On the signature of a Lefschetz fibration coming from an involution

Transcription

1 Toology and its Alications 153 (006) On the signature of a Lefschetz fibration coming from an involution Ki-Heon Yun Deartment of Mathematics, Konkuk University, 1 Hwayang-dong, Gwangjin-gu, Seoul , Reublic of Korea Received 18 July 005; acceted 18 July 005 Abstract In this article we show that the signature of a Lefschetz fibration coming from a secial involution as a roduct of right-handed Dehn twists deends only on the number of genus on the involution axis. We investigate the geograhy of such Lefschetz fibrations and we identify it with a blow u of a ruled surface. We also get a geograhy of the Lefschetz fibration coming from a finite order element of maing class grou as a comosition of two secial involutions. 005 Elsevier B.V. All rights reserved. MSC: 57N13; 57R17 Keywords: Lefschetz fibration; Signature; Maing class grou 1. Introduction The study of symlectic toology in dimension four is closely related to the study of Lefschetz fibration over S which is determined by monodromy factorization. A relation in the maing class grou as a roduct of right-handed Dehn twists gives a monodromy factorization of a Lefschetz fibration. This work was suorted by Grant No. R from KOSEF. address: kyun@member.ams.org (K.-H. Yun) /$ see front matter 005 Elsevier B.V. All rights reserved. doi: /j.tool

2 K.-H. Yun / Toology and its Alications 153 (006) Korkmaz [7] found a relation in the maing class grou involving g + 4 (res., g + 10) right-handed Dehn twists when the genus g of the surface is even (res., odd). Gurtas [5,6] generalized it and he found an involution φ of a Riemann surface Σ g as a roduct of g + 3h + right-handed Dehn twists where h is the number of genus on the involution axis. Gurtas asked whether the signature σ(x φ ) of the Lefschetz fibration X φ S is 4(h + 1). It is known that the signature is 4forh = 0 case [7] but it is not known for h 1. The article is organized as follows. In Section, we briefly review a Lefschetz fibration and a right-handed Dehn twists exression of an involution of tye (l, k,r) (Definition 4) which was introduced by Gurtas. We show that for a fixed g = l + k + r and h = l + r,all involutions of tye (l, k,r) are related by a sequence of Hurwitz moves and conjugations. In Section 3, we get a signature formula for the Lefschetz fibration coming from an involution of tye (l, k,r). Furthermore we also identify such a Lefschetz fibration. Theorem 1. Let φ : Σ g Σ g be the right-handed Dehn twists exression of an involution of tye (l, k,r). Let X φ D be the Lefschetz fibration corresonding to φ and X φ S be the Lefschetz fibration corresonding to φ. Then σ(x φ ) = 4(h+1) and σ(x φ ) = (h + 1). Furthermore, X φ is diffeomorhic to (Σ k S )#4(l + r + 1)CP. In Section 4, we study a signature formula of a Lefschetz fibration over S coming from a finite order element of a maing class grou as a comosition of two involutions. As an alication, we study a -fold cyclic covering π : Σ (g 1)+1 Σ g of the Riemann surface Σ g and a right-handed Dehn twists exression ψ of the π rotation ma on Σ (g 1)+1 which sends the ith handle to the (i +1)st handle of Σ (g 1)+1. When is odd, ψ is a comosition of two involutions of tye (g, ( 1)(g 1), 0).If is even, then ψ is a comosition of an involution of tye (g, ( )(g 1), (g 1)) and an involution of tye (1,(g 1), 0). Theorem. The Lefschetz fibration X ψ S is a simly connected 4-manifold which has the following toological invariants: c 1 (X ψ ) = 4( )(g 1) and χ h(x ψ ) = 1 8{ c 1 (X ψ ) + 4(g + 1)}.. Preliminaries Definition 3. Let X be a comact, oriented smooth 4-manifold. A (smooth) Lefschetz fibration is a roer smooth ma π : X B where B is a comact connected oriented surface and π 1 ( B) = X such that (1) the set of critical oints C ={ 1,,..., n } of π is non-emty and lies in int(x) and π is injective on C; () about each i and π( i ), there are local comlex coordinate charts agreeing with the orientations of X and B such that π can be exressed as π(z 1,z ) = z 1 + z.

3 1996 K.-H. Yun / Toology and its Alications 153 (006) A Lefschetz fibration is relatively minimal if no fiber contains an embedded shere of self intersection 1. Let Σ g be a Riemann surface of genus g. For any simle closed curve c on Σ g, t c is the right-handed Dehn twist along the curve c. Letφ : Σ g Σ g be an orientation reserving diffeomorhism such that φ = t an t a t a1 for some simle closed curves a 1,a,...,a n on Σ g. Then we can construct a Lefschetz fibration X φ over D with boundary (Σ g [0, 1])/ (x,1) (φ(x),0) by attaching n -handles along a i with framing 1 corresonding to the surface framing, i.e. X φ = {{ {{ Σ g D } a1 ( D D )} } an ( D D )}. Moreover if we have t an t an 1 t a t a1 = id, then we can construct a Lefschetz fibration X φ id (Σ g D ) over S with generic fiber Σ g and n singular fibers. The simle closed curve a i, i = 1,,...,n, is called vanishing cycle and the orientation reserving diffeomorhism t an t an 1 t a t a1 is called monodromy factorization of the Lefschetz fibration. For a given Lefschetz fibration π : X B, monodromy factorization is not uniquely determined. Let us define t a (t b ) = t ta (b) which is t a t b ta 1 as a ma. If φ is a monodromy factorization of a Lefschetz fibration, then its conjugation g(φ) by using an orientation reserving diffeomorhism g : Σ g Σ g is also a monodromy factorization of the Lefschetz fibration. If two monodromy factorization φ 1 and φ are related by a sequence of the following two Hurwitz moves t ai+1 t ai t ai+1 (t ai ) t ai+1 and t ai+1 t ai t ai ta 1 i (t ai+1 ), then φ 1 and φ give the isomorhic Lefschetz fibration. We call φ 1 is Hurwitz equivalent to φ or φ 1 φ if φ 1 can be changed to φ by a finite sequence of Hurwitz moves or conjugations. Let X φ1 S and X φ S be two Lefschetz fibrations with monodromy factorization φ i : F F as a roduct of right-handed Dehn twists. The fiber sum X φ1 # f X φ is defined by (X φ1 int(f D )) g (X φ int(f D )) where g : (X φ int(f D )) (X φ1 int(f D )) is a fiber reserving and orientation reversing diffeomorhism. The Euler characteristic of a Lefschetz fibration X φi S with generic fiber F is given by χ(x φ1 ) = χ(f) + n, where n is the number of vanishing cycles in φ i, and χ(x φ1 # f X φ ) = χ(x φ1 ) + χ(x φ ) χ(f). The Lefschetz fibration X φ corresonding to a monodromy factorization as a roduct of right-handed Dehn twists is a symlectic 4-manifold and so it has an almost comlex structure. Let us define c1 (X φ) = 3σ(X φ ) + χ(x φ ) and χ h (X φ ) = 1 4 (σ (X φ) + χ(x φ )). From now on we introduce the involution and its right-handed Dehn twists exression which was studied by Gurtas in [5,6]. Definition 4. Let Σ g be a Riemann surface of genus g. An orientation reserving diffeomorhism φ : Σ g Σ g of Σ g = L M R is called an involution of tye (l, k,r) if it is a π rotation along the axis as in Fig. 1 where l = genus(l) 0, k = genus(m) 0 and r = genus(r) 0 as in Fig.. Note that g = l + k + r and the number of genus on the involution axis is h = l + r.letl = t c1 t c t cl, R = t c(l+r)+1 t c(l+r) t c(l+1) and M = t B0 t B1 t Bk t cl+1 by using the simle closed curves in Fig. and let L, R be the reversed word of L, R, resectively.

4 K.-H. Yun / Toology and its Alications 153 (006) Fig. 1. The involution tye (l, k,r). Fig.. Surface decomosition and its vanishing cycles: (a) L,(b)R,(c)M,(d)M R, L M. Theorem 5. [5] The orientation reserving diffeomorhism φ = (R R) (L L) M defined on Σ g as a roduct of right-handed Dehn twists is an involution of tye (l, k,r). Proof. The four boundary circles of M arefixedbythemam but if we consider the influence of M on L or R, then its role is the same as t d1 t d to R and t d3 t d4 to L. The hyerellitic involution on R which gives a π rotation to R is the ositive word R R t d1 t d. Similarly the hyerellitic involution on L is given by L L t d3 t d4. Therefore what we need to do to get the involution of tye (l, k,r) is that first we aly M which gives π rotation to M Σ g while fixing M and then aly the ma (R R) (L L) which gives π rotation to L and R. Remark 6. (1) Sometimes we call the right-handed Dehn twists exression in Theorem 5 as the involution of tye (l, k,r). () The right-handed Dehn twists exression of φ which aears in [5] is Hurwitz equivalent to the exression in Theorem 5. (3) The right-handed Dehn twist exression of φ which aears in [6] is an easy modification of Theorem 5. Therefore we can aly most statements aeared in this article including signature formula and toological invariants to the case.

5 1998 K.-H. Yun / Toology and its Alications 153 (006) Corollary 7. Let φ : Σ g Σ g be the right-handed Dehn twists exression of the involution of tye (l, k,r) and h = l + r. Then φ is Hurwitz equivalent to {t h+1 t h t t t 3 t h+1 t B0,0 t B0,1 t B0,k t c0,1 } where t i = t ci, c 0,1 = (t h+1 t h t )(c 1 ) and B 0,i = (R L)(B i ) for each i = 0, 1,,...,k. Proof. By Theorem 5, φ = (R R) (L L) (t B0 t B1 t Bk t l+1 ), where R = t h+1 t h t l+ and L = t 1 t t l. Then φ { (R R) (L L) (t B0 t B1 t Bk t l+1 ) } { (R L ) (R L) (t B0 t B1 t Bk t l+1 ) } { (R L ) ( t B0,0 t B0,1 t B0,k (R L) t l+1 )} { (L t l+1 R ) L t B0,0 t B0,1 t B0,k R } because R L = L R, Hurwitz moves and cyclic ermutations of φ. We observe that θ 1 := L t l+1 R = t 1 t t h+1 and θ 1 (c i ) = c i+1 for i = 1,,...,h. Therefore we have φ { (L t l+1 R ) L t B0,0 t B0,1 t B0,k R } { t l+1 t l t θ 1 t B0,0 t B0,1 t B0,k R } { R t l+1 t l t θ 1 t B0,0 t B0,1 t B0,k } { t h+1 t t 1 t t 3 t h+1 t B0,0 t B0,1 t B0,k } { t c0,1 t h+1 t 3 t t 3 t h+1 t B0,0 t B0,1 t B0,k } { t h+1 t 3 t t 3 t h+1 t B0,0 t B0,1 t B0,k t c0,1 } by Hurwitz moves and cyclic ermutations. Remark 8. For a fixed g = l + k + r and h = l + r with different l and r, the righthanded Dehn twists exression of the involution of tye (l, k,r) looks retty different. But Corollary 7 imlies that if φ 1 and φ are any two involution of tye (l, k,r) with the same g and h, then the two Lefschetz fibrations X φ 1 S and X φ S are isomorhic. 3. A Lefschetz fibration coming from an involution We first collect some well-known results in signature comutation of Lefschetz fibration. The roof may be found in the corresonding references. Theorem 9. [] Let X be a four manifold that admits a hyerellitic Lefschetz fibration of genus g over S. Let m and s = [g/] h=0 s h be the numbers of non-searating and searating vanishing cycles in the monodromy factorization of this fibration, resectively, where s h

6 K.-H. Yun / Toology and its Alications 153 (006) denote the number of searating vanishing cycles that searate the genus g surface into two surfaces one of which has genus h. Then the signature of X is σ(x)= g + 1 g + 1 m + [g/] h=1 ( ) 4h(g h) g s h. Definition 10. Let X be the genus g Lefschetz fibration over D with monodromy factorization φ and let X be the 4-manifold obtained from X by attaching a -handle along a simle closed curve γ on Σ g {t} X. We define σ(φ,γ)= σ(x ) σ(x). Generally, let φ = t an t an 1 t a1 and let ψ i = t ai ψ i 1 and ψ 0 = ψ. Then σ(ψ,φ) is defined by σ(ψ,φ)= n i=1 σ(ψ i 1,a i ). Theorem 11. [9] Let X be a 4-manifold which admits a genus g Lefschetz fibration over D or S and let φ = t an t an 1 t a t a1 be a monodromy factorization of the fibration. Let φ 0 = id and φ i = t ai t ai 1 t a t a1. Then σ(x)= σ(id,φ)= n i=1 σ(φ i 1,a i ). Proosition 1. Let φ i = t ci,ni t ci,ni 1 t c i, t ci,1 be an involution of Σ g and f : Σ g Σ g be a diffeomorhism such that f(φ ) φ 1 = id where f(φ ) = t f(c,n ) t f(c,n 1) t f(c, ) t f(c,1 ). Let X φi D and X f(φ ) φ 1 S be the corresonding Lefschetz fibrations. Then the signature satisfies σ(x f(φ ) φ 1 ) = σ(x φ1 ) + σ(x φ ). Proof. Let us consider the Lefschetz fibration π : X f(φ ) φ 1 S.LetD 1 beaclosed disk which contains n 1 critical values corresonding to the vanishing cycles c 1,j, j = 1,,...,n 1, in its interior and no other critical values on D 1. Then π 1 (D 1 ) = X φ1 and π 1 (S int(d 1 )) = X f(φ ). Since f(φ ) 1 = φ 1,wehaveX f(φ ) φ 1 = X φ1 Xφ1 X f(φ ). Now we aly the Novikov additivity of signature, we get σ(x f(φ ) φ 1 ) = σ(x φ1 ) + σ(x f(φ )). Since φ and f(φ ) give equivalent Lefschetz fibration, we have σ(x φ ) = σ(x f(φ )). Therefore we get the conclusion. Now we will rove the signature formula σ(x φ ) = 4(h+1) of the Lefschetz fibration X φ S where φ is the involution of tye (l, k,r). We decomose the Riemann surface Σ g by M S where S = L R and decomose the ositive word φ by φ LR M where φ LR = (R R) (L L). From this we observe that all vanishing cycles of M are located on M and all vanishing cycles of φ LR are located on S. Definition 13. Let D be a shere with 4 unctures and let D = t y t cl+1 as in Fig. 3. By relacing M by D and by relacing the word M by D, we get a new Riemann surface Σ of genus h = g k and a right-handed Dehn twists exression φ = φ LR D of an involution on Σ.

7 000 K.-H. Yun / Toology and its Alications 153 (006) Fig. 3. D. Fig. 4. Fig. 5. Standard homotoy generators for L, R and M. Proosition 14. For any vanishing cycle γ aears in L or R, M(γ) = D(γ). Proof. If γ is neither the simle closed curve c l nor c (l+1), then M(γ) = γ = D(γ) because γ is disjoint from all vanishing cycles in M and in D. If γ is either the simle closed curve c l or c (l+1), then M(γ) = D(γ) as in Fig. 4. Remark 15. Let Σg,n r be a genus-g surface with n marked oints and r boundary comonents. Let Γg,n r be the maing class grou of Σr g,n. Let us consider the oriented generators of π 1 (Σk 4 ) as in Fig. 5. Then each vanishing cycles in Fig. (c) can be written as B 0 = β 1,1 β 1, β 1,k d 1 d 1 4, B i 1 = α 1,i β 1,i β 1,i+1 β 1,k+1 i γ 1,k+1 i α 1,k+1 i, 1 i k, B i = α 1,i β 1,i+1 β 1,i+ β 1,k i γ 1,k i α 1,k+1 i, 1 i k 1, B k = α 1,k γ 1,k α 1,k+1, γ i =[α 1,1,β 1,1 ][α 1,,β 1, ] [α 1,i,β 1,i ]d 3 d 1 1.

8 K.-H. Yun / Toology and its Alications 153 (006) If we caing off the four boundary circles by four disks, then it is the standard one which was studied by Korkmaz [7]. We also notice that d 1 d and d 3 d 4 bound a surface in Σ g. Therefore in the homology level, d 1 + d = 0 and d 3 + d 4 = 0. Now we will check that the signature contribution σ(id,m) of M on the Lefschetz fibration X φ D is the same as the signature contribution σ(id,d)of D on the Lefschetz fibration X φ D. We consider the three cases (1) l + r = 0, () (l = 0 and r 1) or (r = 0 and l 1), (3) l 1 and r 1 searately. Proosition 16. For the l + r = 0 case, σ(id,m)= = σ(id,d) Proof. From the construction, M is an element of hyerellitic maing class grou with (k + 1) non-searating vanishing cycles and 1 searating vanishing cycle and M = id. Therefore σ(id,m)= σ(x M ) = 1 σ(x M ) = 1 { k + 1 ( ( ) 4 k (k k) (k + 1) + 4k + 1 4k + 1 ) } 1 = by the Endo s formula and Proosition 1. Since D is a roduct of two searating vanishing cycles, σ(id,d)=. Proosition 17. For the l 1 and r 1 case, σ(id,m)= 0 = σ(id,d). Proof. Let e 1 = c l and e = c (l+1) which are located artially on M with orientation such that e 1 B i =+1 and e B i =+1 for all i = 1,,...,k by using the orientation of B i given in Remark 15. We will define ψ j by ψ 0 = id, ψ 1 = t cl+1 and ψ i = t Bk+ i t Bk+3 i t Bk t cl+1 for i =, 3,...,k +. Then σ(id,m)= σ(id,c l+1 ) + k+ i= σ(ψ i 1,B k i+ ).Now we will comute it by using Ozbagci s algorithm [9]. σ(id,c l+1 ) = 0 because c l+1 is a non-searating vanishing cycle. σ(ψ i 1,B k i+ ) = 0fori =, 4,...,k because i ([l ]) = B k i+ and i ([m ]) = β 1,k i+ ψ i 1 (β 1,k i+ ) = β 1,k i+ β 1,k i+ = 0. σ(ψ i 1,B k i+ ) = 0 for i = 3, 5,...,k + 1 because i ([l ]) = B k i+ and i ([m ]) = ψ i 1 (α 1,k i+ ) α 1,k i+ = α 1,k i+ α 1,k i+ = 0. Now we want to check that σ(ψ k+1,b 0 ) = 0. We have i ([l ]) = B 0 and i ([m ]) = e 1 ψ k+1 (e 1 ) = B 0 + d 4 d 3 or i ([m ]) = e ψ k+1 (e ) = B 0 d 1 + d.fromthe following homology relations, α 1,i = ψ k+1 (α 1,i ) = α 1,i (B 1 + B + +B i 1 ) for i = 1,,...,k and β 1,i = ψ k+1 (β 1,i ) = β 1,i + (B 1 + B + +B i ) for i = 1,,...,k, we get B i = 0 for all i = 1,,...,k in H 1 ( X ψk+1 int(ν(b 0 )); R). From Remark 15 we also have the following homology relations:

9 00 K.-H. Yun / Toology and its Alications 153 (006) B i = α 1,i + (β 1,i+1 + β 1,i+ + +β 1,k i ) + γ 1,k i + α 1,k i+1 for 1 i k 1, B i 1 = α 1,i + (β 1,i + β 1,i+1 + +β 1,k i+1 ) + γ 1,k i+1 + α 1,k i+1 for 1 i k, B k = α 1,k + γ 1,k + α 1,k+1. From these we have 0 = B i 1 B i = β 1,i + β 1,k i+1 + γ 1,k i+1 γ 1,k i = β 1,i + β 1,k i+1 because γ 1,k i+1 γ 1,k i bounds a surface for all i = 1,,...,k. Therefore β := β 1,1 + β 1, + +β 1,k = 0. Since d 1 + d = 0,d 3 + d 4 = 0 (Remark 15) and the two exressions of i ([m ]) give d 1 + d 4 = d + d 3,wehaved 1 = d = d 3 = d 4. From Fig. 6, we have i ( [m ] ) = B 0 + d 4 d 3 = (β d 4 + d 1 ) + d 4 d 3 = d 1 d 3 = 0 and so σ(ψ k+1,b 0 ) = 0. Therefore σ(id,m)= 0. Now we will check that σ(id,d)= 0. Since c l+1 is a non-searating vanishing cycle, σ(id,c l+1 ) = 0. Let e 1 and e be the same as before, then i ( [m ] ) = t cl+1 (e 1 ) e 1 = (e 1 + c l+1 ) e 1 = c l+1 = t cl+1 (e ) e = (e c l+1 ) e = c l+1. Therefore i ([m ]) = 0 and so σ(t cl+1,y)= 0. From this σ(id,d)= 0. Fig. 6. (a) ψ k+1 (c l ),(b)ψ k+1 (c (l+1) ),(c)β := β 1 β β k,(d)b 0,(e)e 1 ψ k+1 (e 1 ), (f) e ψ k+1 (e ).

10 K.-H. Yun / Toology and its Alications 153 (006) Proosition 18. For the (l = 0 and r 1) or (r = 0 and l 1) case, σ(id,m)= 1 = σ(id,d). Proof. We will consider the case when the two boundary circles d 1 and d of Fig. 5 are caed off by two disks. The other case is the same. We will use the same notation as in Proosition 17. The only difference from Proosition 17 is the comutation of σ(ψ k+1,b 0 ).Fromi ([l ]) = B 0 = β d 4 = d 4 and i ([m ]) = e 1 ψ k+1 (e 1 ) = β d 3 = d 3,wehavei ([l ]+[m ]) = (d 3 + d 4 ) = 0 because it bounds a surface. Therefore σ(ψ k+1,b 0 ) = 1 and σ(id,m)= 1. For a roof of σ(id,d)= 1, we need to check that σ(t cl+1,y)= 1. Since i ([l ]) = y = d 4 and i ([m ]) = e 1 t cl+1 (e 1 ) = d 4,wehavei ([l ]+[m ]) = 0 and σ(t cl+1,y)= 1. So σ(id,d)= 1. Proosition 19. Let ψ be one of L L or R R. Then σ(m,ψ)= σ(d,ψ). Proof. It is based on the Ozbagci s result in [9], esecially Theorem 4 and Proosition 5. The signature σ(ψ,γ)is determined by i ([l ]) = γ, i ([m ]) = e ψ(e) e γ and the ratio of [l ] and [m ] in the ker{i : H 1 ( ν(γ ); R) H 1 ( X ψ int(ν(γ )); R)}. Proosition 14 imlies M L = D L and M R = D R. For each vanishing cycle γ in ψ, we can select a simle closed curve e on L or R such that e γ = 1. By using this e, i ([l ]) and i ([m ]) in the comutation of σ(m,ψ)have to be the same as i ([l ]) and i ([m ]) in the comutation of σ(d,ψ). Therefore σ(m,ψ)= σ(d,ψ). Theorem 0. Let φ : Σ g Σ g be the right-handed Dehn twists exression of involution of tye (l, k,r). Let X φ D be the Lefschetz fibration corresonding to φ and X φ S be the Lefschetz fibration corresonding to φ. Then σ(x φ ) = 4(h + 1) and σ(x φ ) = (h + 1). Proof. Let φ = (R R) (L L) M be an orientation reserving diffeomorhism defined on Σ g and let Σ be the Riemann surface obtained by relacing M by D and let φ = (R R) (L L) D be an orientation reserving diffeomorhism defined on Σ. Then from Proositions 16 19, we get σ(x φ ) = σ(id,m)+ σ(m,φ LR ) = σ(id,d)+ σ(d,φ LR ) = σ(x φ ), where φ LR = (R R) (L L). Now we will consider X φ. There are 4l non-searating vanishing cycles on L, 4r nonsearating vanishing cycles on R and non-searating vanishing cycles on D. Therefore φ is a roduct of 4(l + r) + = 4h + non-searating vanishing cycles and φ is an element of hyerellitic maing class grou and φ = id. So we have the signature σ(x φ ) = h + 1 h + 1 {(4h + ) } = 4(h + 1) by using Theorem 9. Therefore by Proosition 1, σ(x φ ) = σ(x φ ) = 1 σ(x φ ) = (h + 1) and σ(x φ ) = σ(x φ ) = 4(h + 1).

11 004 K.-H. Yun / Toology and its Alications 153 (006) Fig. 7. Korkmaz s involution. Corollary 1. [7] Let us consider the ositive word θ = t b 1,0 t b 1,1 t b 1,k t b 1,k+1 t a t b by using the vanishing cycles in Fig. 7. Then σ(x θ ) = 8. Proof. If we rove that θ is an involution of tye (1, k,0), then by Theorem 0 we easily rove that σ(x θ ) = 4(1 + 1) = 8. Let B j be the standard vanishing cycle in Fig., then it is clear that (t a t c )(B j ) = b 1,j for each j = 0, 1,...,k and b 1,k+1 = (t a t b )(c). We also note that if a b =±1, then Therefore t a t b t a t a t b (t a ) t b (t a t b )(t a ) t a t b t ta t b (a) t a t b t b t a t b. θ t b 1,0 t b 1,1 t b 1,k t b 1,k+1 t a t b t b 1,0 t b 1,1 t b 1,k (t a t b )(t c ) t a t b t a t b t b 1,0 t b 1,1 t b 1,k t a t b t c t a t b t b 1,0 t b 1,1 t b 1,k t a (t b t c t b ) t a t b 1,0 t b 1,1 t b 1,k t a (t c t b t c ) t a, because b c =±1 t ta t c (B 0 ) t ta t c (B 1 ) t ta t c (B k ) t a t c t b t c t a t a t c t B0 t B1 t Bk t b t c t a t a t c t a (t a (t c )) t a t B0 t B1 t Bk t b, see Proosition 14 t a (t c ) ta t a(t c ) t B0 t B1 t Bk t b ( t a tc ta t ) c t B0 t B1 t Bk t b which is a conjugation of the involution of tye (1, k,0). Now we investigate toological invariants and geograhy of Lefschetz fibrations X φ S where φ is an involution of tye (l, k,r).

12 K.-H. Yun / Toology and its Alications 153 (006) Proosition. Let φ : Σ g Σ g be an involution of tye (l, k,r) and let X φ S be the corresonding Lefschetz fibration. Then b 1 (X φ ) = k, c1 (X φ) = 4(g 1) and χ h (X φ ) = 1 8 c 1 (X φ ) + 1 (h + 1). Proof. Since φ is an involution of tye (l, k,r), φ is a roduct of 4l + 4r + k + = 4h + k + = g + 3h + right-handed Dehn twists. Therefore χ(x φ ) = ( g) + (g + 3h + ) = 8 g + 6h and σ(x φ ) = 4(h + 1) by Theorem 0. From this we get and c1 (X φ ) = χ(x φ ) + 3σ(X φ) = 4(g 1) χ h (X φ ) = 1 χ(xφ ) + σ(x 4( φ ) ) = 1 8 c 1 (X φ ) + 1 (h + 1). Now we will rove that b 1 (X φ ) = k. LetN be the normal subgrou of π 1 (Σ g ) generated by all vanishing cycles of φ, then π 1 (X φ ) = π 1 (Σ g )/N. We will use the homotoy basis as in Fig. 5 and we consider L, M and R searately. On L, c 1 = α 1 = α 1, c i = β i for i = 1,,...,l, c i+1 = α i α 1 i+1 = α i α 1 i+1 for i = 1,,...,l 1. On R, c (l+r)+1 = α h = α h, c (l+r) (i 1) = β h i+1 for i = 1,,...,r, c (l+r) (i 1) = α h i α 1 h i+1 = α h i α 1 h i+1 for i = 1,,...,r 1. On M, k k c l+1 = α lα 1 [β 1,j,α 1,j ]=α l α 1 [α 1,k+j,β 1,k+j ]=γ 1 l+1 j=1 l+1 j=1 B k = α 1,k γ 1,k α 1,k+1, B 0 = β 1,1 β 1, β 1,ki αl 1 α 1 l+1, B j 1 = α 1,j β 1,j β 1,j+1 β 1,k+1 j γ 1,k+1 j α 1,k+1 j, 1 j k, B j = α 1,j β 1,j+1 β 1,j+ β 1,k j γ 1,k j α 1,k+1 j, 1 j k 1. Note that γ 1,j = γ 1,k = 0 in the homology level for each j. From these, we get the following homology relations: α 1 = α = =α h = 0, α 1 = α = =α h = 0, β 1 = β = =β h = 0, β 1,j + β 1,k+1 j = 0 for j = 1,,...,k, α 1,j + α 1,k+1 j = 0 for j = 1,,...,k. 1,k,

13 006 K.-H. Yun / Toology and its Alications 153 (006) Therefore we have k free homology generators of H 1 (X φ ; Z) located on M.Sowehave b 1 (X φ ) = k. Remark 3. Let φ : Σ g Σ g be an involution of tye (l, k,r). Ifg>0 and g is an odd number, then 1 8 c 1 (X φ ) + 1 χ h(x φ ) 1. If g>0 and g is an even number, then 1 8 c 1 (X φ ) + 1 χ h(x φ ) 1. Remark 4. From Proosition, b 1 = k = g h. Since χ(x φ ) = b 1 + b = (g h) + b = 8 g + 6h, we have b (X φ ) = 4h + 6. Therefore b + (X φ ) = b + σ = 1 and b (X φ ) = b σ = 4h + 5. Definition 5. [3] M(h,k) be the desingularization of double cover of Σ k S branched over (Σ k {q 1,q }) ({ 1,,..., h } S ) where {q 1,q } S and { 1,,..., h } Σ k. Proosition 6. [3,4] M(h,k) is diffeomorhic to (Σ k S ) #4hCP. Theorem 7. Let φ : Σ g Σ g be the right-handed Dehn twists exression of the involution of tye (l, k,r) and let X φ S be the corresonding Lefschetz fibration. Then X φ is diffeomorhic to (Σ k S ) #4(l + r + 1)CP. Proof. By Corollary 7 it is enough to consider the involution φ of tye (h, k,0) and we will check that φ is a monodromy factorization of a Lefschetz fibration M(h+ 1,k) S with generic fiber Σ h+k. Let M (1,k) be the double cover of Σ k S branched over (Σ k {q 1,q }) ({ h+1, h+} S ) Σ h S and let M (h + 1, 0) be the double cover of S S branched over (S {q 1,q }) ({ 1,,..., h+1, (h+1) } S ).Letν be the double cover of D S branched over (D {q 1,q }) ({ h+1 } S ) or branched over (D {q 1,q }) ({ (h+1) } S ) as in the shaded art of Fig. 8(a), (b). Then we get M (h + 1,k)= ( M (1,k) int(ν) ) ( M (h + 1, 0) int(ν) ) which is a double cover of Σ k S branched over ( Σk {q 1,q } ) ( { 1,,..., (h+1) } S ) Σ k S

14 K.-H. Yun / Toology and its Alications 153 (006) Fig. 8. Branch set. as in Fig. 8(c). We consider it as a fibration over S and first resolve ( h+1,q 1 ) and ( h+1,q ) which gives a comonent of [Σ k ]+[S ]. After that we resolve ( h+,q i ) which gives local monodromy M by [8,10] and then we resolve ( h,q i ), ( h 1,q i ),...,( 1,q i ) which gives t h t h 1 t 1 t 1 t t h on the local monodromy of the singular fiber [1]. Therefore each singular fiber gives local monodromy φ and M(h+ 1,k) S considered as a Lefschetz fibration with generic fiber Σ h+k has monodromy factorization φ. Therefore X φ is diffeomorhic to M(h+ 1,k). Remark 8. There is a known roof of Theorem 7 for k given by Stisicz [10]. If X is a symlectic 4-manifold with b + (X) = 1 then X is diffeomorhic to a blow u of a ruled surface or b 1 (X) {0, }. Ifφ is an involution of tye (h, k,0) with k, then X φ is a symlectic 4-manifold with b 1 4 and b + = 1. Therefore it is of the form (Σ m S )#ncp and by comaring b 1, b + and b we get m = k and n = 4(h + 1). For k = 0, φ is the standard form of the hyerellitic involution and it is known to Matsumoto [8] for h = 0 and k = 1 case. 4. More examles We first exlain how to comute the signature of a Lefschetz fibration X φ S where φ is a comosition of two involutions such that φ = id. Proosition 9. Let φ 1, φ be involutions written as a roduct of right-handed Dehn twists. Then ( φ (φ 1 ) φ 1 ) n φ (φ 1 ) f n+1 f n f n 1 f f 1, where f 0 = φ 1, f 1 = φ (φ 1 ) and f i = f i 1 (f i ) for i. Proof. From the construction of f i, each f i is a conjugation of φ 1 and so f i f i = id for each i. Therefore f i f i commutes with any other ma. Since (f i+1 f i ) n i f i+1 ( f i+1 (f i ) f i+1 ) n i fi+1 we have ( f i+1 (f i ) f i+1 ) n (i+1) fi+1 (f i ) f i+1 f i+1 (f i+ f i+1 ) n (i+1) f i+ f i+1,

15 008 K.-H. Yun / Toology and its Alications 153 (006) ( φ (φ 1 ) φ 1 ) n φ (φ 1 ) (f 1 f 0 ) n f 1 = (f f 1 ) n 1 f f 1 (f 3 f ) n f 3 f f 1 (f n f n 1 ) f n fn 1 f f 1 f n+1 f n f n 1 f f 1. Theorem 30. Let φ 1, φ be two involutions of Σ g which are written as a ositive word and φ = φ φ 1 be an element of maing class grou with finite order. Let X φ and X φ be the Lefschetz fibration over D and over S, resectively. Then the signature satisfies σ(x φ ) = {σ(x φ1 ) + σ(x φ )}. Proof. For = 1, it is the above Proosition1. For = : since φ = id, φ φ φ 1 φ φ 1 φ (φ 1 ) φ φ φ 1 φ (φ 1 ) φ 1 φ φ and we have φ (φ 1 ) φ 1 = id. Therefore by Proosition 1 σ(x φ ) = σ(x φ # f X φ (φ 1 ) φ 1 ) = σ(x φ ) + σ(x φ (φ 1 ) φ 1 ) = ( σ(x φ ) + σ(x φ1 ) ). For = n + 1 Z 3,wehave φ n+1 φ n φ ={φ (φ 1 ) φ φ φ 1 } n (φ φ 1 ) { } n φ (φ 1 ) φ 1 φ φ 1 (φ φ ) n { φ (φ 1 ) φ 1 } n φ (φ 1 ) φ (φ φ ) n f n+1 f n f n 1 f f 1 φ (φ φ ) n (f n+1 φ ) f n f n 1 f f 1 (φ φ ) n by Proosition 9. Since each f i is a conjugation of φ 1, fi = id and X fi is isomorhic to X φ1 as Lefschetz fibration. Therefore f n+1 φ = id and X φ = { } X fn+1 φ # f X f n # f X f # f # f X n 1 f # f X 1 φ # f # f X φ. }{{} n So by Proosition 1 σ(x φ ) = σ(x fn+1 ) + σ(x φ ) + n σ(x fi ) + nσ (X φ ) i=1 = (n + 1) { σ(x φ1 ) + σ(x φ ) } = { σ(x φ1 ) + σ(x φ ) }. For = n + Z 3,wehave

16 K.-H. Yun / Toology and its Alications 153 (006) φ n+ = φ n+1 φ (f n+1 φ ) fn f n 1 f f 1 (φ φ ) n φ φ 1 (f n+1 φ φ φ 1 ) f n f n 1 f f 1 (φ φ ) n (f n+1 φ 1 ) f n f n 1 f f 1 (φ φ ) n+1 by using the above odd case. Since each f i is a conjugation of φ 1, X fi is isomorhic to X φ1 as Lefschetz fibration. Therefore X φ = { } X fn+1 φ 1 # f X f n # f X f # f # f X n 1 f # f X 1 φ # f # f X φ }{{} n+1 and by Proosition 1 σ(x φ ) = σ(x fn+1 ) + σ(x φ1 ) + n σ(x fi ) + (n + )σ (X φ ) i=1 = (n + ) { σ(x φ1 ) + σ(x φ ) } = { σ(x φ1 ) + σ(x φ ) }. From now on, we will consider the -fold cyclic covering π : Σ (g 1)+1 Σ g, g, which is obtained by cutting Σ g along c g+1 and attach coies side by side as in Fig. 9. The π rotation ma of Σ (g 1)+1 which sends ith handle to (i +1)st handle can be written as a comosition of two orientation reserving involutions. Definition 31. Let ψ = φ φ 1 be a right-handed Dehn twists exression corresonding to the π rotation ma of Σ (g 1)+1 which sends ith handle to (i + 1)st handle where φ 1 and φ are involutions of tye (g, ( 1)(g 1), 0) if is odd. φ 1 is an involution of tye (g, ( )(g 1), g 1) and φ is an involution of tye (1,(g 1), 0) if is even. Theorem 3. The Lefschetz fibration X ψ S is a simly connected 4-manifold which has the following toological invariants: and c1 (X ψ ) = 4( )(g 1) χ h (X ψ ) = 1 4 ( g + g g 4) Proof. We observe that = 1 8{ c 1 (X ψ ) + 4(g + 1)}. (φ φ 1 ) = ψ 1 (φ ) ψ 1 (φ 1 ) ψ (φ ) ψ (φ 1) ψ (φ ) ψ (φ 1 ) φ φ 1.

17 010 K.-H. Yun / Toology and its Alications 153 (006) Fig. 9. Let c i,j be the simle closed curves located on the ith handle as in Fig. 9 and a i,j, b i,j be the standard homotoy generators. Then ψ k 1 (c 1,j ) = c k,j for all k = 1,,..., and j = 1,,...,g + 1. Let N be the normal subgrou generated by all vanishing cycles. Then c 1,j N for all j = 1,,...,g + 1 and c i,1 = a i,1 N, c i,g+1 = a i,g N, c i,j = b i,j N for j = 1,,...,g, c i,j+1 = a i,j a 1 i,j+1 N for j = 1,,...,g 1. So N = π 1 (Σ (g 1)+1 ) and π 1 (X ψ ) = π 1 (Σ (g 1)+1 )/N = 1. Therefore the Lefschetz fibration X ψ is simly connected. Now we will rove the toological invariants. If is odd, then ψ is a roduct of (4g + ( 1)(g 1) + ) = (3g + g + 3) right-handed Dehn twists and if is even, then ψ is a roduct of {(4g +4(g 1)+( )(g 1)+)+(4+(g 1)+)}= (3g + g + 3) right-handed Dehn twists. From this χ(x ψ ) = (g + g + 5) and by Theorems 0 and 30 we have { ( (g + 1) (g + 1)) = 4(g + 1) if is odd, σ(x ψ ) = ( ((g 1) + 1) (1 + 1)) = 4(g + 1) if is even. Therefore c1 (X ψ ) = 4( )(g 1) and χ h (X ψ ) = 1 ( g + g g 4 ) 4 = 1 { } 4( )(g 1) + 4(g + 1) 8 = 1 8{ c 1 (X ψ ) + 4(g + 1)}.

18 K.-H. Yun / Toology and its Alications 153 (006) Fig. 10. Corollary 33. Assume all conditions of Theorem 3 and assume 3, then we have c 1 (X ψ 8( ) ( ) = χh (X 1 ) ) ψ. Proof. It is clear from the following equations: χ h = 1 { } 4( )(g 1) + 4(g + 1) 8 = 1 { 4( )(g 1) + 8 = 1 { } 4( )(g 1 + ) } c1 ( 1) + = 8( ) c 1 +. Acknowledgement The author would like to thank András Stisicz for useful comments and Jongil Park for helful conversations and suggestions. References [1] D. Auroux, Fiber sums of genus Lefschetz fibrations, Turkish J. Math. 7 (1) (003) [] H. Endo, Meyer s signature cocycle and hyerellitic fibrations, Math. Ann. 316 () (000)

19 01 K.-H. Yun / Toology and its Alications 153 (006) [3] R. Fintushel, R. Stern, Families of simly connected 4-manifolds with the same Seiberg Witten invariants, Toology 43 (6) (004) [4] R. Gomf, A. Stisicz, 4-Manifolds and Kirby Calculus, Graduate Studies in Mathematics, vol. 0, American Mathematical Society, Providence, RI, [5] Y. Gurtas, Positive Dehn twist exressions for some new involutions in maing class grou, math.gt/ [6] Y. Gurtas, Positive Dehn twist exressions for some new involutions in the maing class grou II, math.gt/ [7] M. Korkmaz, Noncomlex smooth 4-manifolds with Lefschetz fibrations, Internat. Math. Res. Notices (3) (001) [8] Y. Matsumoto, Lefschetz fibrations of genus two a toological aroach, in: Toology and Teichmüller Saces, Katinkulta, 1995, World Sci. Publishing, River Edge, NJ, 1996, [9] B. Ozbagci, Signatures of Lefschetz fibrations, Pacific J. Math. 0 (1) (00) [10] A. Stisicz, Singular fibers in Lefschetz fibrations on manifolds with b + = 1, Toology Al. 117 (1) (00) 9 1.