Infinitely many universally tight torsion free contact structures with vanishing Ozsváth-Szabó contact invariants
IIn(cid:28)nitely many universally tight torsion freecontact structures with vanishingOzsvÆth(cid:21)Szab(cid:243) contact invariants
Patrick MassotDecember 2009
AbstractOzsvÆth(cid:21)Szab(cid:243) contact invariants are a powerful way to prove tight-ness of contact structures but they are known to vanish in the presence ofGiroux torsion. In this paper we construct, on in(cid:28)nitely many manifolds,in(cid:28)nitely many isotopy classes of universally tight torsion free contactstructures whose OzsvÆth(cid:21)Szab(cid:243) invariant vanishes. We also discuss therelation between these invariants and an invariant on T and constructother examples of new phenomena in Heegaard(cid:21)Floer theory. Along theway, we prove two conjectures of K Honda, W Kazez and G Mati¢ abouttheir contact topological quantum (cid:28)eld theory. Almost all the proofs inthis paper rely on their gluing theorem for sutured contact invariants. Introduction
Contact topology studies isotopy classes of contact structures . These classescome in two main (cid:29)avors: overtwisted and tight, the latter being further dividedinto universally tight and virtually overtwisted. Up to now, besides homotopicaldata, there are only two algebraic objects which have been successfully used toclassify such isotopy classes on a general 3(cid:21)manifold. The (cid:28)rst one is Girouxtorsion introduced in [Gir00], its de(cid:28)nition is recalled in Section 1. It is eithera non-negative integer or in(cid:28)nite and always in(cid:28)nite for overtwisted classes.It is invariant under isomorphisms, not only isotopies. It shares the mono-tonicity property of symplectic capacities [HZ94] on one hand and the (cid:28)nite-ness property of 3(cid:21)manifolds complexity [Mat90] on the other hand. Indeed, if ( M, ξ ) ⊂ ( M (cid:48) , ξ (cid:48) ) then Tor( ξ ) ≤ Tor( ξ (cid:48) ) and, for (cid:28)xed M and n , there are only(cid:28)nitely many isomorphism classes of contact structures on M whose torsion isat most n . Another way to put it is to say that (cid:28)nite torsion determines contactstructures up to isomorphism and a (cid:28)nite ambiguity. More generally, it plays animportant role in the coarse classi(cid:28)cation of tight contact structures [CGH09]. all manifolds in this paper are oriented and all contact structures are positive a r X i v : . [ m a t h . G T ] D ec he second object, based on open book decompositions [Gir02], is OzsvÆth(cid:21)Szab(cid:243) contact invariants introduced in [OS05] which live in the Heegaard(cid:21)Floerhomology of the ambient manifold. They come in various (cid:29)avors depending ona choice of coe(cid:30)cients. These invariants are a powerful tool to detect tightnessand obstructions to (cid:28)llability by symplectic or complex manifolds. Its mainproperties 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 Girouxtorsion is non zero, the contact invariant over Z coe(cid:30)cients vanishes (we give anew proof of this result in Section 5). Here we prove that the converse does nothold.Main theorem (Section 5). Every Seifert manifold whose base has genus atleast three supports in(cid:28)nitely many (explicit) isotopy classes of universally tighttorsion free contact structures whose OzsvÆth(cid:21)Szab(cid:243) invariant over Z coe(cid:30)cientsvanishes.In the above theorem, the genus hypothesis cannot be completely droppedbecause, for instance, on the sphere S and the torus T , all torsion free contactstructures have non vanishing OzsvÆth(cid:21)Szab(cid:243) invariants. However, it may holdfor genus two bases. Note that the class of Seifert manifolds is the only onewhere isotopy classes of contact structures are pretty well understood. So thetheorem says that examples of universally tight torsion free contact structureswith vanishing OzsvÆth(cid:21)Szab(cid:243) invariant exist on all manifolds we understand,provided there is enough topology (the base should have genus at least three). Inthis statement, isotopy classes cannot be replaced by conjugacy classes becauseof the (cid:28)niteness property explained above. Along the way we prove Conjecture7.13 of [HKM08].Our examples also provide a corollary in the world of Legendrian knots.OzsvÆth(cid:21)Szab(cid:243) theory provides invariants for Legendrian or transverse knotsin di(cid:27)erent (related) ways, see [SV09] and references therein. In the standardcontact 3(cid:21)spheres there are still two seemingly distinct ways to de(cid:28)ne suchinvariants but, in general contact manifolds, the known invariants all come fromthe sutured contact invariant of the complement of the knot according to themain theorem proved by V VØrtesi and A Stipsicz in [SV09]. In this paper theycall strongly non loose those Legendrian knots in overtwisted contact manifoldswhose complement is tight and torsion free. Corollary 1.2 of that papers statesthat 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 over-twisted contact manifold, a null-homologous strongly non loose Legendrian knotwhose sutured invariant vanishes (the construction is explicit).After studying the relationship between OzsvÆth(cid:21)Szab(cid:243) invariants and Girouxtorsion, we now turn to a more speci(cid:28)c relation between these invariants andan invariant de(cid:28)ned only on the 3(cid:21)torus. E Giroux proved that any two incom-pressible prelagrangian tori of a tight contact structure ξ on T are isotopic. We2an then de(cid:28)ne the Giroux invariant G ( ξ ) ∈ H (T ) / ± to be the homologyclass of its prelagrangian incompressible tori. Note that there is a (cid:16)sign ambigu-ity(cid:17) because these tori are not naturally oriented. Translated into this language,Giroux proved that two tight contact structures on T are isotopic if and only ifthey have the same Giroux invariant and the same Giroux torsion, see [Gir00].This invariant is clearly Di(cid:27) (T ) (cid:21)equivariant. Since this group acts transitivelyon primitive elements of H (T ) , we see that all these elements are attained by G . This also proves that all tight contact structures on T which have the sametorsion are isomorphic. This classi(cid:28)cation of tight contact structures on T anda result by Y Eliashberg shows that torsion free contact structures on T areexactly the Stein (cid:28)llable ones.Theorem 2 (Section 3). There is a unique up to sign H (T ) (cid:21)equivariant iso-morphism between (cid:100) HF (T ) and H (T ) ⊕ H (T ) (on the ordinary cohomologyside, H sends H to zero and H to H by slant product). Under this isomor-phism, the OzsvÆth(cid:21)Szab(cid:243) invariant of a torsion free contact structure on T issent to the PoincarØ dual of its Giroux invariant.Note that, on T = R / Z , cohomology classes can be represented by con-stant di(cid:27)erential forms and 1(cid:21)dimensional homology classes by constant vector(cid:28)elds. The slant product of the above theorem is then identi(cid:28)ed with the interiorproduct of vector (cid:28)elds with 2(cid:21)forms.The statement about torsion free contact structures is based on the interac-tion between the action of the mapping class group and (cid:28)rst homology groupof T on its OzsvÆth(cid:21)Szab(cid:243) homology and ordinary cohomology. It sheds somelight on the sign ambiguity of the contact invariant since the sign ambiguity ofthe Giroux invariant is very easy to understand.Corollary 3. There are in(cid:28)nitely many isomorphic contact structures whoseisotopy classes are pairwise distinguished by the OzsvÆth(cid:21)Szab(cid:243) invariant.Theorem 2 proves, via gluing, a conjecture of Honda, Kazez and Mati¢ aboutthe sutured invariants of S (cid:21)invariants contact structures on toric annuli. Thisconjecture is stated in [HKM08][top of page 35] and will be discussed in Section3 Proposition 20.Theorem 2 also have some consequence for the hierarchy of coe(cid:30)cients be-cause Z coe(cid:30)cients can distinguish only (cid:28)nitely many isotopy classes of contactstructures (since (cid:100) HF ( Y ; Z ) is always (cid:28)nite).Corollary 4. There exists a manifold on which the OzsvÆth(cid:21)Szab(cid:243) invariantover integer coe(cid:30)cients distinguishes in(cid:28)nitely many more isotopy classes ofcontact structures than the invariant over Z coe(cid:30)cients.In the same spirit, we prove that twisted coe(cid:30)cients are more powerful than Z coe(cid:30)cients even when the latter give non vanishing invariants.Proposition 5 (see Propositions 20). There exist a sutured manifold with twocontact structures having the same non vanishing OzsvÆth(cid:21)Szab(cid:243) invariant over Z coe(cid:30)cients but which are distinguished by their invariants over twisted coe(cid:30)-cients. 3n Section 1 we review the work of Giroux on certain contact structureson circle bundles, the easy extension of this work to Seifert manifolds and tor-sion calculations. In Section 2 we review OzsvÆth(cid:21)Szab(cid:243) contact invariants. InSection 3 we prove Theorem 2. In Section 4 we review the work of Honda,Kazez and Mati¢ on their contact TQFT and upgrade their SFH groups cal-culations to twisted coe(cid:30)cients. In Section 5, by far the longest, we prove[HKM08][Conjecture 7.13] and the main theorem above.
This section contains preliminary results in contact topology. We (cid:28)rst recall thecrucial de(cid:28)nition of Giroux torsion. The kπ -torsion of a contact manifold ( V, ξ ) was de(cid:28)ned in [Gir00][De(cid:28)nition 1.2] to be the supremum of all integers n ≥ such that there exist a contact embedding of (cid:0) T × [0 , , ker (cos( nkπz ) dx − sin( nkπz ) dy ) (cid:1) , ( x, y, z ) ∈ T × [0 , into the interior of ( V, ξ ) or zero if no such integer n exists. Of course all kπ (cid:21)torsions can be recovered from the π (cid:21)torsion. However when we don’t specify k we mean π -torsion. This is due to the fact that only π -torsion is known tointeract with symplectic (cid:28)llings and OzsvÆth(cid:21)Szab(cid:243) theory.A multi-curve in an orbifold surface B is a 1(cid:21)dimensional submanifold prop-erly embedded in the regular part of B . When B is closed, we will say that amulticurve is essential in B if none of its components bound a disk containingat most one exceptional point.Since we want to extend results from circle bundles to Seifert manifolds andmost surface orbifolds are covered (in the orbifold sense) by smooth surfaces,the following characterization will be useful.Lemma 6. Let Γ be a multicurve in a closed orbifold surface B whose (orbifold)universal cover is smooth. The following statements are equivalent:1. Γ is essential;2. Γ lifts to an essential multicurve in all smooth (cid:28)nite covers of B .3. Γ lifts to an essential multicurve in some smooth (cid:28)nite cover of B .Proof. We (cid:28)rst prove (the contrapositive of) (1) = ⇒ (2). Let Γ be a essentialmulticurve in B and π be an orbifold covering map from a smooth surface ˆ B to B . Suppose that a component of the inverse image of Γ bounds an embeddeddisk ˆ D in ˆ B . Its image in B is a topological disk D and we only need to provethat this disk contains at most one exceptional point. Using multiplicativity ofthe orbifold Euler characteristic under the orbifold covering map from ˆ D to D ,we get χ ( D ) > . This proves that D contains at most one exceptional pointsbecause its Euler characteristic is − s + (cid:80) si =1 /α i with α i ≥ if it has s χ ( D ) ≤ − s/ . So (1) implies (2). Since (2) obviouslyimply (3), we are left with proving (the contrapositive of) (3) implies (1).Assume that Γ is not essential and let D be a connected component of thecomplement of Γ in B which is a disk with at most one exceptional point.In any (cid:28)nite cover ˆ B of B , this disk lifts to a collection of disks bounded bycomponents of the lift of Γ and containing at most one exceptional point. So Γ is non essential in all (cid:28)nite covers of B .The following is the essential de(cid:28)nition of this section.De(cid:28)nition 7 (obvious extension of [Gir01]). A contact structure is partitionedby a multi-curve Γ in B if it transverse to the (cid:28)bers over B \ Γ and if the surface π − (Γ) is transverse to ξ and its characteristics are (cid:28)bers.Example 8 ([Lut77, KT91], see also [Nie]). Let V → B be a Seifert manifoldand Γ be a non empty multi-curve in B whose class in H ( B, ∂B ; Z ) is trivial.There is a S (cid:21)invariant contact structure on V which is partitioned by Γ . Thiscontact structure is unique up to isotopy among S (cid:21)invariant contact structures.The following theorem relies on [Mas08][Theorem A] and on easy extensionsor consequences of the fourth part of [Gir01]. Of course it also uses a lot theresults of [Gir00]. The two papers by Giroux can also be replaced by the Hondaversions [Hon00a, Hon00b]. This theorem could be easily improved to say thingsabout Seifert manifolds with non empty boundary but we won’t need such im-provements. Recall that a closed Seifert manifold is small if it has at most threeexceptional (cid:28)bers and its base has genus zero. Otherwise it is called large. Inparticular the bases of large Seifert manifolds admit essential multi-curves. Wedenote by e ( V ) the rational Euler number of a Seifert manifold V . See [Mas08]for the conventions used here for Seifert invariants and Euler numbers. In thestatement we exclude for convenience the ((cid:28)nitely many) Seifert manifolds whichare torus bundles over the circles (see for instance [Hat] to get the list).Theorem 9. Let V be a closed oriented Seifert manifold over a closed orientedorbifold surface.1. A contact structure on V partitioned by a multi-curve Γ is universally tightif and only if one of the following holds:(a) Γ is empty(b) V is large and Γ is essential(c) V is a Lens space (including S and S × S ), e ( V ) ≥ , Γ is con-nected and each component of its complement contains at most oneexceptional point.2. Any universally tight contact structure on V is isotopic to a partitionedcontact structure. 5. Suppose V is not a torus bundle over the circle. Let ξ be a contact structureon V partitioned by an essential multi-curve Γ . Let n be the greatestinteger such that there exist n closed components of Γ in the same isotopyclass of curves. The Giroux torsion of ξ is zero if Γ is empty and at most (cid:98) n (cid:99) otherwise.4. Let ξ and ξ be contact structures on V partitioned by non empty multi-curves denoted by Γ and Γ respectively. If Γ and Γ are isotopic then ξ and ξ are so. If ξ and ξ are isotopic and universally tight then Γ and Γ are isotopic.We (cid:28)rst comment on some consequences of this theorem which have not muchto 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 manifoldswhich carry universally tight contact structures. This list did not appear inthe literature while the (much subtler) list of Seifert manifolds which carrytight contact structures (maybe virtually overtwisted) was obtained (with muchmore work) by P Lisca and A Stipsicz in [LS]. In addition, the road taken inthat paper to prove existence on large Seifert manifold is much heavier thanusing the above theorem (but the point of that paper is small manifolds).Corollary 10. A closed Seifert manifold V admits a universally tight contactstructure if and only if one of the following holds:1. V is large2. V is a Lens space (including S and S × S )3. V has three exceptional (cid:28)bers which can be numbered such that its Seifertinvariants are (0 , − , ( α , β ) , ( α , β ) , ( α , β )) with β α > m − am , β α > am , and β α > m − m for some relatively prime integers < a < m .The above theorem also proves that all universally tight contact structureson Seifert manifolds interact nicely with the Seifert structure.Corollary 11. If ξ is a universally tight contact structure on a closed Seifertmanifold V then there exist a locally free S action on V such that ξ is eithertransverse to the orbits or invariant.Note that the alternative in the above corollary is not exclusive. A contactstructure which is both invariant and transverse to the orbits of a locally free S action exists exactly when e ( V ) < , this was proved by Y Kamishima andT Tsuboi in [KT91]. There is only one isomorphism class of contact structureof this type when they exist. This class is of Sasaki type and sometimes calledthe canonical isomorphism class of contact structures on V .6roof of Theorem 9. We now outline the main di(cid:27)erences between Theorem 9and the parts which are already written in [Gir01]. First it should be notedthat, when V is either a Lens space or a solid torus with a standard Seifert(cid:28)bration, everything is well understood thanks to the classi(cid:28)cation theorems of[Gir00] (see also [Hon00a]). So we don’t consider these Seifert manifolds in thefollowing.1) Let ξ be a contact structure on a closed V partitioned by Γ . If Γ isempty then ξ is transverse to the (cid:28)bers hence universally tight according to[Mas08][Theorem A] (this direction follows rather directly from Bennequin’stheorem). If V is large and Γ is essential then the base B of V is covered (inthe orbifold sense) by a smooth surface Σ and there is a corresponding circlebundle ˆ V → Σ covering (honestly) V . The pulled back contact structure ispartitioned by the inverse image of Γ which is essential according to Lemma 6so ξ is universally tight according to [Gir01] ((cid:28)rst line of page 252).Conversely, assume that ξ is universally tight and partitioned by a non emptymulti-curve Γ . Assume (cid:28)rst the base of V is covered by a smooth surface ofgenus at least one (for instance if V is large). The manifold V then is coveredby 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 toisotopy, which is essential. Since the lift of Γ is such a curve, it is essential andLemma 6 implies that Γ is also essential. In particular V is large.If no such cover of the base exists (and V is not a Lens space) then its base B is a sphere with exceptional points of order (2 , , n ) , (2 , , , (2 , , or (2 , , (see [Thu][Theorem 13.3.6]). In each case B is covered by S and all curves inthe regular locus of B bounds a disk whose pre-image in S is disconnected so ξ 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 number if it is isotopic to a contact structure forwhich there exists a Legendrian regular (cid:28)ber whose contact framing coincideswith the (cid:28)bration framing. If this property is not satis(cid:28)ed then [Mas08][TheoremA] ensures that any universally tight ξ is isotopic to a contact structure parti-tioned by the empty multi-curve (i.e. transverse to the (cid:28)bers). We now assumethat ξ has non negative maximal twisting number and has been isotoped so thatit admits a Legendrian (cid:28)ber L as above. Let K be a wedge of circles based at L in the smooth part of B (seen as the space of all (cid:28)bers) let R be a smallregular neighborhood of K . We can choose K an R such that the complement R (cid:48) of R in B is made of disks containing exactly one exceptional point. Thetechniques of [Gir01] prove that ξ is isotopic to a contact structure which, over R is partitioned by a multicurve Γ R which intersects all boundary componentsof R . We now assume this property. Let V (cid:48) denote the (non necessarily con-nected) Seifert manifold over R (cid:48) and ξ (cid:48) the restricted contact structure. Since Γ R intersects all components of ∂R , each component T of the boundary of V (cid:48) contains a Legendrian regular (cid:28)ber which is either a closed leaf or a circle ofsingularities of the characteristic foliation ξ (cid:48) T . If ξ (cid:48) is universally tight then the some texts say zero twisting number in this case ξ (cid:48) is ∂ (cid:21)isotopic to a contact structure partitioned by some Γ R (cid:48) extending Γ R and we are done. More precisely, for each component W of V (cid:48) , this classi(cid:28)-cation guaranties the existence of exactly one isotopy class of universally tightcontact structure coinciding with ξ (cid:48) on ∂W (cid:48) when W contains no exceptional(cid:28)ber and two otherwise. In the latter case, the two classes correspond to thetwo isotopy classes of arcs extending Γ R inside the base of W (which is a diskwith one exceptional point).So it remains to prove that if ξ has non negative maximal twisting numberand is universally tight then each solid torus W isotopic to a (cid:28)bered one hasa universally tight induced contact structure. This is obvious if the universalcover ˜ W of W naturally embeds into the universal cover ˜ V of V . This ˜ V canbe built in two stages: (cid:28)rst one takes the (orbifold) universal cover of the base B and pulls back the Seifert (cid:28)bration and then one unwraps the (cid:28)bers as muchas possible. The sought embedding of ˜ W obviously exist when the (cid:28)bers can becompletely unwrapped. Due to the classi(cid:28)cation of orbifolds surfaces the onlyproblematic case if one excludes Lens spaces is when ˜ V is S with its (smooth)Hopf (cid:28)bration. But, by de(cid:28)nition of tightness, any tight contact structure on S has negative twisting number with respect to the Hopf (cid:28)bration so this casedoes not happen here (the property of having non negative twisting number isobviously inherited by (cid:28)nite covers using lifts of isotopies).3) Since we assume that V is not a torus bundle over the circle, all incom-pressible tori are isotopic to (cid:28)bered ones (see e.g. [Hat]).Suppose (cid:28)rst that ξ is partitioned by the empty multicurve (i.e. is transverseto all (cid:28)bers). It was proved in [Mas08][Theorem A] that such a contact structurehas negative maximal twisting number. Suppose by contradiction that it hasnon vanishing π (cid:21)torsion. Up to isotopy of ξ there is an annulus in the basewhich is foliated by circles ( C t ) t ∈ [0 , such that, • For all t , the torus T t above C t in V is prelagrangian. • The directions of the Legendrian foliations of the T t go all over the pro-jective line .During this full turn around the projective line, the Legendrian direction meetsthe (cid:28)ber direction and there are Legendrian curve whose contact framing co-incides with the (cid:28)bration framing so we get a contradiction with the maximaltwisting number estimate.We now assume that ξ is partitioned by a non empty multicurve Γ and thatno two components of Γ are isotopic. Incompressible (cid:28)bered tori correspondto essential curves in the base orbifold B . To any such curve C correspond anorbifold covering of B by an open annulus ˆ B and the Seifert (cid:28)bration lifts to atrivial (smooth) circle (cid:28)bration ˆ V . The lifted contact structure is partitionedby the inverse image of Γ which is made of as many essential circles as therewere components of Γ isotopic to C (at most n ) and lines properly embeddedin ˆ B . If there exist a contact embedding of a toric annulus with its standardtorsion contact structure in V then it lifts to ˆ V inside some K × S with K ⊂ ˆ B (cid:98) n (cid:99) knowing the partition we have over K . This argumentis not new, it was explained to me (around 2005) by E Giroux.4) The (cid:28)rst part is a straightforward extension of [Gir01][Lemma 4.7]. Sup-pose now that ξ and ξ are isotopic. If V is not large then we are in case (c) ofthe (cid:28)rst point so that Γ and Γ are trivially isotopic. So we now assume that V is large. In particular Γ and Γ are essential. By de(cid:28)nition, they are isotopicin B if and only if they are isotopic in the smooth surface R obtained from B by removing a small open disk around each exceptional point. By de(cid:28)nition ofessential curves, no component of Γ or Γ is parallel to the boundary of R . Ac-cording to W Thurston, Γ and Γ are isotopic if and only if they have the samegeometric intersection number with all closed curves in B [Thu88][Propositionpage 421]. These geometric intersections number have a contact topology in-terpretation explained in [Gir01][Section 4.E] which proves they are invariantunder contact structures isotopy exactly as in the circle bundle case. In this section we review sutured Heegaard(cid:21)Floer homology and the contactinvariants which lives in it.Heegaard(cid:21)Floer homology was introduced by P OzsvÆth and Z Szab(cid:243) in[OS04b] and extended to sutured manifold by A JuhÆsz in [Juh06]. In thefollowing we will often silently identify a closed manifold M with the suturedmanifold ( M \ B , S ) and use sutured Floer theory (SFH) also in this case.We denote the universal twisted SFH( − M, Γ; Z [ H ( M ; Z )]) by SFH( − M, Γ) and, whenever there is no ambiguity on the manifold M we are considering, wedenote Z [ H ( M ; Z )] by L .According to [GH][Lemma 10], if a contact invariant vanishes in SFH thenit vanishes for all coe(cid:30)cients rings.Theorem 12 (OzsvÆth(cid:21)Szab(cid:243), Honda(cid:21)Kazez(cid:21)Mati¢, Ghiggini(cid:21)Honda(cid:21)Van HornMorris). Let ( M, Γ) be a balanced sutured manifold. To each contact struc-ture ξ on ( M, Γ) , one can associate a contact invariant which is a set c ( ξ ) in SFH( − M, − Γ) / ± and a twisted contact invariant which is a set c ( ξ ) in SFH( − M, − Γ) / L × satisfying the following properties:1. the set c ( ξ ) is invariant under ∂ (cid:21)isotopy of ξ
2. if ξ is overtwisted then c ( ξ ) = 0
3. if ξ has non zero torsion then c ( ξ ) = 0
4. if M is closed and ξ is weakly (cid:28)llable then c ( ξ ) (cid:54) = 0
5. if M is closed and ξ is strongy (cid:28)llable then c ( ξ ) (cid:54) = 0
9. if ( M (cid:48) , Γ (cid:48) ) is a sutured submanifold of ( M, Γ) and ξ is a contact structureon ( M \ M (cid:48) , Γ ∪ Γ (cid:48) ) then there exists a linear map Φ ξ : SFH( − M (cid:48) , − Γ (cid:48) ) → SFH( − M, − Γ) such that, for any contact structure ξ (cid:48) on ( M (cid:48) , Γ (cid:48) ) , one has c ( ξ ∪ ξ (cid:48) ) = Φ ξ ( c ( ξ (cid:48) )) . If every connected component of M \ int ( M (cid:48) ) intersect ∂M then there areanalogous maps over Z coe(cid:30)cients. They are denoted without underlines.7. if ( M (cid:48) , ξ (cid:48) ) is a contact submanifold of ( M, ξ (cid:48) ∪ ξ ) then c ( ξ (cid:48) ) = 0 implies c ( ξ ∪ ξ (cid:48) ) = 0 and analogously over Z coe(cid:30)cients.The construction of the contact invariants (and the isotopy invariance) canbe found in [OS05] for the closed case and [HKM07] in general. The fact that itvanishes for overtwisted contact structures was (cid:28)rst proved for the closed caseand untwisted coe(cid:30)cients in [OS05] and follow in general from the last propertyand the explicit calculation of the twisted contact invariant of a neighborhood ofan overtwisted disk found in [HKM07]. The assertion about torsion was provedin [GHV]. Both assertions about (cid:28)llings are consequences of [OS04a][Theorem4.2], using the fact that, for strong (cid:28)llings, the coe(cid:30)cient ring in this theorem re-duces to Z (see also [Ghi06][Theorem 2.13] for an alternative proof of the strong(cid:28)lling property). The gluing properties are proved in [HKM08] for untwistedcoe(cid:30)cients and extended to twisted coe(cid:30)cients in [GH]. The gluing maps areunique up to multiplication by an invertible element of the relevant coe(cid:30)cientsring. Such maps will be called HKM gluing maps.There is one piece of structure of Heegaard(cid:21)Floer theory which doesn’t seemto have been explicitly discussed in our context up to now: the mapping classgroup action. Any di(cid:27)eomorphism of a 3(cid:21)manifold M acts on any variant of HF ( M ) . Here we need to be precise about what depends on the way a Heegaarddiagram is embedded inside a manifold and what does not depend on it. Theusual way to do that is to consider embedded Heegaard diagrams as pairs madeof a self-indexing Morse function with unique minima and maxima and oneof its Morse(cid:21)Smale pseudo-gradients. Given such a pair ( f, X ) , the Heegaardsurface is f − (3 / and the Heegaard circles are the intersections of the stableor unstable disks of the index 1 and 2 critical points. We denote the groupassociated to ( f, X ) by HF ( f, X ) (we can use here (cid:100) HF , HF + ,. . . ). Let ϕ be adi(cid:27)eomorphism of M . Then [OS06][Theorem 2.1] gives an isomorphism Ψ : HF ( f, X ) → HF ( f ◦ ϕ, ϕ ∗ X ) which is well de(cid:28)ned up to sign. But of course the di(cid:27)eomorphism ϕ also givesan isomorphism between the corresponding abstract Heegaard diagrams whichthen gives an isomorphism Φ between Heegaard(cid:21)Floer groups. The action of ϕ we don’t claim to do anything new in this paragraph, but we can’t (cid:28)nd a reference for it HF ( f, X ) is de(cid:28)ned to be Φ − ◦ Ψ . It is obvious from the construction thatthe contact invariant is equivariant under this action. What is not obvious isthat isotopic di(cid:27)eomorphisms have the same action so that we get an action ofthe mapping class group. This has been checked by P OzsvÆth and A Stipsiczin the context of knot Floer homology in [OS]. In this paper we don’t use thisinvariance but use speci(cid:28)c di(cid:27)eomorphisms. Actually this invariance shouldnever be needed in contact geometry since we already know that the contactinvariant is a contact structure isotopy invariant so that di(cid:27)eomorphism isotopyinvariance is automatic on the subgroup spanned by contact invariants in any (cid:100) HF or HF + . In this section we prove Theorem 2 from the introduction. The following easylemma is the key algebraic trick.Lemma 13. If an isomorphism
Φ : (cid:100) HF (T ) → H (T ) ⊕ H (T ) is H (T ) (cid:21)equivariant then it conjugates the SL actions of both sides.Proof. In this proof we drop T from the notations. We denote by ρ the canoni-cal action of SL on H . Let ρ and ρ be two representations of SL on H ⊕ H which are compatible with the H action, that is: ∀ g ∈ SL , γ ∈ H , m ∈ H ⊕ H , ( ρ ( g ) γ ) ρ i ( g ) m = ρ i ( g ) ( γm ) . We want to prove that ρ = ρ since this, applied to the standard action andto the action transported by Φ , will prove the proposition.We (cid:28)rst prove that, for all g ∈ SL , ρ ( g ) and ρ ( g ) agree on H . Thekey property of the H action is that it separates all elements of H : for all m (cid:54) = m (cid:48) ∈ H , there exists γ in H such that γm = 0 and γm (cid:48) (cid:54) = 0 .Suppose by contradiction that there exists g ∈ SL and m ∈ H such that ρ ( g ) m (cid:54) = ρ ( g ) m . According to the separation property, there exists γ (cid:48) in H such that γ (cid:48) ρ ( g ) m = 0 and γ (cid:48) ρ ( g ) m (cid:54) = 0 . Setting γ = ρ ( g ) − ( γ (cid:48) ) , we get ρ ( g ) γρ ( g ) m = 0 and ρ ( g ) γρ ( g ) m (cid:54) = 0 , so ρ ( g )( γm ) = 0 and ρ ( g )( γm ) (cid:54) = 0 ,which is absurd since ρ ( g ) and ρ ( g ) are both isomorphisms.We now prove that the representations agree on H . For all m (cid:48) ∈ H , thereexists m ∈ H and γ ∈ H such that m (cid:48) = γm . So for any g ∈ SL and i = 1 , ,we get ρ i ( g ) m (cid:48) = ρ i ( g )( γm ) = ρ ( g ) γρ i ( g ) m and we know that ρ ( g ) m = ρ ( g ) m thanks to the (cid:28)rst part so ρ ( g ) m (cid:48) = ρ ( g ) m (cid:48) .Proof of Theorem 2. The existence of such an isomorphism is Proposition 8.4of [OS03]. The above lemma proves that, for any Φ as in the statement andany x ∈ (cid:100) HF , x and Φ( x ) have the same stabilizer under the action of SL . Theuniqueness of Φ follows since primitive elements of H ⊕ H are characterizedup to sign by their stabilizers. Linearity of Φ then guaranties that the sign iscommon to all elements. 11e now prove that the PoincarØ dual of the Giroux invariant and the imageof the OzsvÆth(cid:21)Szab(cid:243) invariant coincide on torsion free contact structures. Firstremark that the OzsvÆth(cid:21)Szab(cid:243) invariant belongs to (cid:100) HF − / (cid:39) H because theHopf invariant of tight contact structures on T is / . So both invariants areprimitive elements of H . We prove that the stabilizer of G ( ξ ) is contained inthat of c ( ξ ) using equivariance of both invariants and the fact that G is a totalinvariant. For any g in SL and ξ a torsion free contact structure, we have gG ( ξ ) = G ( ξ ) ⇐⇒ G ( gξ ) = G ( ξ ) ⇐⇒ gξ ∼ ξ = ⇒ c ( gξ ) = c ( ξ ) ⇐⇒ gc ( ξ ) = c ( ξ ) so we have the annouced inclusion of stabilizers and this gives c ( ξ ) = G ( ξ ) . We now review the contact TQFT of Honda(cid:21)Kazez(cid:21)Mati¢. Let Σ be a nonnecessarily connected compact oriented surface with non empty boundary. Let F be a (cid:28)nite subset of ∂ Σ whose intersection with each component of ∂ Σ isnon empty and consists of an even number of points. We assume that thecomponents of ∂ Σ \ F are labelled alternatively by + and − . This labelling willalways be implicit in the notation (Σ , F ) . The contact TQFT associates to each (Σ , F ) the graded group V (Σ , F ) = SF H ( − (Σ × S ) , − ( F × S )) (strictly speaking, one should replace F by a small translate of F along ∂ Σ inthis formula).In this construction one can use coe(cid:30)cients in Z or twisted coe(cid:30)cients(including the trivial twisting which leads to Z coe(cid:30)cients). We denote by V (Σ , F ) the version twisted by Z [ H (Σ × S )] .Proposition 14. Let (Σ , F ) be a surface with marked boundary points as aboveand M be any coe(cid:30)cient module for the sutured manifold (Σ × S , F × S ) . Wehave, for any coherent orientations system: V (Σ , F ; M ) (cid:39) ( M ( − ⊕ M (1) ) ⊗ ( F/ − χ (Σ)) . The subscripts ( − and (1) refer to the grading.Proof. The analogous statement over Z coe(cid:30)cients was proved in [HKM08] us-ing product annuli decomposition, [FJR][Proposition 7.13]. This technology isnot yet available over twisted coe(cid:30)cients but one can actually draw explicitadmissible sutured Heegaard diagrams with vanishing di(cid:27)erential for these su-tured manifolds. We will sketch how to construct them and draw pictures forthe three cases where we actually use this computation below.12e (cid:28)rst recall what is an (embedded) Heegaard diagram for a (balancedconnected) sutured manifold ( V, Γ) . It consists of a surface S properly embeddedin V and circles α , β , . . . , α k , β k in S such that: • ∂V = Γ • if we denote by V + the connected component of V \ S containing R + ,there exist open disks properly embedded in V + , called compression disks,bounded by the α circles and such that V + ∪ R + retracts by deformationon R + • the analogous statement holds for V − and R − with the β circles.We now return to the proposition. Let g be the genus of Σ , r the numberof boundary components and n = F/ . The sutured manifold we study willbe denoted by ( V, Γ) for concision. We rule out the trivial ( g = 0 , r = 1 , n = 1) case from this discussion as it needs (easy) special treatment. Assume (cid:28)rstthat r = 1 and n = 1 . Let a , . . . , a g be a system of disjoints arcs properlyembedded in Σ which cuts Σ to a disk. Let P , . . . , P g be tubes around thearcs a i × { θ } for some (cid:28)xed θ ∈ S . We can assume that each P i meets theboundary of V in its positive part R + . Let S (cid:48) be the union of R + and the tubes P i . The surface S obtained by pushing S (cid:48) to make it properly embedded in V is a Heegaard surface for ( V, Γ) .Each tube P i naturally bounds a regular neighborhood D × [ − , of thearc a i . Let α i be the boundary of D × { } in each P i . Let β i be the union of {± } × [ − , and two arcs in R + so that β i and half of P i becomes isotopicto a (cid:28)bered annulus in V . See (cid:28)gure 1 for the case ( g = 1 , r = 1 , n = 1) .We then have a Heegaard diagram ( S, α, β ) for ( V, Γ) . We now explain whathappens when we add some extra boundary components (i.e. r > ). Foreach extra component T j we add two tubes P g + j and P (cid:48) g + j around horizontalarcs a g + j × { θ } and a (cid:48) g + j × { θ } . We choose these arcs so that they can becompleted by arcs in the positive part of ∂ Σ to get a circle isotopic to the newboundary component. See (cid:28)gure 2 for the case ( g = 0 , r = 2 , n = 2) where theextra boundary component is the front one. We add circles α g + j , α (cid:48) g + j , β g + j and β (cid:48) g + j to the diagram as above. When there are extra marked points on theboundary (i.e. n > r ), we add one tube P g + r − k between two positive partsof the relevant boundary component. We add the corresponding circles to thediagram. See (cid:28)gure 3 for the case ( g = 0 , r = 1 , n = 3) where the extra suturesare the front ones. In this paragraph, whenever we started from the trivial case ( g = 0 , r = 1 , n = 1) which was ruled out above, we can use as a starting pointthe degenerate diagram with Heegaard surface R + and no circle.The constructed diagrams have g + 2( r −
1) + ( n − r ) circles of each typeand α i ∩ β j = 2 δ ij . Hence the chain complex has rank n − χ (Σ) . So theproposition follows from the admissibility of these diagrams and the vanishingof the associated di(cid:27)erentials.Each arc a i , ≤ i ≤ g can be extend to a loop ¯ a i and each pair of arcscorresponding to extra boundary components can be extended to a loop l j ,13igure 1: Heegaard surface for V = (T \ D ) × S with two vertical sutures.Everything lives inside a cube minus a neighborhood of its vertical edges. Thecube’s faces are pairwise glued to get (T \ D ) × S . The picture show the(almost) Heegaard surface S (cid:48) of the proof. The blue annulus in the back left isthe negative part R − of the boundary of V . The β compression disks are insidethe front, back, left and right faces of the cube (which get glued to two annuliin V ). ≤ r − such that the collection of tori ¯ a i × S and l j × S gives a basis of H ( V, Z ) . This basis can be realized by periodic domains using the α and β circles associated to the corresponding arcs. So we have a basis of H ( V, Z ) associated to disjoint periodic domains, each having both positive and negativecoe(cid:30)cients. Since they have disjoint support, any linear combination of thesedomains will be admissible and the diagram is admissible.To compute the di(cid:27)erential we note that each region of the complement ofthe circles in S 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 circlesor to the base region or to an annulus. One can then use Lipshitz’s formula toprove that the Heegaard(cid:21)Floer di(cid:27)erential vanishes.A dividing set for (Σ , F ) is a multi-curve K in Σ (see De(cid:28)nition 7). Thecomplement of a dividing set in Σ splits into two (non connected) surfaces R ± according to the sign of their intersection with ∂ Σ . The graduation of a dividingset is de(cid:28)ned to be the di(cid:27)erence of Euler characteristics χ ( R + ) − χ ( R − ) .A dividing set K is said to be isolating if there a connected component ofthe complement of K which does not intersect the boundary of Σ .To each dividing set K for (Σ , F ) is associated the contact invariant of thecontact structures partitioned by K . All such contact structures are eitherisotopic according to Theorem 9 or overtwisted so they have the same invariant.14igure 2: Heegaard surface for V = ( I × S ) × S with two vertical sutures.Top and bottom are glued. Left and right are glued. The boundary of V is theunion of the transparent tori (drawn as rectangles). The Heegaard surface isthe union of two vertical annuli and two horizontal tubes. A compression diskbounded by a β curve is shown in green.These invariants belong to the graded part given by the graduation of K .Theorem 15 ([HKM08]). Over Z coe(cid:30)cients, the following are equivalent:1. c ( K ) (cid:54) = 0 c ( K ) is primitive3. K is non isolatingOver Z coe(cid:30)cients, (3) = ⇒ (2) = ⇒ (1).Conjecture 7.13 of [HKM08] states that the assertions in this theorem areequivalent over Z coe(cid:30)cients. What remains to be proved is that isolatingdividing sets have vanishing invariant. This (and more) will be proved in Section5. In this section we prove the main theorem from the introduction and the fol-lowing theorem which (cid:28)nishes o(cid:27) the proof of Conjecture 7.13 of [HKM08]. Weuse the de(cid:28)nitions and notations of the previous section.Theorem 16. If K is isolating then c ( K ) = 0 over Z (cid:21)coe(cid:30)cients.15igure 3: Heegaard surface for V = D × S with three vertical sutures. Topand bottom are glued. The boundary of V is the transparent torus (drawn asan annulus). The Heegaard surface is the union of two vertical annuli and twohorizontal tubes. A compression disk bounded by a β curve is shown in green.Note that the analogous statement over twisted coe(cid:30)cients is known to befalse. For instance if we consider on T a contact structure partitioned by fouressential circles and remove a small disk meeting one of these circles along anarc then we get an isolating dividing set on a punctured torus whose twistedinvariant is sent to a non vanishing invariant according to Theorem 12 since thecorresponding contact structures on T are weakly (cid:28)llable.De(cid:28)nition 17. We say that dividing sets K , K and K are bypass-related ifthey coincide outside a disk D where they consists of the dividing sets of Figure4. Figure 4: Bypass relationThe following lemma is essentially proved in [HKM08] in the combination ofproofs of Lemma 7.4 and Theorem 7.6. We write a proof here to explain whytwisted coe(cid:30)cients come for free. 16emma 18. If K , K and K are bypass-related then, for any representatives ˜ c i ∈ c ( K i ) , there exist a, b ∈ L × such that ˜ c = a ˜ c + b ˜ c . The same holds over Z coe(cid:30)cients.Proof. The (cid:28)rst part of the proof concentrate on the disk where the dividing setsdi(cid:27)er. Let ˜ c Di be representatives of the contact invariants of the three dividingsets on a disk D involved in De(cid:28)nition 17. Note that H ( D × S ) is trivial sowe now work over Z coe(cid:30)cients and suppress the underlines.Because the c Di ’s all belong to the same rank 2 summand of V ( D, F D ) thereare integers λ , µ and ν not all zero such that λ ˜ c D = µ ˜ c D + ν ˜ c D . (1)We denote by K ± the dividing sets of Figure 5 and by c ± their contactinvariants. Figure 5: Dividing sets used to prove Lemma 18Label the points of F D clockwise by , . . . , starting with the upper rightpoint. Let Φ j , j = 1 , , , denote a HKM gluing map obtained by attaching aboundary parallel arc between points j and j + 1 . The gluing maps have thefollowing e(cid:27)ects: Φ : c D (cid:55)→ c + , c D (cid:55)→ c + , c D (cid:55)→ (2) Φ : c D (cid:55)→ , c D (cid:55)→ c − , c D (cid:55)→ c − (3) Φ : c D (cid:55)→ c + , c D (cid:55)→ , c D (cid:55)→ c + (4)Using these equations and the facts that c ± are non zero in a torsion free group(see Proposition 14), we get (2) = ⇒ λ = ± µ (3) = ⇒ µ = ± ν (4) = ⇒ λ = ± ν and they are all non zero so we can divide equation 1 by λ to get ˜ c D = ε ˜ c D + ε ˜ c D . (5)with ε = µ/λ and ε = ν/λ . 17e now return to our full dividing sets. Let D be the disk where the K i ’sdi(cid:27)er. Denote by F D the (common) intersection of the K i ’s with ∂D . Let ξ , ξ and ξ be contact structures partitioned by K , K and K respectively andcoinciding with some ξ b outside D × S .Let Φ : V ( D, F D ) → V (Σ , F ) be a HKM gluing map associated to ξ b .According to Theorem 12, there exist invertible elements a i of L such that Φ(˜ c Di ) = a i ˜ c i for all i . We now apply Φ to equation 5 and put a = ε a a − and b = ε a a − Using this Lemma, we can reprove the main result of [GHV].Proposition 19 ([GHV]). Contact structures with positive Giroux torsion havevanishing contact invariant over Z coe(cid:30)cients.Figure 6: Dividing sets for Propositions 19 and 20. Left and right sides of eachsquares should be glued to get annuli.Proof. Let ( A, F A ) be an annulus with two marked points on each boundarycomponent and consider the dividing sets of Figure 6. We will denote by ξ , ξ and ξ contact structures partitioned by the corresponding K i . Using the diskwhose boundary is dashed, one sees that K is bypass-related to K and K .We denote ( A × S , F A × S ) by ( N, Γ) .Let ξ b be a basic slice on a toric annulus ( N (cid:48) , Γ (cid:48) ) . We glue ( N, Γ) and ( N (cid:48) , Γ (cid:48) ) to get a new toric annulus. Using the obvious decomposition of H ( N ) and thecorresponding one for H ( N ∪ N (cid:48) ) , we want the dividing slopes to be ∞ (thisis the slope of the S factor) and respectively. By changing the sign of thebasic slice, we can assume that ξ ∪ ξ b is universally tight. It follows from theclassi(cid:28)cation of tight contact structures on toric annuli that a contact manifoldhas positive Giroux torsion if and only if it contains a copy of ξ ∪ ξ b . Thereforewe only need to prove that c ( ξ ∪ ξ b ) vanishes.Let Φ = Φ ξ b be a corresponding HKM gluing map. The structures ξ ∪ ξ b and ξ ∪ ξ b are ∂ (cid:21)isotopic and they are basic slices. Using invariance underisotopy, we get c ( ξ ∪ ξ b ) = c ( ξ ∪ ξ b ) . Let ˜ c b be a representative of this commoncontact invariant. Let ˜ c and ˜ c be representatives of c ( K ) and c ( K ) such that ˜ c b = Φ(˜ c ) = Φ(˜ c ) . Such representatives exist according to the gluing property.We also take any representative ˜ c ( K ) ∈ c ( K ) and denote by ˜ c ( ξ ∪ ξ b ) its imageunder Φ . This image belong to c ( ξ ∪ ξ b ) according to the gluing property.18emma 18 gives ε , ε ∈ {± } such that ˜ c ( K ) = ε ˜ c + ε ˜ c . We then apply Φ to this equation to get: ˜ c ( ξ ∪ ξ b ) = ( ε + ε )˜ c b . (6)Let ( W, ξ W ) be a standard neighborhood of a Legendrian knot ( W is a solidtorus). We now glue ( W, ξ W ) along the boundary component of N ∪ N (cid:48) whichis in ∂N so that meridian curves have slope . The structure ξ W ∪ ξ ∪ ξ b isovertwisted whereas ξ W ∪ ξ ∪ ξ b (and ξ W ∪ ξ ∪ ξ b which is isotopic to it) isa standard neighborhood of a Legendrian curve so can be embedded into Stein(cid:28)llable closed contact manifolds. Let Φ W be a gluing map associated to ξ W .Applying Φ W to equation 6 and using the vanishing property of overtwistedcontact structures, we get ε + ε )Φ W (˜ c b ) . Using that Φ W (˜ c b ) is non zero and the fact that the relevant SFH group has notorsion (see [Juh08][Proposition 9.1]) we get ε + ε = 0 . Returning to Equation6, we then get c ( ξ ∪ ξ b ) = 0 .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 A . Then thethick parts of ∂A can be glued by translation to make the punctured torus ofthe bottom row where the sides of the squares are glued by translation and theglued part of ∂A is dashed.Proposition 20. Let ( A, F A ) be an annulus with two points on each boundarycomponent. Let T be one of the components of ∂A × S and t = e [ T ] ∈ Z [ H ( A × S )] . Let K A , K A and K A be the dividing sets of Figure 7 and let ˜ c A , ˜ c A and ˜ c A be any representatives of their contact invariants in V ( A, F A ) .19. There exist invertible elements a and b in L such that: ˜ c A = a ˜ c A + b ( t − c A .
2. Twisted invariants distinguish K A , K A and K A . Over Z coe(cid:30)cients, c ( K A ) and c ( K A ) are independent but c ( K A ) = c ( K A ) .3. Let τ be the right handed Dehn twist along the core of A . There exist ˜ c ∈ c ( K A ) and ˜ c ∈ c ( K A ) such that for any n ∈ Z , ˜ c + n ˜ c ∈ c ( τ n K A ) .The second part of this proposition was proved over Z coe(cid:30)cients in Section7.5 of [HKM08]. The last part was conjectured in [HKM08][top of page 35].Proof. The statement of the proof contains A superscripts everywhere in viewof its application to Proposition 21 but we don’t use them in this proof since itwould clutter all formulas.Thanks to grading, the twisted invariants ˜ c , ˜ c and ˜ c all live in the samerank two summand of V ( A, F A ) so there exist λ, µ, ν ∈ L , not all zero, such that λ ˜ c = µ ˜ c + ν ˜ c . (7)We now use two HKM gluing maps: Φ (resp. Φ ) corresponding to gluingthe dividing set K (resp. K ) from the bottom in Figure 7. We will denoteloosely by K ∪ K for instance the result of gluing K on the bottom of K .For any ξ in partitioned by K we can perform a generalized Lutz twist on theunique torus which is foliated by Legendrian (cid:28)bers and the result is partitionedby K ∪ K so the main result of [GH] gives Φ (˜ c ) = d ( t − c for some invertibleelement d . Since contact structures partitioned by K ∪ K are overtwisted, weget Φ (˜ c ) = 0 . And K ∪ K is isotopic to K so there is some invertible e suchthat Φ (˜ c ) = e ˜ c . So when we apply Φ to equation 7 we get: λd ( t − c = νe ˜ c .A similar argument for Φ gives invertible elements f and g such that: µf ( t − c + νg ˜ c = 0 . Since
SFH(
A, F A ) is a free module over the integral domain L and ˜ c and ˜ c arenon zero (the corresponding contact structures embed into Stein (cid:28)llable contactmanifolds), we get λd ( t −
1) = νeµf ( t −
1) + νg = 0 . so that ν = λe − d ( t − and µ = − f − ge − dλ . Since λ , µ and ν are not allzero, we get that λ is non zero. Setting a = − f − ge − d and b = e − d , equation7 gives the announced relation.We now prove the second point. We have already met morphisms sending ˜ c , ˜ c and ˜ c to elements not related to each other by invertible elements of L .So the invariants c ( K i ) are pairwise distinct. Going to Z coe(cid:30)cients sends t −