CLASSIFICATION OF TOTALLY REAL ELLIPTIC LEFSCHETZ FIBRATIONS VIA NECKLACE DIAGRAMS

CLASSIFICATION OF TOTALLY REAL ELLIPTIC LEFSCHETZ FIBRATIONS VIA NECKLACE DIAGRAMS NERMİN SALEPCİ Abstract. We shw that ttally real elliptic Lefschetz fibratins admitting a real sectin are classified by their real lci which can be encded in terms f a cmbinatrial bject that we call a necklace diagram. By means f necklace diagrams, we btain an eplicit list f certain classes f ttally real elliptic Lefschetz fibratins. 1. Intrductin As is well knwn, a Lefschetz fibratin is a prjectin frm an riented cnnected smth 4-manifld nt an riented cnnected smth surface such that there eist finitely many critical pints arund which ne can chse cmple charts n which the prjectin takes the frm (z 1, z 2 ) z1 2 + z2. 2 Regular fibers f a Lefschetz fibratin are riented clsed smth surfaces f genus g, while singular fibers have nly ndes. In the present wrk, we cnsider nly thse fibratins whse fiber genus is 1. We call such fibratins elliptic Lefschetz fibratins. Withut lss f generality, we assume that each singular fiber cntains nly ne nde and that n fiber cntains a self intersectin -1 sphere (fibratins with the latter prperty are called relatively minimal). We study real elliptic Lefschetz fibratins; that is t say, elliptic Lefschetz fibratins whse ttal and base spaces have real structures which are cmpatible with the fiber structure. A real structure n an riented smth 4-manifld is defined as an rientatin preserving invlutin whse fied pint set (called the real part) has dimensin 2, if it is nt empty. Tw real structures are cnsidered the same if they differ by a cnjugatin by an rientatin preserving diffemrphism. It is wrth mentining here that nt every 4-manifld admits such an invlutin. Eamples f 4-maniflds which d nt admit real structures can be fund in [3]. Likewise, a real structure n a smth riented surface is defined as an rientatin reversing invlutin. Obviusly every surface admits a real structure. Besides, the classificatin (up t cnjugatin by an rientatin preserving diffemrphism) f real structures n surfaces is knwn. There are tw invariants that determine the cnjugacy class f a real structure: its type (separating/nn-separating) and the number f the cmpnents f its real part. A real structure is called separating if the real part divides the surface int tw disjint halves; therwise, it is called nn-separating. Thrughut the present wrk, we fcus n fibratins ver S 2, and the real structure cnsidered n S 2 is the ne induced frm the cmple cnjugatin, dented cnj, n CP 1. By definitin f real Lefschetz fibratins, fibers ver the real part S 1 f The authr was partially supprted by the Eurpean Cmmunity s Seventh Framewrk Prgramme ([FP7/2007-2013] [FP7/2007-2011]) under grant agreement n [258204], as well as by the French Agence natinale de la recherche grant ANR-08-BLAN-0291-02. 1

2 NERMİN SALEPCİ cnj inherit real structures frm the real structure f the ttal space. Such fibers are called real (fibers). A real elliptic Lefschetz fibratin has 3 types f real regular fibers. They are distinguished by the number f real cmpnents that can be 0, 1, 2 (nly the structure with 2 real cmpnents is separating n T 2 ). Fr the sake f simplicity, mst f the time we assume that the real part S 1 f (S 2, cnj ) is riented. Fibratins with such a feature are called directed. Mrever, we cnsider mainly fibratins which admit a real sectin (a sectin cmpatible with the real structures). But the cases f nn-directed fibratins as well as f fibratins withut a real sectin are als cvered. The nly essential cnditin impsed n fibratins is that all the critical values are real. Fibratins with this prperty are called ttally real. Our main interest is the tplgical classificatin f ttally real elliptic Lefschetz fibratins. Tw real Lefschetz fibratins will be cnsidered ismrphic if they differ by rientatin preserving equivariant diffemrphisms. Recall that the classificatin f elliptic Lefschetz fibratins ver S 2 has been knwn fr ver 30 years. It is due t B. Mishezn and R. Livné [4] that (nn-real, relatively minimal) elliptic Lefschetz fibratins ver S 2 are classified (up t ismrphism) by the number f the critical values. The latter is divisible by 12 and the class f elliptic Lefschetz fibratins with 12n critical values is dented by E(n), n N. Furthermre, E(1) is ismrphic t the fibratin CP 2 #9CP 2 CP 1, btained by blwing up a pencil f cubics in CP 2, and E(n) = E(n 1) F E(1) where F stands fr the fiber sum. In this nte, we give the real versin f this result fr ttally real elliptic Lefschetz fibratins. The classificatin is btained by means f certain cmbinatrial bjects that we call necklace diagrams. Necklace diagrams are cmbinatrial cunterparts f real Lefschetz chains intrduced in [6]. Main results f this wrk are presented as Therem 4.1 and Therem 7.1 in which we treat the cases f directed ttally real elliptic Lefschetz fibratins admitting a real sectin and, respectively, thse fibratins pssibly withut a real sectin. As immediate crllaries (Crllary 4.4, respectively, Crllary 7.2) f these therems, we btain that nn-directed ttally real elliptic Lefschetz fibratins admitting a sectin are classified by their necklace diagrams (defined up t symmetry and with the identity mndrmy), while thse fibratins pssibly withut a real sectin are classified by the symmetry classes f their refined necklace diagrams with the identity mndrmy. As a cnsequence f Crllary 4.4, we btain an eplicit list f ttally real E(1) and E(2) that admit a real sectin. We investigate the algebraic realizatins f these fibratins and shw that certain fibratins n the list are nt algebraically realizable. We als cnsider sme peratins n the set f necklace diagrams: mild/harsh sums, flip-flps and metamrphses. These peratins allw us t cnstruct new necklace diagrams frm the given nes. By means f these peratins, we cnstruct an eample f a real Lefschetz fibratin which cannt be written as a fiber sum f tw real fibratins (see Prpsitin 6.5). Acknwledgements. The material presented here is etracted frm my thesis. I am deeply indebted t my supervisrs Sergey Finashin and Viatcheslav Kharlamv fr their guidance and limitless supprt. I we many thanks t Andy Wand wh wrte the prgram t get the eplicit list f necklace diagrams, wh als edited my present and past articles as a native english speaker. I thank Ale Degtyarev fr his precius cmments n the first manuscript and fr prductive discussins.

CLASSIFICATION OF TOTALLY REAL ELLIPTIC LEFSCHETZ FIBRATIONS 3 The article has been finalized during my visit t the Mathematisches Frschungsinstitut Oberwlfach as an Oberwlfach Leibniz fellw. I wuld like t thank the institute fr prviding me equisite wrking cnditins. 2. Real lci f real elliptic Lefschetz fibratins and necklace diagrams Let π : X S 2 be a directed real elliptic Lefschetz fibratin. Cnsider the restrictin, π R : X R S 1, f π t the real part X R f X. By definitin, fibers f π R are the real parts f the real fibers f π. The base S 1 is riented (as π : X S 2 is directed), whereas the ttal space X R is either empty r a surface nt necessarily riented nr cnnected. By definitin f real Lefschetz fibratins, the map π R is an S 1 -valued Mrse functin n X R whse regular fibers can be S 1, S 1 S 1 r the empty set. On the ther hand, singular fibers are either a wedge f tw circles (this ccurs in the case when the critical pint is f inde 1) r a disjint unin f S 1 with an islated pint r just an islated pint (these cases ccur when the critical pint is f inde 0 r 2). As an immediate cnsequence, we nte that the real part X R is nt empty if π has a real critical value. (We cnsider fibratins with real critical values, s X R will never be empty thrughut this article.) Fr the sake f simplicity, we first fcus n fibratins which admit a real sectin. By a real sectin, we understand a sectin s : S 2 X cmmuting with the real structures. Eistence f a real sectin assures that fibers f π R are never empty, s there are nly tw pssible tplgical types fr regular fibers: S 1 r S 1 S 1. We nw intrduce a decratin n the base S 1 f π R : X R S 1 as fllws. First, label the critical values f π R by r accrding t the parity f indices f the crrespnding critical pints. Namely, if the crrespnding critical pint is f inde 1, label the critical value by, therwise by. (Nte that π R has critical values as lng as π has real critical values.) Secnd, cnsider a labeling n the set f regular intervals, S 1 \ {critical values}. Over each regular interval the tplgy f the fibers f π R is fied; mrever, it alternates as we pass thrugh a critical value. We label regular intervals ver which fibers have tw cmpnents by dubling the interval, see Figure 1. Intervals ver which the fibers are a cpy f S 1 remain unlabeled. The riented base S 1 tgether with such a decratin is called an riented uncated necklace diagram. Fig. 1. An eample f an uncated necklace diagram. We cnsider sme standard pieces ut f which all pssible uncated necklace diagrams can be built. It is cnvenient fr us t take pieces with tw cnsecutive critical values (in rder t avid the prblem f matching real structures). Let us chse a regular value n S 1 (fr sme later use we chse the pint n an unlabeled regular interval). With respect t this pint and the rientatin f S 1, we have 4

4 NERMİN SALEPCİ instances fr a pair f tw critical values. In rder t simplify the decratin, fr each instance we intrduce a new ntatin as shwn in Figure 2. Fig. 2. Prtins f the X R and the pieces f uncated necklace diagrams, and crrespnding necklace stnes. The riented S 1 decrated using elements f the set S = {,, >, <} is called an riented necklace diagram (an eample is shwn in Figure 3). We call the elements f the set S (necklace) stnes and the pieces f the circle between the stnes (necklace) chains. Tw riented necklace diagrams are cnsidered identical if they cntain the same types f stnes ging in the same cyclic rder. Fig. 3. A necklace diagram. It is bvius frm the cnstructin that riented necklace diagrams are invariants f directed real elliptic Lefschetz fibratins. Fr nn-directed real Lefschetz fibratins, we d nt have a preferable rientatin n the necklace diagram. Nndirected fibratins, hence, determine a pair f riented necklace diagrams related by a mirrr symmetry in which -type and -type stnes remain unchanged, while >-type and <-type stnes interchanged. 3. Mndrmy representatins f stnes Mndrmies f real Lefschetz fibratins alng certain lps (namely, lps n which the real structure acts as a reflectin) can be written as a cmpsitin f tw real structures, see [5]. In particular, the mndrmy arund a real singular fiber can be written as t a = c c where t a dentes the psitive Dehn twist alng the vanishing cycle a n a (nn-real) marked fiber F = T 2 and c, c are the real structures pulled back frm the nearby real fibers. (If the fibratin is directed, we

CLASSIFICATION OF TOTALLY REAL ELLIPTIC LEFSCHETZ FIBRATIONS 5 chse the marked fiber s that c and c are the real structures acquired frm the left and, respectively, right real fibers). Recall that mndrmy f elliptic Lefschetz fibratins are elements f the mapping class grup Map(T 2 ) (the grup f istpy classes f rientatin preserving diffemrphisms) f the trus. It is knwn that Map(T 2 ) is ismrphic t the grup f rientatin preserving autmrphisms f H 1 (T 2, Z) = Zu Zv. The latter is knwn t be ismrphic t SL(2, Z). Let T 2 = S 1 S 1 s that u = S 1 {0} and v = {0} S 1. Then, we cnsider tw presentatins f Map(T 2 ) SL(2, Z): SL(2, Z) = { ( 1 1 [t u ] = 0 1 = { = ( 0 1 1 0 ) ) and [t v ] = and y = ( 1 0 1 1 ( 0 1 1 1 ) : [t u][t v][t u] = [t v][t u][t v] ([t u][t v]) 6 = id ) : 2 = y 3, 4 = id} where t w dentes the Dehn twists alng w and [t w ] is the matri representatin f the induced ismrphism t w. One can switch frm the first presentatin t the secnd by setting = [t u ][t v ][t u ] = [t v ][t u ][t v ] and y = [t u ][t v ]. Each real structure, c : T 2 T 2, induces an ismrphism c n H 1 (T 2, Z) = Zu Zv that defines tw rank 1 subgrups H±(T c 2 ) = {γ H 1 (T 2, Z) : c γ = ±γ}. (If c has ne real cmpnent, then H 1 (T 2, Z)/ < H+, c H c >= Z/2Z, therwise, H 1 (T 2, Z) = H+ c H.) c Mrever, a real structure c n a real fiber F defines a pair f bases ±(u, v) s that u ± v generates H±. c This defines a cannical identificatin f H 1 (F, Z) t H 1 (T 2, Z). T each decratin arund a critical value q, we assign the transitin matri P q frm a basis f H+ c H c H 1 (T 2, Z) t a basis f H+ c H c H 1 (T 2, Z) where c, c are left and, respectively, right real structures n the real fibers near F q. Lemma 3.1. Fr each decratin arund a critical value, we get the fllwing matrices defined up t sign. ( ) ( ) 1 0 2 0 P < = 1 2, P 1 2 > =, ( ) ( 1 1) 2 1 1 1 P < = 1 2, P 0 1 > =. 0 2 Remark 3.2. By definitin f real Lefschetz fibratins, arund each critical pint and its image, ne can chse equivariant lcal (clsed) charts n which the fibratin is equivariantly ismrphic t either f ξ ± : (E ±, cnj ) (D ɛ, cnj ), where E ± = {(z 1, z 2 ) C 2 : z 1 ɛ, z 2 1 ± z2 2 ɛ 2 } and D ɛ = {t C : t ɛ 2 }, 0 < ɛ < 1 with ξ ± (z 1, z 2 ) = z1 2 ± z2. 2 Frm the mdels ξ ± ne can see immediately that in the case f ξ + (this is the mdel fr the critical pint f inde 0, 2) there are tw types f real regular fibers distinguished by their real parts. In bth types, ne can chse an invariant representative f the vanishing cycle, and the actin f the real structure n the invariant representative can be either the antipdal map r the identity. On the ther hand, in the case f ξ (this is the case f critical pints f inde 1) the real structure acts n the invariant representative f the vanishing cycle as a reflectin. Cnsequently, in the frmer situatin (which crrespnds t the decratin ) the class f the vanishing cycle gives an element in H+, c }

6 NERMİN SALEPCİ while in the latter case (the case crrespnding t the decratin ) the class f the vanishing cycle gives an element f H c. (A detailed discussin abut the lcal mdels can be fund in [6, Sectin 3] where the abve claims are depicted in Figure 2.) Prf: Eplicit calculatins are made fr P < and P > ; the ther cases are similar. Let q be a critical value. We cnsider a sufficiently small ɛ-neighbrhd D q S 2 f q such that cnj (D q ) = D q and D q {critical values} = q and chse a nn-real pint m n D q. By means f the shrtest paths n D q frm m t the left q ɛ and right q + ɛ real pints f D q, we pull the real structures n F q±ɛ back t F m. Fi an auiliary identificatin T 2 with the fiber F m and let c : T 2 T 2 (respectively, c : T 2 T 2 ) be the real structure btained by pulling back the real structure n F q ɛ (respectively, F q+ɛ ) s that the (lcal) mndrmy is the cmpsitin c c. The case f ( <) : as discussed in Remark 3.2, the critical value f the type prvides a generatr f H c H 1 (T 2, Z). Let b dente the crrespnding vanishing cycle and β the hmlgy class f b. Set < β >= H c and chse a generatr α fr H+ c such that α β = 2 (since c has ne real cmpnent). Frm the lcal mndrmy decmpsitin, we get c = t b c ; therefre, c (α) = t b (c (α)) = α 2β c (β) = t b (c (β)) = β. Obviusly, the class α + c (α) = 2α 2β H+ c, while β c (β) H c. We set α = α β 2 and β = β s that ( we have ) B = (α, β ) with < α >= H+ c and < β >= H c. Then, P < = 1 1 0 2 1 2. The case f (> ): we repeat the same with the difference that the vanishing cycle a gives a generatr α fr H+, c and we chse a generatr β fr H c s that α β = 1 (the left real structure c is separating). Frm the lcal mndrmy decmpsitin, we get c (α) = α c (β) = t a (c (β)) = β α. Therefre, α = α and ( β = ) β c (β) = 2β +α is a basis f H+ c and, respectively, f H c 1 1 ; thus, P > = 0 2. (As α β > 0 if and nly if ( α) ( β) > 0, the matrices are defined up t sign.) Remark 3.3. Intuitively, the mndrmy assigned t a decratin arund a critical value can be interpreted as the half f the mndrmy arund the real critical value. Namely, it is the mndrmy assigned t the path shwn in Figure 4. It is straightfrward t check that P < = MP> M 1 and P < = MP> M 1 where ( M = M 1 = 1 0 0 1 ). Mrever, ne bserves that P > P < = (P > )(MP 1 > M) = [t v ] B where [t v ] B dentes the matri f t v with respect t the base B = (α, β). Similarly, ne has P > P < = [t u ] B as well as P < P > = [t v ] B and P < P > = [t u ] B where B = (α, β ).

CLASSIFICATION OF TOTALLY REAL ELLIPTIC LEFSCHETZ FIBRATIONS 7 MP 1 > M P > Fig. 4. An eample f a decmpsitin f the mndrmy assciated t the decratin f a real critical value. Fllwing Lemma 3.1, t each necklace stne, we assign the fllwing prducts (defined up t sign). ( ) 1 0 P = P < P > =, 2 1 P = P < P > = P > = P < P > = 1 2 ( 1 2 0 1 ), ( 1 1 1 3 P < = P < P > ( ) 3 1 = 1 2. 1 1 Because we chse the marked pint n an unlabeled interval, the matrices have cefficients in 1 2Z. If the marked pint was chsen n a labeled interval, the matrices wuld be elements f PSL(2, Z). (The reasn why we prefer a pint n an unlabeled interval is t get a nice relatin between necklace stnes and the real part, see Remark 4.6.) The subgrup generated by {P, P, P >, P < } is cnjugate t PSL(2, Z) = {, y : 2 = y 3 = id}. T be able t wrk with PSL(2, Z), we cnsider the fllwing lemma. ( ) Lemma 3.4. Let R = 1 1 1 2 1 1 and P = R 1 PR. Then, fr each necklace stne, we btain the fllwing factrizatin. P = yy P = yy P > = y 2 P < = y 2 Prf: It fllws frm the bservatin that P > = [t u ], P < = [t v ], while P = [t u ][t v ][t u ] 1, P = [t u ] 1 [t v ][t u ]. The matrices P, P, P >, P < are called mndrmies f stnes. The prduct (with respect t a chsen marked pint and t the rientatin) f mndrmies f necklace stnes is called the mndrmy relative t the marked pint f the riented necklace diagram. The mndrmy f an riented necklace diagram is, thus, defined as the cnjugacy class f its mndrmy relative t a marked pint. Prpsitin 3.5. Let π : X S 2 be a directed ttally real elliptic Lefschetz fibratin admitting a real sectin. Then, the mndrmy f the riented necklace diagram assciated with π is the identity in PSL(2, Z). ),

8 NERMİN SALEPCİ Prf: Let {s 1,..., s k } be the rdered set f stnes f an riented necklace diagram (rder is given with respect t the rientatin and a marked pint) and P = P 1 P 2... P k be the mndrmy f the necklace diagram relative t the marked pint where each P i is the mndrmy f the stne s i. By reinterpreting Remark 3.3 in terms f the matrices P = R 1 PR assciated t necklace stnes, the mndrmy alng a curve surrunding all real critical values can be written ( as ) the prduct P 1 P 2... P k T P 1 k... P 1 2 P 1 1 T, where T = T 1 = R 1 0 1 MR = 1 0 (see Figure 5). T P 1 1 T T P 1T k P 1 P k Fig. 5. A decmpsitin f the ttal mndrmy. 0 1 1 0 0 1 1 0 If there is n nn-real critical value, the mndrmy alng the curve we cnsider is( identical t ( the ttal ) mndrmy f the fibratin which is the identity. Thus, P )P 1 = id (since everything cnjugate t the identity is the identity). The equality assures that P is a diagnal matri, hence it is the identity in PSL(2, Z). Remark 3.6. An imprtant bservatin is that P = P and P < = P >. Hence, if a necklace diagram has the identity mndrmy, then the necklace diagram btained frm the riginal by replacing each -type stne with -type stne, and each >-type stne with <-type stne and vice versa, has als the identity mndrmy. Necklace diagrams btained in this manner are called dual necklace diagrams. Remark 3.7. Althugh fr the mment, we fcus n fibratins admitting a real sectin (hence, we cnsider nly the case f real fibers with nn-empty real part), it is wrth mentining the case f real fibers with empty real part. It is well knwn that the real structures with n real cmpnent and with tw real cmpnents are istpic t each ther as rientatin reversing diffemrphisms, althugh they are tw nn-istpic real structures. As a result, they induce the same ismrphism n the hmlgy grup; hence, mndrmy calculatins remain the same if the real structure with tw real cmpnents is replaced by a real structure with n real cmpnent. 4. The crrespndence therems and cnsequences Therem 4.1. There eists a ne-t-ne crrespndence between the set f riented necklace diagrams with 6n stnes and with the identity mndrmy and the set f ismrphism classes f directed ttally real elliptic Lefschetz fibratins E(n), n N, that admit a real sectin.

CLASSIFICATION OF TOTALLY REAL ELLIPTIC LEFSCHETZ FIBRATIONS 9 Prf: The discussins in the previus sectins and Prpsitin 3.5 result in an injectin frm the set f classes f directed ttally real elliptic Lefschetz fibratins admitting a real sectin t the set f riented necklace diagrams with identity mndrmy. We want t shw that this injectin is als well-defined and surjective. T this end, first recall that the decratin n critical values determines the istpy class f the vanishing cycle. (Thus, it determines the (nn-real) ismrphism class f a small neighbrhd f the singular fiber.) Whereas, the decratin n regular intervals determines the type f the real structure n the fibers ver the intervals. The decratin arund a critical value, hence, dictates a cnjugacy class f the pair (c, a) where c is the real structure n the chsen nearby real fiber and a is the vanishing cycle such that c(a) = a. In [6], we call such a pair a real cde and shw that the ismrphism class f an equivariant neighbrhd f a real singular fiber is determined by the cnjugacy class f a real cde (see [6, Sectin 3]). An riented necklace diagram, therefre, defines an rdered sequence f the cnjugacy classes f real cdes up t cyclic rder. Such a sequence is called a real Lefschetz chain in [6], the result, thus, fllws frm [6, Prpsitin 8.5]. Remark 4.2. It is easy t be cnvinced that the real cde (c, a) determines the ismrphism type f an equivariant neighbrhd f a singular fiber. In the case f elliptic fibratins admitting a real sectin, it is als easy t see that rdered sequence f real cdes cnsidered up t cyclic rder is enugh t recver the ismrphism type f the crrespnding ttally real fibratin. This is because the ttally real fibratins can be cnstructed by successive fiber sums (cnnected sums preserving the fiber structure) f equivariant neighbrhds. The crucial pint is that the fiber sum peratin is well-defined due t the cntractibility f the cmpnents f the space diffemrphisms the trus with a marked pint. (A detailed discussin can be fund in [6] where we cver the cases f higher genus fibratins as well as genus 1 fibratins withut a sectin.) Remark 4.3. As mentined in [6, Remark 3.6], in the case when the regular fibers are tri, there are 6 cnjugacy classes f real cdes. Whereas, arund each critical value we have 4 different decratins { <, >, <, > }. These decratins are eactly 4 f the 6 pssible real cdes n T 2. The ther tw cases appear in the case f nn-eistence f a real sectin that we discuss in the last sectin. Crllary 4.4. There eists a bijectin between the set f symmetry classes f nn-riented necklace diagrams with 6n stnes and with the identity mndrmy and the set f ismrphism classes f nn-directed ttally real E(n), n N which admit a real sectin. Each necklace diagram defines a decmpsitin f the identity in PSL(2, Z) int a prduct f 6n elements that are chsen frm the set f mndrmies f necklace stnes. There is a simple algrithm t find all necklace diagrams assciated with E(n). Applying the algrithm, we btain the cmplete list f necklace diagrams f E(1). Later Andy Wand wrte a cmputer prgram which wrks fr n = 1, 2. The fllwing therem cncerns n = 1. Therem 4.5. There eist precisely 25 ismrphism classes f nn-directed ttally real E(1) admitting a real sectin. These classes are characterized by the nnriented necklace diagrams presented in Figure 6.

10 NERMİN SALEPCİ Fig. 6. List f necklace diagrams f ttally real E(1) admitting a real sectin. Prf: By Crllary 4.4, it is enugh t find the list f symmetry classes f 6- stne necklace diagrams whse mndrmy is the identity. Each necklace diagram defines a decmpsitin f the identity in PSL(2, Z) = {, y : 2 = y 3 = id} int a prduct f elements yy, yy, y 2, y 2. Let S = yy, C = yy, L = y 2, R = y 2. Then, first cnsider all the wrds f length 6 f letters S, C, L, R such that the prduct is the identity; then, (t get list f all riented necklace diagrams which have the identity mndrmy) qutient ut the wrds which are equivalent t each ther up t cyclic rdering; finally, qutient ut the symmetry classes: in terms f wrds cmpsed f the letters S, C, L, R, symmetry classes can be interpreted as fllws: tw wrds are cnsidered t be equivalent if up t cyclic rdering, ne is the reversed f the ther with each L is replaced by R, and vice versa. Fr eample, CLLSRR LLSRRC. It is wrth mentining here that there are 8421 necklace diagrams f 12 stnes with the identity mndrmy. Belw, in Prpsitin 4.9, we give an eplicit list f necklace diagrams crrespnding t certain classes. Later, we als eplre sme interesting eamples. Remark 4.6. The tplgical invariants f X R can be read frm the necklace diagram f π : X S 2. Namely, β 0 (X R ) = β 2 (X R ) = + 1 and β 1 (X R ) = 2( + 1) where β i dentes the i th Betti number (with Z 2 cefficients) and, dente the number f -type and, respectively, -type stnes f the necklace diagram assciated with π R. Cnsequently, we have the Euler characteristic χ(x R ) = 2( ), and the ttal Betti number β (X R ) = 2( + ) + 4. Recall that in general we have β (X R ) β (X) (knwn as Smith inequality). It is knwn that β (E(n)) = 12n ([2]), s the Smith inequality implies that β (E(n) R ) 12n.

CLASSIFICATION OF TOTALLY REAL ELLIPTIC LEFSCHETZ FIBRATIONS 11 Prpsitin 4.7. Each 6n-stne necklace diagram with the identity mndrmy cntains at least tw arrw-type stnes. Prf: As mentined abve β (E(n) R ) 12n. Meanwhile, we have β (E(n) R ) = 2( + ) + 4. Thus, + 6n 2 n a 6n-stne necklace diagram, s there are at least tw arrw-type stnes. Definitin 4.8. A real structure c X n X is called maimal if β (X R ) = β (X). A necklace diagram is maimal if + = 6n 2. There are 4 maimal E(1) whse necklace diagrams that are depicted n the tp line f Figure 6. Fr n = 2 we have: Prpsitin 4.9. There are 10 ismrphism classes f maimal nn-directed ttally real E(2) admitting a real sectin. Crrespnding necklace diagrams are given in Figure 7. Fig. 7. List f necklace diagrams f maimal ttally real E(2) admitting a real sectin. 5. Applicatins f necklace diagrams In this sectin, we study the algebraic realizatin f the ttally real elliptic Lefschetz fibratins admitting a real sectin. The crucial bservatin is that any algebraic elliptic Lefschetz fibratin E(n) admitting a real sectin can be seen as the duble branched cvering f a Hirzebruch surface f degree 2n, branched at the eceptinal sectin and a trignal curve disjint frm the sectin. In [7], S. Yu. Orevkv intrduced a real versin f Grthendieck s dessins d enfants fr

12 NERMİN SALEPCİ the trignal curves, which are disjint frm the eceptinal sectin, n Hirzebruch surfaces. We apply his result by cnverting language f real dessin d enfants t the language f necklace diagrams. 5.1. Trignal curves n Hirzebruch surfaces. The Hirzebruch surface, H(k), f degree k is a cmple surface equipped with a prjectin, π k : H(k) CP 1, which defines a CP 1 -bundle ver CP 1 with a unique eceptinal sectin s such that s s = k. In particular, H(0) = CP 1 CP 1 and H(1) is CP 2 blwn up at ne pint. Each Hirzebruch surface H(k) can be btained frm H(0) by a sequence f blwups fllwed by blw-dwns at a certain set f pints. If these pints are chsen t be real, then the resulting Hirzebruch surface has a real structure inherited frm the real structure cnj cnj n H(0). This will be the real structure f ur cnsideratin. With respect t this real structure, the real part f H(k) is a trus if k is even; therwise it is a Klein bttle. In this nte, we nly cnsider nnsingular curves, s by a trignal curve n a Hirzebruch surface H(k) we understand a smth algebraic curve C H(k) such that the restrictin f the bundle prjectin, π k : H(k) CP 1, t C is f degree 3. A trignal curve n H(k) is called real if it is invariant under the real structure f H(k). 5.2. Real dessins d enfants f trignal curves. We chse affine (cmple) crdinates (, y) fr H(k) such that the equatin = cnst crrespnds t fibers f π k and y = is the eceptinal sectin s. Then, with respect t such affine crdinates any (algebraic) trignal curve can be given by a plynmial f the frm y 3 + u()y + v() where u and v are real ne variable plynmials such that deg u = 2k and deg v = 3k. The discriminant f y 3 + u()y + v() = 0 with respect t y is 4u 3 27v 2. Let D = 4u 3 + 27v 2. The fractin j = 4u3 D is the j-invariant f a trignal curve C H(k). The j-invariant defines a real ratinal functin j : CP 1 CP 1 whse ples are the rts f D, zers are the rts f u (taken with multiplicity 3), and the slutins f j = 1 are the rts f v (taken with the multiplicity 2). Let us clr RP 1 as in Figure 8. Then, the inverse image j 1 (RP 1 ) turns naturally int an riented clred graph n CP 1. Since j is real, the graph is symmetric with respect t the cmple cnjugatin n CP 1. Arund the vertices the graph lks as shwn in Figure 9. (Detailed discussin n j-invariant f trignal curves can be fund in [1], [7].) 0 1 Fig. 8. Clring f RP 1. The fllwing therem gives the cnditins which are sufficient fr real algebraic realizability f a graph and the eistence f respective plynmials u, v, D. Therem 5.1. [7] Let Γ S 2 be an embedded riented graph where each f its edges is ne f the three kinds:,, and sme f its vertices are clred by the elements f the set {,, }, while thers remain unclred. Let Γ satisfy

CLASSIFICATION OF TOTALLY REAL ELLIPTIC LEFSCHETZ FIBRATIONS 13 Fig. 9. The graph arund the inverse images f zers f D, v, u, respectively. the fllwing cnditins: the graph Γ is symmetric with respect t an equatr f S 2, which is included int Γ; the valency f each verte is divisible by 6, and the incident edges are clred alternatively by incming, and utging ; the valency f each verte is divisible by 4, and the incident edges are clred alternatively by incming, and utging ; the valency f each verte is 2, and the incident edges are clred alternatively by incming, and utging ; the valency f each nn-clred verte is even, and the incident edges are f the same clr; each cnnected cmpnent f S 2 \ Γ is hmemrphic t an pen disk whse bundary is clred as a cvering f RP 1 (clred and riented as in Figure 8) and the rientatins f the bundaries f neighbring disks are ppsite. Then, there eists a real ratinal functin j = 4u3 D whse graph is Γ. (And thus, there eist a nn-singular real algebraic trignal curve assciated t the j-invariant.) Definitin 5.2. A graph n S 2 satisfying the si cnditins f the abve therem is called a real dessin d enfant. Remark 5.3. Let us accentuate the fact that there is n relatin between the decratins, f the critical values that we use t intrduce the necklace diagrams and the clring f the vertices f the real dessin d enfants cnsidered in this sectin. 5.3. Relatin t the real scheme. The real scheme f a trignal curve impses strng restrictins n the arrangement f the real rts f u, v and D. Fr eample, the zers f D crrespnd t the pints where the trignal curve is tangent t the fibers f π k : H(k) CP 1. A typical crrespndence fr certain mdel pieces f the curve is shwn in Figure 10. Because the graph is symmetric with respect t the equatr, we cnsider the part f the graph lying n ne f the semi-disks. As we mentin in the previus sectin, necklace diagrams encde the tplgy f the real part (ecept rientability) f E(n) which admit a real sectin. Indeed, the real part f ttally real E(n), admitting a real sectin, cnsists f spherical

14 NERMİN SALEPCİ Fig. 10. Segments f the curve crrespnding t fragments f the minimal graphs. cmpnents (the number f which is ) and a higher genus cmpnent which is an rientable surface f genus + 1 if n is even; a nn-rientable surface with 2( + 1) crss-caps, therwise. Definitin 5.4. A segment f a necklace chain is called essential if it is ne f the segments shwn in Figure 11. Fig. 11. Essential segments. 5.4. Applicatins. Prpsitin 5.5. If a real elliptic Lefschetz fibratin, E(n), admitting a real sectin is algebraic then the crrespnding necklace diagram has the fllwing prperties: there are nt mre than 2n essential segments, the sum f the number f essential segments and the number f arrw-type stnes cannt be greater than 6n. Prf: Fr a trignal curve n H(2n) defined by y 3 +u()y+v(), deg u = 2 2n and deg v = 3 2n. Thus, the real dessin d enfant can have at mst 4n vertices clred by and at mst 6n vertices clred by. The result fllws frm the bservatin that each essential interval crrespnds t a graph fragment which cntains at least tw type vertices and at least ne type verte, while each arrwtype stne crrespnds t a fragment having at least ne type verte, see [1]. By listing the necklace diagrams vilating the cnditins f Prpsitin 5.5, we get: Crllary 5.6. The ttally real elliptic Lefschetz fibratins crrespnding t the necklace diagrams depicted in Figure 12 are nt realized algebraically.

CLASSIFICATION OF TOTALLY REAL ELLIPTIC LEFSCHETZ FIBRATIONS 15 Fig. 12. The last diagram vilates the secnd cnditin, while all the thers vilate the first ne. Lemma 5.7. If a ttally real elliptic Lefschetz fibratin admitting a real sectin is algebraic then the ttally real elliptic Lefschetz fibratin whse necklace diagram is dual t the necklace diagram f the frmer is als algebraic. Prf: The crucial bservatin is that althugh the real parts f fibratins assciated with dual necklace diagrams are tplgically different, trignal curves appearing as the branching set f cverings E(n) H(2n) are the same. Duality f necklace diagrams crrespnds, indeed, t the tw different liftings f the real structure f H(2n) t E(n), see Figure 13. Thus, the result fllws. - + - + + + + - - - Fig. 13. Fr each trignal curve n H(2n), there are tw real structures f E(n). Therem 5.8. All ttally real E(1) admitting a real sectin are algebraic ecept thse fibratins whse necklace diagram is ne f the diagrams listed in Figure 12. Prf: By Therem 5.1 it is enugh t cnstruct real dessins d enfants crrespnding t necklace diagrams which are nt prhibited by Prpsitin 5.5. Fllwing Lemma 5.7, we nly need t cnsider necklace diagrams with. Figures 14-15 shw the required real dessins d enfants. (In the figures, the real part is depicted arund necklace diagrams. The dtted inner circle stands fr the lift f the eceptinal sectin. Because f the symmetry, we nly draw a half f the graph.) 6. Necklace calculus and further applicatins In this sectin, we cnsider certain peratins n the set f necklace diagrams. These peratins allw us t cnstruct new necklace diagrams frm the given nes. 6.1. Necklace sums. A necklace sum is basically the cnnected sum f the underlying riented circle and it refers t the fiber sum f the crrespnding real Lefschetz fibratins. We cnsider tw types f necklace sums which we call mild sum and harsh sum. T perfrm a mild sum, we cut each necklace diagram at a pint n the chain then reglue the diagrams crsswise respecting the rientatin. The harsh sum, n the ther hand, is btained by cutting necklace diagrams at

16 NERMİN SALEPCİ Fig. 14. Real dessin d enfants. a stne and regluing them accrding t the table shwn in Figure 16. It fllws frm their definitin that bth the mild sum and the harsh sum d nt change the

CLASSIFICATION OF TOTALLY REAL ELLIPTIC LEFSCHETZ FIBRATIONS 17 Fig. 15. Real dessin d enfants. mndrmy f the diagram. Evidently, the Euler characteristic f the real part X R is additive with respect t the mild sum; hwever, it is nt always additive with respect t the harsh sum. Let us nte als that we can als cnsider necklace sum f nn-riented necklace diagrams by fiing auiliary rientatins n diagrams. Fig. 16. Table f the harsh sum Eamples f mild and harsh sums are given in Figure 17. Remark 6.1. There are tw types f necklace chain segments (essential, nnessential). It is nt hard t see that the mild sum preserves algebraic realizability if the pints where the sum is taken are chsen n the same type f chain segments and if the segments after the sum remain f the same type; r if they are chsen n different types f chain segments. In ther wrds, algebraic realizability is preserved if we make a mild sum at tw essential (respectively, nn-essential) intervals, in a way that after gluing we btain again tw essential (respectively, nn-essential) intervals; r if we make sum at an essential and a nn-essential intervals. As fr the harsh sum, we nte that it preserves algebraic realizability if the number f neither -type nr -type stne decreases after the sum.

18 NERMİN SALEPCİ Fig. 17. Eamples f the mild and the harsh sums. Prpsitin 6.2. Fr each n, maimal necklace diagrams eist and each ttally real elliptic E(n) represented by a maimal necklace diagram is algebraic. Prf: It is easy t see that the harsh sum f tw maimal necklace diagrams where the sum is perfrmed at arrw-type stnes f the ppsite directins is maimal. Mrever, by the remark abve the harsh sum perfrmed at tw arrw-type stnes f ppsite directins preserves algebraic realizability. Nte als that as Therem 5.8 asserted all maimal necklace diagrams f 6 stnes are algebraic. T finish the prf, we shw that all maimal necklace diagrams f n stnes are btained as harsh sums f maimal necklace diagrams f 6 stnes. We will prve the claim by inductin n n. The first step is t check the claim fr n = 2. As we have the eplicit list f maimal diagrams f 12 stnes, we see immediately that any maimal necklace diagram f 12 stnes can be btained as the harsh sum f maimal necklace diagrams f 6 stnes. Nw, let us assume that fr n = k the claim is true. T prve the claim fr n = k + 1, nte that the mndrmies f -type stnes and -type stnes d nt have any cancelatin. The fact that the mndrmy f the necklace diagram is the identity and that there is n cancellatin between the mndrmy f -type stnes and the mndrmy f -type stnes impse certain cnditins n the pssible arrangements f stnes arund an arrw-type stne n a maimal necklace diagram. By checking the pssibilities f the neighbrhd f an arrw-type stne, we see that required cancelatins appear nly in the cases where the arrangements cme frm the maimal necklace diagrams f 6 stnes, s the claim fllws frm the inductive step. 6.2. Flip-flps and metamrphses. Let N k dente the set f (riented) necklace diagrams with k stnes and with the identity mndrmy, and let N (i,j) k be the subset cnsisting f diagrams with (, ) = (i, j). We define tw kinds f peratins n N k : flip-flp and metamrphsis. The flip-flp preserves (, ) and it cincides with canceling and creating handles n the real part E(n) R. On the ther hand, the metamrphsis decreases r by ne and it appertains t a ndal defrmatin f E(n) R.

CLASSIFICATION OF TOTALLY REAL ELLIPTIC LEFSCHETZ FIBRATIONS 19 We define flip-flp as the peratin which swaps the segments shwn belw. Because the segments have the same mndrmy, the ttal mndrmy des nt change. Eamples f flip-flp are shwn in Figure 18. flip-flp : N k (i,j) N k (i,j) Fig. 18. Eamples f flip-flps. As and remain unchanged after a flip-flp, the Euler characteristic and the ttal Betti number f the crrespnding E(n) R d nt change. Thus, tplgical type f E(n) R is nt affected by the flip-flp. In Figure 19, we interpret the effect f a flip-flp n E(n) R. Canceling the handles Recreating the handles Canceling the handles Recreating the handles Fig. 19. The effect f flip-flp n the real part. We cnsider tw types f metamrphses, m 1, m 2, f necklace diagrams. They mdify the abve segments as fllws. Eamples f metamrphses are depicted in Figure 20. m 1 : N k (i,j) N k (i 1,j) m 2 : N k (i,j) N k (i,j 1) Since r are mdified by metamrphses, the tplgical type f the crrespnding E(n) R changes. Recall that fr fibratins which admit a real sectin

20 NERMİN SALEPCİ m 1 m 2 m 1 m 2 Fig. 20. Eamples f metamrphses. each -type stne crrespnds t a spherical cmpnent while each -type stne drives a handle. Indeed, a sphere cmpnent r a genus disappears after a metamrphsis (r appears after a inverse metamrphsis). In Figure 21, we depict the effects f metamrphses n E(n) R. Fig. 21. The effect f metamrphses, m 1, m 2, t the real part. 6.3. Applicatins. Belw, in Figure 22 and Figure 23, we present tw graphs (fr E(1) and E(2), respectively) whse vertices crrespnd t necklace diagrams with fied (, ), and edges t necklace metamrphses m 1, m 2. As we mentin befre the real part f ttally real E(n), admitting a real sectin, cnsists f spherical cmpnents (the number f which is ) and a higher genus cmpnent which is an rientable surface f genus + 1 if n is even; a nn-rientable surface with 2( + 1) crss-caps, therwise. Each pair (, ) and the parity f n, thus, defines the tplgical type f E(n) R. (0,4) (4,0) m 2 m 1 (0,3) m 2 (1,1) m 1 (3,0) (0,2) (2,0) (0,1) (1,0) (0,0) Fig. 22. Metamrphsis graph f E(1) R. Eamining the list f necklace diagrams f 6 stnes we btain the fllwing:

CLASSIFICATION OF TOTALLY REAL ELLIPTIC LEFSCHETZ FIBRATIONS 21 (1,9) (5,5) (9,1) m 2 m 1 m 2 m 1 m 2 m 1 m 2 m 1 (0,9) (2,6) (5,4) (6,2) (9,0) (1,8) (4,5) (8,1) (0,8) (8,0) (1,7) (3,5) (4,4) (5,3) (7,1) (0,7) (7,0) (1,6) (2,5) (3,4) (4,3) (5,2) (6,1) (0,6) (1,5) (2,4) (3,3) (4,2) (5,1) (0,5) (1,4) (2,3) (3,2) (4,1) (0,4) (0,3) (1,3) (2,2) (3,1) (1,2) (2,1) (0,2) (1,1) (2,0) (0,1) (1,0) (0,0) (4,0) (3,0) (6,0) (5,0) Fig. 23. Metamrphsis graph f E(2) R. Prpsitin 6.3. All 6-stne necklace diagrams listed in Figure 6 can be btained frm the maimal nes by a sequences f metamrphses, inverse metamrphses r flip-flps. Mrever, the list f necklace diagrams which are btained frm maimal diagrams nly by a sequence f metamrphses and eventually an inverse metamrphsis, cincide with the list f diagrams f algebraic fibratins. Prpsitin 6.4. There eist 12-stne necklace diagrams (with the identity mndrmy) that cannt be btained frm the maimal necklace diagrams by necklace peratins. Prf: Eamples, shwn in Figure 24, are fund by investigating the list f 12- stne necklace diagrams basically with (, ) = (9, 1), (9, 0), (8, 1), (8, 0). (By duality it is enugh t cnsider the case f.) The fibratins assciated with the diagrams shwn in Figure 24 are algebraic. Fig. 24. Diagrams that cannt be btained by necklace peratins. As a crllary f the abve prpsitin we claim that all ttally real algebraic E(1) admitting a real sectin can be btained frm the maimal nes by a sequences f ndal defrmatins, while there are ttally real algebraic E(2) which cannt be btained in this way. Prpsitin 6.5. There eist 12-stne necklace diagrams (with the identity mndrmy) which are nt a necklace sum f tw 6-stne necklace diagrams listed in Figure 6.

22 NERMİN SALEPCİ Prf: In Figure 25, we cnstruct a nn-decmpasable eample applying a mild sum fllwed by a flip-flp. Let us als nte that such eamples can be prduced the same way fr any n > 1. Fig. 25. An eample f cnstructin f a nn-decmpsable necklace diagram. Using the list f necklace diagrams f 6 stnes (see Figure 6) and by analyzing pssible divisins f the pair (, ), we see that the 12-stne necklace diagram shwn in Figure 25 cannt be divided int tw 6-stne necklace diagrams with the identity mndrmy. Crllary 6.6. There eist ttally real E(2) which cannt be written as a fiber sum f tw ttally real E(1). 7. Ttally real elliptic Lefschetz fibratins withut a real sectin and refined necklace diagrams In this sectin, we eplre the case f ttally real elliptic Lefschetz fibratins π : X S 2 which d nt admit a real sectin and intrduce refined necklace diagrams assciated with them. A refinement f a necklace diagram is btained by replacing each -type stne with ne f the fllwing refined stnes,,,,. If the refined necklace diagram is identical t the underlying necklace diagram then the crrespnding real Lefschetz fibratin admits a real sectin. Eamples f refinements f a necklace diagram are shwn in Figure 26. Fig. 26. Refinement f a necklace diagram. Frm Remark 3.2, it is clear that if a real structure n a fiber f π has n real cmpnent, then the nearby critical values can nly be f type. In ther wrds, eistence r lack f a real sectin influences nly -type necklace stnes. Bth f the refined stnes f type, crrespnd t the case where the real structure n the real fibers ver the interval between the tw critical values has 2

CLASSIFICATION OF TOTALLY REAL ELLIPTIC LEFSCHETZ FIBRATIONS 23 real cmpnents. Already the real part X R f X distinguishes the cases f and, see Figure 27. (In Figures 27 and 28 belw, the dtted part depicts the traces f the curves n which the inherited real structure acts as the antipdal map.) As ntatin suggested has t d with the case where there is a real sectin, and hence, nly -type refined stnes refer t a spherical cmpnent f X R. Fig. 27. Real part and assciated refined stnes. Recall that if the cnditin that the fibratin admits a real sectin is discarded, then the fibers f π R may als be empty (which happens when the real structure n the real fibers f π has n real cmpnent). We intrduce the refined stnes and which crrespnd t the case where the real structure n the real fibers f π has n real cmpnent. As depicted in Figure 28, the real part f X R des nt distinguish the tw situatins assciated with,. The difference between and (as well as between and ) can indeed be cnceived by cmparing the equivariant istpy classes f the tw vanishing cycles crrespnding t the tw critical values f the necklace stne. In the case f (respectively, ) the equivariant istpy classes f the tw vanishing cycles are the same, while in the case f (respectively, ) the tw vanishing cycles are f different equivariant classes. (A mre detailed discussin can be fund in [6, Sectin 8].) Fig. 28. Real part and assciated refined stnes. As mentined in Remark 3.7, there is n difference between real structures with 2 real cmpnents and real structures with n cmpnent n the hmlgical level. As a cnsequence, the calculatin f the mndrmy is nt affected by the refinement. Thus, we have the fllwing therem. Therem 7.1. There is a ne-t-ne crrespndence between the set f riented refined 6n-stne necklace diagrams whse mndrmy is the identity and the set f ismrphism classes f directed ttally real E(n), n N. Prf: The prf is analgus t the prf f Therem 4.1. It is bvius frm its cnstructin that the refinements f necklace diagram is eactly the decratin

24 NERMİN SALEPCİ f the real Lefschetz chains, see [6, Figure 10]. Thus, we relate refined necklace diagrams with the decrated real Lefschetz fibratins, cmplete invariants f ttally real elliptic Lefschetz fibratins, presented in [6, Sectin 8]. The result, thus, fllws frm Therem 8.1 and Prpsitin 8.2 f [6]. Crllary 7.2. There is a ne-t-ne crrespndence between the set f symmetry classes f nn-riented refined 6n-stne necklace diagrams whse mndrmy is the identity and the set f ismrphism classes f ttally real E(n), n N. Belw, fr each fied (, ), we give the number f pssible refinements f 6-stne necklace diagrams. (, ) = (1, 1) there are 12 refined necklace diagrams; (, ) = (1, 0) there are 8 refined necklace diagrams; (, ) = (2, 0) there are 46 refined necklace diagrams; (, ) = (3, 0) there are 84 refined necklace diagrams; (, ) = (4, 0) there are 251 refined necklace diagrams. References [1] Degtrayev, A.; Itenberg I.; Kharlamv V. On Defrmatin types f real elliptic surfaces. American Jurnal f Mathematics, Vlume 130, Number 6, 2008, 1561-1627. [2] Gmpf, R.E., A.I. Stipsicz 4-manifls and Kirby calculus. Amer. Math. Sc. Grad. Stud. in Math. 20 Rhde Island, (1999). [3] Kulikv, S.; Kharlamv, V. On real structures n rigid surfaces. (Russian) Izv. Rss. Akad. Nauk Ser. Mat. 66 (2002), n. 1, 133 152; translatin in Izv. Math. 66 (2002), n. 1, 133150. [4] Mishezn, B. Cmple surfaces and cnnected sums f cmple prjective planes. Lecture ntes in Math. 603 Springer Verlag(1977). [5] Salepci, N. Real classes in the mapping class grup f T 2, Tplgy and its Applicatins, Vlume 157, Issue 16, 2010, 2480-2590. [6] Salepci, N. Invariants f ttally real Lefschetz fibratins, t appear in Pacific J. f Math. [7] Orevkv, S.Yu. Riemann eistence therem and cnstructin f real algebraic curves. Ann. Fac. Sci. Tuluse Math. (6)12, 2003, n. 4, 517-531. Institut Camille Jrdan, Université Lyn I, 43, Bulevard du 11 Nvembre 1918 69622 Villeurbanne Cede, France E-mail address: salepci@math.univ-lyn1.fr