isomorphism class of contact structure of this type when they exist. This class is of Sasaki type and sometimes called the canonical isomorphism class of contact structures on .

Proof of Theorem 9.

We now outline the main differences between Theorem 9 and the parts which are already written in [Gir01]. First it should be noted that, when is either a Lens space or a solid torus with a standard Seifert fibration, everything is well understood thanks to the classification theorems of [Gir00] (see also [Hon00a]). So we don’t consider these Seifert manifolds in the following.

1) Let be a contact structure on a closed partitioned by . If is empty then is transverse to the fibers hence universally tight according to [Mas08][Theorem A] (this direction follows rather directly from Bennequin’s theorem). If is large and is essential then the base of is covered (in the orbifold sense) by a smooth surface and there is a corresponding circle bundle covering (honestly) . The pulled back contact structure is partitioned by the inverse image of which is essential according to Lemma 6 so is universally tight according to [Gir01] (first line of page 252).

Conversely, assume that is universally tight and partitioned by a non empty multi-curve . Assume first the base of is covered by a smooth surface of genus at least one (for instance if is large). The manifold then is covered by a circle bundle over that surface as above. We get from [Gir01][Theorem 4.4] that the lifted contact structure is partitioned by a multi-curve, unique up to isotopy, which is essential. Since the lift of is such a curve, it is essential and Lemma 6 implies that is also essential. In particular is large.

If no such cover of the base exists (and is not a Lens space) then its base is a sphere with exceptional points of order , , or (see [Thu][Theorem 13.3.6]). In each case is covered by and all curves in the regular locus of bounds a disk whose pre-image in is disconnected so is virtually overtwisted according to [Gir01][Proposition 4.1 and Lemma 4.7].

2) Recall that a contact structure on a Seifert manifold is said to have non-negative maximal twisting number2 if it is isotopic to a contact structure for which there exists a Legendrian regular fiber whose contact framing coincides with the fibration framing. If this property is not satisfied then [Mas08][Theorem A] ensures that any universally tight is isotopic to a contact structure partitioned by the empty multi-curve (i.e. transverse to the fibers). We now assume that has non negative maximal twisting number and has been isotoped so that it admits a Legendrian fiber as above. Let be a wedge of circles based at in the smooth part of (seen as the space of all fibers) let be a small regular neighborhood of . We can choose an such that the complement of in is made of disks containing exactly one exceptional point. The techniques of [Gir01] prove that is isotopic to a contact structure which, over is partitioned by a multicurve which intersects all boundary components of . We now assume this property. Let denote the (non necessarily connected) Seifert manifold over and the restricted contact structure. Since intersects all components of , each component of the boundary of contains a Legendrian regular fiber which is either a closed leaf or a circle of singularities of the characteristic foliation . If is universally tight then the classification of universally tight contact structures on solid tori directly implies that is –isotopic to a contact structure partitioned by some extending and we are done. More precisely, for each component of , this classification guaranties the existence of exactly one isotopy class of universally tight contact structure coinciding with on when contains no exceptional fiber and two otherwise. In the latter case, the two classes correspond to the two isotopy classes of arcs extending inside the base of (which is a disk with one exceptional point).

So it remains to prove that if has non negative maximal twisting number and is universally tight then each solid torus isotopic to a fibered one has a universally tight induced contact structure. This is obvious if the universal cover of naturally embeds into the universal cover of . This can be built in two stages: first one takes the (orbifold) universal cover of the base and pulls back the Seifert fibration and then one unwraps the fibers as much as possible. The sought embedding of obviously exist when the fibers can be completely unwrapped. Due to the classification of orbifolds surfaces the only problematic case if one excludes Lens spaces is when is with its (smooth) Hopf fibration. But, by definition of tightness, any tight contact structure on has negative twisting number with respect to the Hopf fibration so this case does not happen here (the property of having non negative twisting number is obviously inherited by finite covers using lifts of isotopies).

3) Since we assume that is not a torus bundle over the circle, all incompressible tori are isotopic to fibered ones (see e.g. [Hat]).

Suppose first that is partitioned by the empty multicurve (i.e. is transverse to all fibers). It was proved in [Mas08][Theorem A] that such a contact structure has negative maximal twisting number. Suppose by contradiction that it has non vanishing –torsion. Up to isotopy of there is an annulus in the base which is foliated by circles such that,

  • For all , the torus above in is prelagrangian.

  • The directions of the Legendrian foliations of the go all over the projective line .

During this full turn around the projective line, the Legendrian direction meets the fiber direction and there are Legendrian curve whose contact framing coincides with the fibration framing so we get a contradiction with the maximal twisting number estimate.

We now assume that is partitioned by a non empty multicurve and that no two components of are isotopic. Incompressible fibered tori correspond to essential curves in the base orbifold . To any such curve correspond an orbifold covering of by an open annulus and the Seifert fibration lifts to a trivial (smooth) circle fibration . The lifted contact structure is partitioned by the inverse image of which is made of as many essential circles as there were components of isotopic to (at most ) and lines properly embedded in . If there exist a contact embedding of a toric annulus with its standard torsion contact structure in then it lifts to inside some with compact. The classification of tight contact structures on toric annuli forbids torsion higher than knowing the partition we have over . This argument is not new, it was explained to me (around 2005) by E Giroux.

4) The first part is a straightforward extension of [Gir01][Lemma 4.7]. Suppose now that and are isotopic. If is not large then we are in case (c) of the first point so that and are trivially isotopic. So we now assume that is large. In particular and are essential. By definition, they are isotopic in if and only if they are isotopic in the smooth surface obtained from by removing a small open disk around each exceptional point. By definition of essential curves, no component of or is parallel to the boundary of . According to W Thurston, and are isotopic if and only if they have the same geometric intersection number with all closed curves in [Thu88][Proposition page 421]. These geometric intersections number have a contact topology interpretation explained in [Gir01][Section 4.E] which proves they are invariant under contact structures isotopy exactly as in the circle bundle case. ∎

2 Contact invariants in sutured Floer homology

In this section we review sutured Heegaard–Floer homology and the contact invariants which lives in it.

Heegaard–Floer homology was introduced by P Ozsváth and Z Szabó in [OS04b] and extended to sutured manifold by A Juhász in [Juh06]. In the following we will often silently identify a closed manifold with the sutured manifold and use sutured Floer theory (SFH) also in this case.

We denote the universal twisted by and, whenever there is no ambiguity on the manifold we are considering, we denote by .

According to [GH][Lemma 10], if a contact invariant vanishes in then it vanishes for all coefficients rings.

Theorem 12 (Ozsváth–Szabó, Honda–Kazez–Matić, Ghiggini–Honda–Van Horn Morris).

Let be a balanced sutured manifold. To each contact structure on , one can associate a contact invariant which is a set in and a twisted contact invariant which is a set in satisfying the following properties:

  1. the set is invariant under –isotopy of

  2. if is overtwisted then

  3. if has non zero torsion then

  4. if is closed and is weakly fillable then

  5. if is closed and is strongy fillable then

  6. if is a sutured submanifold of and is a contact structure on then there exists a linear map

    such that, for any contact structure on , one has

    If every connected component of intersect then there are analogous maps over coefficients. They are denoted without underlines.

  7. if is a contact submanifold of then implies and analogously over coefficients.

The construction of the contact invariants (and the isotopy invariance) can be found in [OS05] for the closed case and [HKM07] in general. The fact that it vanishes for overtwisted contact structures was first proved for the closed case and untwisted coefficients in [OS05] and follow in general from the last property and the explicit calculation of the twisted contact invariant of a neighborhood of an overtwisted disk found in [HKM07]. The assertion about torsion was proved in [GHV]. Both assertions about fillings are consequences of [OS04a][Theorem 4.2], using the fact that, for strong fillings, the coefficient ring in this theorem reduces to (see also [Ghi06][Theorem 2.13] for an alternative proof of the strong filling property). The gluing properties are proved in [HKM08] for untwisted coefficients and extended to twisted coefficients in [GH]. The gluing maps are unique up to multiplication by an invertible element of the relevant coefficients ring. Such maps will be called HKM gluing maps.

There is one piece of structure of Heegaard–Floer theory which doesn’t seem to have been explicitly discussed3 in our context up to now: the mapping class group action. Any diffeomorphism of a 3–manifold acts on any variant of . Here we need to be precise about what depends on the way a Heegaard diagram is embedded inside a manifold and what does not depend on it. The usual way to do that is to consider embedded Heegaard diagrams as pairs made of a self-indexing Morse function with unique minima and maxima and one of its Morse–Smale pseudo-gradients. Given such a pair , the Heegaard surface is and the Heegaard circles are the intersections of the stable or unstable disks of the index 1 and 2 critical points. We denote the group associated to by (we can use here , ,…). Let be a diffeomorphism of . Then [OS06][Theorem 2.1] gives an isomorphism

which is well defined up to sign. But of course the diffeomorphism also gives an isomorphism between the corresponding abstract Heegaard diagrams which then gives an isomorphism between Heegaard–Floer groups. The action of on is defined to be . It is obvious from the construction that the contact invariant is equivariant under this action. What is not obvious is that isotopic diffeomorphisms have the same action so that we get an action of the mapping class group. This has been checked by P Ozsváth and A Stipsicz in the context of knot Floer homology in [OS]. In this paper we don’t use this invariance but use specific diffeomorphisms. Actually this invariance should never be needed in contact geometry since we already know that the contact invariant is a contact structure isotopy invariant so that diffeomorphism isotopy invariance is automatic on the subgroup spanned by contact invariants in any or .

3 Contact structures on the three torus

In this section we prove Theorem 2 from the introduction. The following easy lemma is the key algebraic trick.

Lemma 13.

If an isomorphism is –equivariant then it conjugates the actions of both sides.

Proof.

In this proof we drop from the notations. We denote by the canonical action of on . Let and be two representations of on which are compatible with the action, that is:

We want to prove that since this, applied to the standard action and to the action transported by , will prove the proposition.

We first prove that, for all , and agree on . The key property of the action is that it separates all elements of : for all , there exists in such that and .

Suppose by contradiction that there exists and such that . According to the separation property, there exists in such that and . Setting , we get and , so and , which is absurd since and are both isomorphisms.

We now prove that the representations agree on . For all , there exists and such that . So for any and , we get and we know that thanks to the first part so . ∎

Proof of Theorem 2.

The existence of such an isomorphism is Proposition 8.4 of [OS03]. The above lemma proves that, for any as in the statement and any , and have the same stabilizer under the action of . The uniqueness of follows since primitive elements of are characterized up to sign by their stabilizers. Linearity of then guaranties that the sign is common to all elements.

We now prove that the Poincaré dual of the Giroux invariant and the image of the Ozsváth–Szabó invariant coincide on torsion free contact structures. First remark that the Ozsváth–Szabó invariant belongs to because the Hopf invariant of tight contact structures on is . So both invariants are primitive elements of . We prove that the stabilizer of is contained in that of using equivariance of both invariants and the fact that is a total invariant. For any in and a torsion free contact structure, we have

so we have the annouced inclusion of stabilizers and this gives . ∎

4 The contact TQFT

We now review the contact TQFT of Honda–Kazez–Matić. Let be a non necessarily connected compact oriented surface with non empty boundary. Let be a finite subset of whose intersection with each component of is non empty and consists of an even number of points. We assume that the components of are labelled alternatively by and . This labelling will always be implicit in the notation . The contact TQFT associates to each the graded group

(strictly speaking, one should replace by a small translate of along in this formula).

In this construction one can use coefficients in or twisted coefficients (including the trivial twisting which leads to coefficients). We denote by the version twisted by .

Proposition 14.

Let be a surface with marked boundary points as above and be any coefficient module for the sutured manifold . We have, for any coherent orientations system:

The subscripts and refer to the grading.

Proof.

The analogous statement over coefficients was proved in [HKM08] using product annuli decomposition, [FJR][Proposition 7.13]. This technology is not yet available over twisted coefficients but one can actually draw explicit admissible sutured Heegaard diagrams with vanishing differential for these sutured manifolds. We will sketch how to construct them and draw pictures for the three cases where we actually use this computation below.

We first recall what is an (embedded) Heegaard diagram for a (balanced connected) sutured manifold . It consists of a surface properly embedded in and circles in such that:

  • if we denote by the connected component of containing , there exist open disks properly embedded in , called compression disks, bounded by the circles and such that retracts by deformation on

  • the analogous statement holds for and with the circles.

We now return to the proposition. Let be the genus of , the number of boundary components and . The sutured manifold we study will be denoted by for concision. We rule out the trivial case from this discussion as it needs (easy) special treatment. Assume first that and . Let be a system of disjoints arcs properly embedded in which cuts to a disk. Let be tubes around the arcs for some fixed . We can assume that each meets the boundary of in its positive part . Let be the union of and the tubes . The surface obtained by pushing to make it properly embedded in is a Heegaard surface for .

Each tube naturally bounds a regular neighborhood of the arc . Let be the boundary of in each . Let be the union of and two arcs in so that and half of becomes isotopic to a fibered annulus in . See figure 1 for the case .

Figure 1: Heegaard surface for with two vertical sutures. Everything lives inside a cube minus a neighborhood of its vertical edges. The cube’s faces are pairwise glued to get . The picture show the (almost) Heegaard surface of the proof. The blue annulus in the back left is the negative part of the boundary of . The compression disks are inside the front, back, left and right faces of the cube (which get glued to two annuli in ).

We then have a Heegaard diagram for . We now explain what happens when we add some extra boundary components (i.e. ). For each extra component we add two tubes and around horizontal arcs and . We choose these arcs so that they can be completed by arcs in the positive part of to get a circle isotopic to the new boundary component. See figure 2 for the case where the extra boundary component is the front one.

Figure 2: Heegaard surface for with two vertical sutures. Top and bottom are glued. Left and right are glued. The boundary of is the union of the transparent tori (drawn as rectangles). The Heegaard surface is the union of two vertical annuli and two horizontal tubes. A compression disk bounded by a curve is shown in green.

We add circles , , and to the diagram as above. When there are extra marked points on the boundary (i.e. ), we add one tube between two positive parts of the relevant boundary component. We add the corresponding circles to the diagram. See figure 3 for the case where the extra sutures are the front ones.

Figure 3: Heegaard surface for with three vertical sutures. Top and bottom are glued. The boundary of is the transparent torus (drawn as an annulus). The Heegaard surface is the union of two vertical annuli and two horizontal tubes. A compression disk bounded by a curve is shown in green.

In this paragraph, whenever we started from the trivial case which was ruled out above, we can use as a starting point the degenerate diagram with Heegaard surface and no circle.

The constructed diagrams have circles of each type and . Hence the chain complex has rank . So the proposition follows from the admissibility of these diagrams and the vanishing of the associated differentials.

Each arc , can be extend to a loop and each pair of arcs corresponding to extra boundary components can be extended to a loop , such that the collection of tori and gives a basis of . This basis can be realized by periodic domains using the and circles associated to the corresponding arcs. So we have a basis of associated to disjoint periodic domains, each having both positive and negative coefficients. Since they have disjoint support, any linear combination of these domains will be admissible and the diagram is admissible.

To compute the differential we note that each region of the complement of the circles in which is not the base region is either a rectangle or an annulus. In addition each rectangle is adjacent to either a rectangle using the same circles or to the base region or to an annulus. One can then use Lipshitz’s formula to prove that the Heegaard–Floer differential vanishes. ∎

A dividing set for is a multi-curve in (see Definition 7). The complement of a dividing set in splits into two (non connected) surfaces according to the sign of their intersection with . The graduation of a dividing set is defined to be the difference of Euler characteristics .

A dividing set is said to be isolating if there a connected component of the complement of which does not intersect the boundary of .

To each dividing set for is associated the contact invariant of the contact structures partitioned by . All such contact structures are either isotopic according to Theorem 9 or overtwisted so they have the same invariant. These invariants belong to the graded part given by the graduation of .

Theorem 15 ([Hkm08]).

Over coefficients, the following are equivalent:

  1. is primitive

  2. is non isolating

Over coefficients, (3) (2) (1).

Conjecture 7.13 of [HKM08] states that the assertions in this theorem are equivalent over coefficients. What remains to be proved is that isolating dividing sets have vanishing invariant. This (and more) will be proved in Section 5.

5 Vanishing results

In this section we prove the main theorem from the introduction and the following theorem which finishes off the proof of Conjecture 7.13 of [HKM08]. We use the definitions and notations of the previous section.

Theorem 16.

If is isolating then over –coefficients.

Note that the analogous statement over twisted coefficients is known to be false. For instance if we consider on a contact structure partitioned by four essential circles and remove a small disk meeting one of these circles along an arc then we get an isolating dividing set on a punctured torus whose twisted invariant is sent to a non vanishing invariant according to Theorem 12 since the corresponding contact structures on are weakly fillable.

Definition 17.

We say that dividing sets , and are bypass-related if they coincide outside a disk where they consists of the dividing sets of Figure 4.

Figure 4: Bypass relation

The following lemma is essentially proved in [HKM08] in the combination of proofs of Lemma 7.4 and Theorem 7.6. We write a proof here to explain why twisted coefficients come for free.

Lemma 18.

If , and are bypass-related then, for any representatives , there exist such that . The same holds over coefficients.

Proof.

The first part of the proof concentrate on the disk where the dividing sets differ. Let be representatives of the contact invariants of the three dividing sets on a disk involved in Definition 17. Note that is trivial so we now work over coefficients and suppress the underlines.

Because the ’s all belong to the same rank 2 summand of there are integers , and not all zero such that

(1)

We denote by the dividing sets of Figure 5 and by their contact invariants.

Figure 5: Dividing sets used to prove Lemma 18

Label the points of clockwise by starting with the upper right point. Let , , denote a HKM gluing map obtained by attaching a boundary parallel arc between points and . The gluing maps have the following effects:

(2)
(3)
(4)

Using these equations and the facts that are non zero in a torsion free group (see Proposition 14), we get

and they are all non zero so we can divide equation 1 by to get

(5)

with and .

We now return to our full dividing sets. Let be the disk where the ’s differ. Denote by the (common) intersection of the ’s with . Let , and be contact structures partitioned by , and respectively and coinciding with some outside .

Let be a HKM gluing map associated to . According to Theorem 12, there exist invertible elements of such that for all . We now apply to equation 5 and put and

Using this Lemma, we can reprove the main result of [GHV].

Proposition 19 ([Ghv]).

Contact structures with positive Giroux torsion have vanishing contact invariant over coefficients.

Figure 6: Dividing sets for Propositions 19 and 20. Left and right sides of each squares should be glued to get annuli.
Proof.

Let be an annulus with two marked points on each boundary component and consider the dividing sets of Figure 6. We will denote by , and contact structures partitioned by the corresponding . Using the disk whose boundary is dashed, one sees that is bypass-related to and . We denote by .

Let be a basic slice on a toric annulus . We glue and to get a new toric annulus. Using the obvious decomposition of and the corresponding one for , we want the dividing slopes to be (this is the slope of the factor) and respectively. By changing the sign of the basic slice, we can assume that is universally tight. It follows from the classification of tight contact structures on toric annuli that a contact manifold has positive Giroux torsion if and only if it contains a copy of . Therefore we only need to prove that vanishes.

Let be a corresponding HKM gluing map. The structures and are –isotopic and they are basic slices. Using invariance under isotopy, we get . Let be a representative of this common contact invariant. Let and be representatives of and such that . Such representatives exist according to the gluing property. We also take any representative and denote by its image under . This image belong to according to the gluing property.

Lemma 18 gives such that

We then apply to this equation to get:

(6)

Let be a standard neighborhood of a Legendrian knot ( is a solid torus). We now glue along the boundary component of which is in so that meridian curves have slope . The structure is overtwisted whereas (and which is isotopic to it) is a standard neighborhood of a Legendrian curve so can be embedded into Stein fillable closed contact manifolds. Let be a gluing map associated to . Applying to equation 6 and using the vanishing property of overtwisted contact structures, we get

Using that is non zero and the fact that the relevant group has no torsion (see [Juh08][Proposition 9.1]) we get . Returning to Equation 6, we then get . ∎

Figure 7: Dividing sets for Proposition 20 and Proposition 21. On the top row, left and right sides of the squares are glued to make the annulus . Then the thick parts of can be glued by translation to make the punctured torus of the bottom row where the sides of the squares are glued by translation and the glued part of is dashed.
Proposition 20.

Let be an annulus with two points on each boundary component. Let be one of the components of and . Let , and be the dividing sets of Figure 7 and let , and be any representatives of their contact invariants in .

  1. There exist invertible elements and in such that:

  2. Twisted invariants distinguish , and . Over coefficients, and are independent but .

  3. Let be the right handed Dehn twist along the core of . There exist and such that for any , .

The second part of this proposition was proved over coefficients in Section 7.5 of [HKM08]. The last part was conjectured in [HKM08][top of page 35].

Proof.

The statement of the proof contains superscripts everywhere in view of its application to Proposition 21 but we don’t use them in this proof since it would clutter all formulas.

Thanks to grading, the twisted invariants , and all live in the same rank two summand of so there exist , not all zero, such that

(7)

We now use two HKM gluing maps: (resp. ) corresponding to gluing the dividing set (resp. ) from the bottom in Figure 7. We will denote loosely by for instance the result of gluing on the bottom of . For any in partitioned by we can perform a generalized Lutz twist on the unique torus which is foliated by Legendrian fibers and the result is partitioned by so the main result of [GH] gives for some invertible element . Since contact structures partitioned by are overtwisted, we get . And is isotopic to so there is some invertible such that . So when we apply to equation 7 we get: .

A similar argument for gives invertible elements and such that:

Since is a free module over the integral domain and and are non zero (the corresponding contact structures embed into Stein fillable contact manifolds), we get

so that and . Since , and are not all zero, we get that is non zero. Setting and , equation 7 gives the announced relation.

We now prove the second point. We have already met morphisms sending , and to elements not related to each other by invertible elements of . So the invariants are pairwise distinct. Going to coefficients sends to zero so the formula of the first point proves that Ozsváth–Szabó invariants over coefficients don’t distinguish and . But they distinguish and as can be seen for instance by using the coefficients version of .

In order to prove the third point we will use the results of Section 3.

Figure 6 shows that , and are bypass related. We start with any representatives for the relevant invariants and Lemma 18 gives us instructions to change signs so that we get and with .

We now stick to these representatives. Using the image of Figure 6 under , we see that , and are bypass related. So Lemma 18 gives signs and such that is in . We set so that is in . We want to prove that . the only other possibility, would give but this is forbidden by Theorem 2 since the corresponding contact structures are sent by gluing the two components of to contact structures on which are distinguished by Ozsváth–Szabó invariants. So is in . The general case follows from an inductive process using the same arguments. ∎

Figure 8: Dividing set for Proposition 21. Sides of the square are glued to make a punctured torus. The dashed circle bounds a disk used to apply Lemma 18
Proposition 21.

Let be a punctured torus, a set of two points on and a dividing set on consisting of a circle and an arc, both boundary–parallel (see Figure 8). Let be the image in