Weierstrass points and Morita Mumford classes on hyperelliptic mapping class groups

Transcription

1 Topoloy and its Applications Weierstrass points and Morita Mumford classes on hyperelliptic mappin class roups Nariya Kawazumi Department of Mathematical Sciences, University of Tokyo, Tokyo , Japan Received 12 December 2000; received in revised form 15 September 2001 Abstract We describe the Morita Mumford classes on the hyperelliptic mappin class roup of enus 2, and prove a slihtly weakened version of Akita s conjectures in the hyperelliptic case. The proof involves the notion of Weierstrass points Elsevier Science B.V. All rihts reserved. MSC: primary 57N05; secondary 14H15, 14H45, 14H55, 57M60, 57R20 Keywords: Morita Mumford classes; Mappin class roups; Hyperelliptic curves; Gap sequences; Bernoulli numbers Introduction Let Σ be a connected closed oriented 2-dimensional C -manifold of enus 2. The mappin class roup M is the path-component roup of the topoloical roup Diff + Σ consistin of all the orientation-preservin diffeomorphisms of Σ. The manifold Σ can be rearded as a compactification of the complex plane curve {z, w C 2 ; w 2 = fz}, where fz is a polynomial of deree 2 + 2or2 + 1 with no multiple roots. The involution of Σ iven by z, w z, w is called the hyperelliptic involution, which we denote by ι. The hyperelliptic mappin class roup is, by definition, the centralizer of the hyperelliptic involution ι: := { ψ M ; ιψι 1 = ψ }. As is known, M 2 = 2,andM,if 3. The rational cohomoloy of the roup is trivial: H ; Q = 0 forany > 0 address: kawazumi@ms.u-tokyo.ac.jp N. Kawazumi /01/$ see front matter 2001 Elsevier Science B.V. All rihts reserved. PII: S

2 364 N. Kawazumi / Topoloy and its Applications [6,12]. Its torsion cohomoloy, however, is very far from trivial. This is illustrated by notable works by homotopy theorists includin Cohen [2,6,7]. It should be remarked that the cohomoloy roup H ; k with coefficients in any field k has been computed by Bödiheimer et al. [4]. The present paper is a sequel of the author s ones [11 13] on the hyperelliptic mappin class roup. Our purpose in this paper is to describe the mth Morita Mumford class on the hyperelliptic mappin class roup e m H 2m ; Z for any m 0, and to prove a slihtly weakened version of Akita s conjectures [1] in the hyperelliptic case. We determine whether the modp reduction of the mth Morita Mumford class on the hyperelliptic mappin class roup vanishes or not for any m and any prime number p. Followin Akita [1] we denote it by e m p H 2m ; Z/p. Theorem A. 1 If p divides 2 2,thene m p = 0 for any m 0. 2 If p does not divide ,thene m p = 0 for any m 1. 3 If p does not divide and divides + 1,thene p 2n 1 = 0 and ep 2n is not nilpotent in H ; Z/p for any n 1. 4 Suppose p does not divide 2 2 and divides Letc be the order of 2 in the multiplicative roup Z/p.Ifm 1 modc, thene m p is not nilpotent in H ; Z/p. Ifm 1 mod c,thene m p = 0. The proof will be iven in Section 6. To prove the non-vanishin we make use of some examples computed by Uemura [19]. In [1] Akita has iven three fascinatin conjectures related to Morita Mumford classes on the whole mappin class roup M.Letp be an odd prime. His conjectures are: 1 e m 2 = 0 H M ; Z/2 for all m 0. 2 If m 1 modp 1,thene m p = 0 H M ; Z/p. 3 For any n 1, num B2n 2n e 2n 1 = 1 n 1 den B2n 2n s 2n 1 H 4n 2 M ; Z. Here e m and s m H 2m M ; Z are the mth Morita Mumford class and the mth Newton class on the whole mappin class roup M, respectively. B 2n is the 2nth Bernoulli number. It should be remarked our convention of sin is different from the one in Akita [1] and Morita [16], where 1 m s m is called the Newton class instead. The third conjecture implies the second one because of a theorem of Clausen von Staudt: denb 2n = p 1 2n p. With rational coefficients it is deduced from the Grothendieck Riemann Roch formula. For more details, see [1].

3 N. Kawazumi / Topoloy and its Applications On the hyperelliptic mappin class roup the first and the second conjectures are derived from Theorem A. In Section 5 we prove a slihtly weakened version of the third conjecture in the hyperelliptic case: Theorem B. For any n 1, numb 2n e 2n 1 = 1 n 1 2n denb 2n s 2n 1 H 4n 2 ; Z. In Section 1 we recall the definition of the Morita Mumford classes and the Newton classes on the whole mappin class roup M, and describe the Newton classes in terms of Weierstrass ap sequences. This yields an explicit description of 4s 2n 1 on the hyperelliptic mappin class roup Theorem 5.2. Moreover we ive a fixed-point formula of Newton classes for not necessarily hyperelliptic finite surface symmetries Theorem 1.1. Theorem B follows from the explicit description of 4s 2n 1 and Voronoi s conruence 5.1. Makin use of the conruence, Akita and the author have already proved a similarly weakened version of the third conjecture restricted to any finite subroup of M which acts on the surface in a semi-free way. The proof will be iven in the forthcomin paper. 1. Morita Mumford classes, Newton classes and ap sequences We recall the definitions of the Morita Mumford classes and the Newton classes on the mappin class roup M, and describe the Newton classes in terms of ap sequences. The orientation-preservin diffeomorphism roup Diff + Σ of the manifold Σ acts on the space consistin of all the orientation-preservin complex structures J + Σ on Σ in an obvious way. The Teichmüller space T of enus is, by definition, the quotient space by the action of the identity component Diff 0 Σ of the roup Diff + Σ : T := J + Σ / Diff 0 Σ. Thus the mappin class roup M = Diff + Σ / Diff 0 Σ acts on the Teichmüller space T in a natural way. In [8] Earle and Eells established that the quotient map J + Σ T is a principal Diff 0 Σ -bundle. As is known, the space J + Σ and the Teichmüller space T are both contractible. These facts imply that the identity component Diff 0 Σ is also contractible. Hence we obtain the homotopy equivalence BM = B Diff + Σ. This means the interal cohomoloy rin of the mappin class roup M of enus is isomorphic to the rin of all the interal characteristic classes of continuous families of compact Riemann surfaces of enus. In other words, we may define an interal cohomoloy class on the mappin class roup M as an interal characteristic class of continuous families of compact Riemann surfaces of enus. Let π : X B be an oriented fiber bundle with fiber Σ, whose structure roup is in the diffeomorphism roup Diff + Σ. Then the vector bundle T X/B over X consistin of all the tanent vectors alon the fibers is an oriented R 2 -bundle, which we call the relative tanent bundle. From the facts quoted above there exists a fiberwise orientationpreservin complex structure on π, or equivalently, an almost complex structure on T X/B,

4 366 N. Kawazumi / Topoloy and its Applications which is unique up to homotopy. It should be remarked that any almost complex structure on a 2-dimensional C -manifold is interable. Therefore we may reard π : X B as a continuous family of compact Riemann surfaces of enus. We denote by e = ex the first Chern class of the relative tanent bundle T X/B : e = ex := c 1 T X/B H 2 X; Z. The mth Morita Mumford class e m X of the family X is defined to be the Gysin imae of the m + 1st power of the cohomoloy class e = c 1 T X/B : e m X := π! e m+1 = π! c1 T X/B m+1 H 2m B; Z [16,17]. Here we denote by π! the Gysin map H 2m+2 X; Z H 2m B; Z of the fiber bundle π : X B. This defines a cohomoloy class e m H 2m M ; Z, which we call the mth Morita Mumford class on the mappin class roup M. The holomorphic 1-forms on each fiber π 1 t, t B, forma-dimensional complex vector bundle Λ = Λ X/B := t B H 0 π 1 t; O π 1 t T π 1 t 1.1 over B, which is usually called the Hode bundle associated to the family π : X B.Here we denote by T π 1 t the holomorphic cotanent bundle of each fiber π 1 t, and by O π 1 t T π 1 t = Ω 1 the sheaf of erms of holomorphic 1-forms on the Riemann π 1 t surface π 1 t.ifπ is a holomorphic family of compact Riemann surfaces, then it is equal to a direct imae π ω X/B of the dualizin sheaf ω X/B. By construction the homotopy type of the complex vector bundle Λ X/B over B depends only on the topoloical type of the continuous fiber bundle π : X B. Therefore the mth Newton class, which is the characteristic class corresponds to the symmetric polynomial j X j m,ofthecomplex vector bundle Λ X/B is well-defined, so that the cohomoloy class s m Λ H 2m M ; Z is defined. Throuhout the present paper we call it simply the mth Newton class of the mappin class roup M, and denote it by s m. Our convention of sin is different from that in Akita [1] and Morita [16], where s m Λ = 1 m s m is called the Newton class instead. If we study only the rational cohomoloy, we have no need to consider the Newton classes. In fact, we have s 2n = 0 H M ; Q,and n 1 2n e 2n 1 = 1 s 2n 1 H 4n 2 M ; Q, 1.2 B 2n [16,17]. The latter is deduced from the Grothendieck Riemann Roch formula. As was pointed out by Mumford [17], the cohomoloy of the moduli space of compact Riemann surfaces is closely related to the notion of ap sequences. We briefly review ap sequences in the context of continuous families of compact Riemann surfaces. Let C be a compact Riemann surface of enus 2, L a holomorphic line bundle over C, x C a point, and n a non-neative inteer. The set consistin of all erms of holomorphic sections at x with order n + 1 is a complex linear subspace of O C L x

5 N. Kawazumi / Topoloy and its Applications of codimension n + 1. The quotient vector space is denoted by J n L x, and is called the holomorphic n-jet space of L at x. By definition, the n-jet homomorphism j n x : H 0 C; O C L J n L x maps any holomorphic section to its n-jet at x, i.e., its equivalent class in J n L x. Recall det C = c 1 T C,[C] = 2 2. So there is no non-zero holomorphic 1-form ω defined on the whole C satisfyin ord x ω 2 1. This means the 2 2nd jet homomorphism jx 2 2 : H 0 C; O C T C J 2 2 T C x is injective for any point x C. For a eneric point x C the 1st jet homomorphism jx 1 : H 0 C; O C T C J 1 T C x is an isomorphism. A point x is called Weierstrass if jx 1 is not an isomorphism. Moreover there exist exactly inteers 1 = ν 1 < ν 2 < <ν 2 1 and holomorphic 1-forms θ i on C, 1 i, such that the order of θ i at x is equal to ν i 1. The set ν ={ν 1,ν 2,...,ν } is called the ap sequence of the point x on C. Clearly the ap sequence of a non-weierstrass point is ν 0 := {1, 2,...,}. See, for example, [9, 6.6]. Let π : X B be a continuous family of compact Riemann surfaces of enus 2. For each point x X, consider the n-jet space J n T π 1 πx x of the holomorphic cotanent bundle of the fiber π 1 πx. It is easy to see that JX/B n T X/B := J n T π 1 πx x x X forms a n + 1-dimensional complex vector bundle over X. The nth fiberwise jet homomorphism j n : π Λ X/B JX/B n T X/B is defined in an obvious way, which is a homomorphism of complex vector bundles over X. The2 2nd jet homomorphism j 2 2 : π Λ X/B J 2 2 X/B T X/B is injective. We have a natural extension of complex vector bundles 0 TX/B n+1 JX/B n T X/B J n 1 X/B T X/B 0. Hence the total Chern class of the n-jet bundle is iven by n+1 n+1 c J n X/B T X/B = c T k X/B = 1 ke. 1.3 Let W = W X/B X be the locus of Weierstrass points on the family X : W = W X/B := {x X; x is a Weierstrass point in π 1 πx}. Moreover, for a set of inteers ν = {ν 1,ν 2,...,ν },1= ν 1 <ν 2 < <ν 2 1, we define W ν = W ν X/B := { x X; ν is the ap sequence of the pointx on π 1 πx }. Especially, if ν = ν 0,wehaveW ν 0 = X W. From the definition of ap sequences the jet homomorphism j 2 2 : π Λ X/B J 2 2 X/B T X/B induces an isomorphism of continuous vector bundles π Λ X/B W ν = T ν i X/B W ν. i=1

6 368 N. Kawazumi / Topoloy and its Applications Hence we have π c Λ W ν = i=1 1 ν ie W ν H W ν ; Z,andso π s W m ν = 1 m m ν i e W ν m H 2m W ν ; Z 1.4 i=1 for any m 0. In the case where ν = ν 0 ={1, 2,...,} there exists a cohomoloy class t m H 2m X, X W; Z such that s m π Λ 1 m k e m m = t m H 2m X; Z. 1.5 In eneral, the space W X/B may have hihly complicated sinularities, while W X/B admits a fiberwise tubular neihborhood if the family π is a hyperelliptic fibration, or if it is induced by finite surface symmetries. In these cases we have the Thom isomorphism H X, X W = H 2 W; Z. 1.6 The main purpose of this paper is to study the universal hyperelliptic fibration. The rest of this section, however, is devoted to discussin not necessarily hyperelliptic finite surface symmetries. Suppose a finite roup G acts on a compact Riemann surface C in a holomorphic and faithful way. Then we can consider a continuous family of compact Riemann surfaces π : C G BG over the classifyin space BG of the roup G defined by C G = EG G C. Here EG BG is the universal principal G-bundle. We denote by G x the isotropy subroup at a point x C. WedefineS := {x C; G x {1}}, whichisag-stable finite subset of C. Let{x 1,x 2,...,x l } C be a complete system of representatives of the quotient S/G. Foranyj, 1 j l, wedefine R j := EG G G x j C G, and write simply G j := G xj. Clearly R j = BG j.let ν 1j <ν 2j < <ν j be the ap sequence of the point x j. Then: Theorem 1.1. For any m 1, we have 1 m G/G j s m Λ CG /BG = 1 m 2 2s m Λ CG /BG = i=1 ν ij m l j=1 i=1 cor G G e Rj j m, ν ij m cor G G e Rj j m. Proof. The set consistin of all the Weierstrass points is also a G-stable finite subset of C. Let {x l+1,x l+2,...,x p } C be a complete system of representatives with respect to the action of G on the Weierstrass points whose isotropy roups are trivial. We introduce R j, G j and ν 1j <ν 2j < <ν j for l + 1 j p in a similar way. Then R j is contractible, since G j is trivial.

7 N. Kawazumi / Topoloy and its Applications Let U j H 2 C G ; Z be the imae of the Thom class of R j in H 2 C G,C G R j ; Z, 1 j p. From 1.5 and 1.6 there exist cohomoloy classes t j,m H 2m 2 R j ; Z, 1 j p, satisfyin p 1 m π s m = k e m m + t j,m U j H 2m C G ; Z. j=1 Restrictin it to R j,wehave m ν ij e Rj m = k e m Rj m + t j,m e Rj H 2m R j ; Z. i=1 Hence 1 m U j π s m = k e m Rj m U j + t j,m e Rj U j = Applyin the Gysin map π! to it, we have 1 m m G/G j s m = ν ij cor G G e Rj j m, i=1 as was to be shown. See, for example, [14, pp ]. The Riemann Hurwitz formula implies l G/G j 2 2 mod G. j=1 i=1 ν ij m Hence the second formula follows from the first ones. This completes the proof. When m 2, we obtain t j,m = ν m ij k e m Rj m 1, i=1 e Rj m U j. since G j is a cyclic roup and e Rj H 2 G j ; Z is a enerator. If l + 1 j, then e Rj m 1 H 2m 2 R j ; Z = 0. Hence we have l 1 m π s m = k e m m + ν m ij k e m Rj m 1 U j. 1.7 j=1 i=1 2. Characteristic classes of S 2 -bundles We review characteristic classes of oriented S 2 -bundles. As was shown by Smale [18], the rotation roup SO3 is a deformation retract of the diffeomorphism roup Diff + S 2. Hence we have B Diff + S 2 = BSO3.

8 370 N. Kawazumi / Topoloy and its Applications Thus we may consider the first Pontrjain class p 1 and the Euler class χ of any oriented S 2 - bundle. Let ϖ : P = P SO3 = ESO3 SO3 S 2 B = BSO3 be the universal oriented S 2 -bundle, e H 2 P ; Z the Euler class of the relative tanent bundle T P/B.Thenwe have: Lemma 2.1. e 2 = ϖ p 1 H 4 P ; Z. Proof. Since S 2 = SO3/SO2, wehavep BSO2 and the projection ϖ : P B is equivalentto the map induced by the standard inclusion SO2 SO3. Furthermore the standard action of SO3 on S 2 induces a faithful and transitive action of SO3 on the unit tanent bundle of S 2. Hence the relative tanent bundle T P/B is equal to the universal R 2 - bundle over BSO2. Therefore, if ξ SO3 is the universal R 3 -bundle over BSO3,thenwe have ϖ ξ SO3 = T P/B R. Since we may reard T P/B as a complex line bundle over P, we obtain ϖ p 1 = p 1 T P/B R = p 1 T P/B = c 2 TP/B T P/B = e 2, as was to be shown. As a consequence of this lemma, we obtain: Corollary 2.2. The Morita Mumford classes of the universal oriented S 2 -bundle ϖ : P SO3 BSO3 is iven by e m P SO3 = ϖ! e m+1 { 0, if m is odd, = 2p m/2 1, if m is even, for any m 0.Herep 1 H 4 BSO3; Z is the first Pontrjain class. As is known, the universal coverin of SO3 is the Lie roup SU2. Lemma 2.3. Let B be a connected space, and let ϖ : P B be an oriented S 2 -bundle. Then the followins are equivalent to each other. a The structure roup of the bundle P can be reduced to SU2. b The Euler class χp vanishes: χp= 0 H 3 B; Z. c There exists a complex line bundle L over the total space P, such that ϖ! c 1 L H 0 B; Z = Z is an odd inteer. Proof. An extension of coefficients 0 Z Z Z/2 0 induces an isomorphism β : H 2 BSO3; Z/2 = H 3 BSO3; Z, which maps the second Stiefel Whitney class to the Euler class. This means that the condition a is equivalent to b. The equivalence of the conditions b and c follows from the Gysin sequence of the S 2 -bundle H 2 P ; Z ϖ! H 0 B; Z χ H 3 B; Z.

9 N. Kawazumi / Topoloy and its Applications In the present paper we call such an oriented S 2 -bundle of even type. Finally we consider the universal S 2 -bundle ϖ : P SU2 = ESU2 SU2 S 2 BSU2. If we consider U1 as a subroup of SU2 by the homomorphism z U1 z 0 0 z SU2, 2.1 then we have S 2 = SU2/U1, andsop SU2 = BU1 CP.Lety 0 H 2 P SU2 ; Z = Z be the enerator satisfyin ϖ! y 0 = 1 H 0 BSU2; Z,andL 0 the complex line bundle over P SU2 correspondin to y 0 H 2 P SU2 ; Z.Wehave 2y 0 = c 1 T PSU2 /BSU2 H 2 P SU2 ; Z, 2.2 since the Gysin map ϖ! : H 2 P SU2 ; Z H 0 BSU2; Z is an isomorphism. If c 2 H 4 BSU2; Z is the second Chern class of the universal SU2-bundle, then 2.1 implies ϖ c 2 = c 1 L 0 c 1 L 0 = y 0 2. Hence we have y 0 2 = ϖ c 2 H 4 P SU2 ; Z Hyperelliptic Riemann surfaces In this section we construct the universal family of hyperelliptic Riemann surfaces over the classifyin space B of the hyperelliptic mappin class roup, π : X B. In eneral, let π : Y Z be a continuous family of compact Riemann surfaces of enus. Suppose a discrete roup Γ acts on the family π : Y Z. More precisely, the roup Γ acts on Y and Z continuously such that π is Γ -equivariant and γ : π 1 t π 1 γ t is biholomorphic for each γ Γ and t Z. Assume Z is connected and simply connected. Then, for any t 1,t 2 Z, the diffeomorphism f t 2 t 1 : π 1 t 2 π 1 t 1 iven by parallel translation is uniquely determined up to isotopy. Clearly f γt 2 γt 1 γf t 2 t 1 γ 1 : π 1 γ t 2 π 1 γ t 1 are isotopic to each other. Choose a basepoint b Z and an identification π 1 b = Σ.Thenthe holonomy homomorphism h : Γ M of the Γ -action is defined by hγ := f γb γ π 1 b.the conjuacy class of the homomorphism h does not depend on the choice of a basepoint and an identification. Let EΓ BΓ be the universal principal Γ -bundle. Define Y Γ := EΓ Y/Γ and Z Γ := EΓ Z/Γ. Here the roup Γ acts diaonally on EΓ Y and EΓ Z. π induces a continuous family of compact Riemann surfaces π Γ : Y Γ Z Γ.Wehaveπ 1 Z Γ = Γ, since Z is simply connected. The holonomy homomorphism of the surface bundle π Γ [16, p. 553] is equal to h : Γ M. Rouhly speakin, the hyperelliptic Riemann surfaces are parametrized by the confiuration space of the Riemann sphere P 1. For a positive inteer n and a Riemann surface C, the confiuration space F n C is defined by F n C := { z 1,z 2,...,z n C n ; z i z j i j }. b and

10 372 N. Kawazumi / Topoloy and its Applications The quotient space of the confiuration space of the Riemann sphere P 1 by the diaonal action of the roup PGL 2 C of linear fractional transformations H := F 2+2P 1 /PGL 2 C is equal to the moduli space of hyperelliptic Riemann surfaces of enus with a labellin of the Weierstrass points. Let Z be the universal coverin space of H. We denote by ϖ : P Z the pullback of the complex analytic family of ordered pointed Riemann spheres, F 2+2 P 1 P 1 /PGL 2 C H. Then a roup, which is an extension of the symmetric roup S 2+2 by the fundamental roup π 1 H, acts on the family ϖ : P Z in an obvious way, and the holonomy homomorphism of the action ives an isomorphism of the extended roup onto the mappin class roup of the unordered 2 +2-pointed sphere, Γ Thus it turns out that the roup Γ acts on the complex analytic family ϖ : P Z. The family ϖ : P Z admits canonical analytic sections. The space Z is contractible, since H = F 2 1 P 1 {0, 1, } is aspherical. Hence there exists a fiberwise double branched coverin space Φ : Y P branched alon the sections. The analytic map π : Y Z defined to be the composite of ϖ : Y P and Φ : P Z is the complex analytic family of hyperelliptic Riemann surfaces of enus. A roup, which is an extension of Γ by {±1}, acts on the family π : Y Z in a natural way. The holonomy homomorphism ives an isomorphism of the extended roup onto the hyperelliptic mappin class roup. Hence acts on the family π : Y Z. We may replace B by Z,sinceZ is contractible. We define the universal continuous family of hyperelliptic Riemann surfaces π : X B by X := Y and π := π, whose holonomy homomorphism is conjuate to the inclusion M. Especially we have e m X = e m H B ; Z = H ; Z. The universal family of pointed Riemann spheres ϖ : P B is defined by ϖ := ϖ and P := P.ThisisanorientedS 2 -bundle over the classifyin space B. The coverin Φ : Y P induces a fiberwise double branched coverin Φ : X P. Let C be a hyperelliptic Riemann surface of enus. It is a compactification of the complex plane curve {z, w C 2 ; w 2 = fz} for some polynomial fzof deree As is well known, the 1-forms z i dz/w,0 i 1, provide a basis of the vector space H 1 C; O C T C, so that the canonical map associated to the holomorphic cotanent bundle T C Φ T C : C P 1, x [ dz/w x : z 1 dz/w x : : z 1 dz/w ] x

11 N. Kawazumi / Topoloy and its Applications coincides with the composite of the double coverin C P 1 iven by z, w z and the Veronese embeddin P 1 P 1, [z : 1] [1 : z : :z 1 ].IfL is the Hopf line bundle over P 1, the pullback Φ T C L is naturally isomorphic to the cotanent bundle T C. Such a construction can be applied to the universal family π : X B see, for example, [11]. We denote by Λ = Λ the Hode bundle Λ X /B, and by ϖ : PΛ B the fiber bundle on B with fiber P 1 defined by projectifyin the dual of the vector bundle Λ. The fiberwise canonical map Φ K : X P Λ decomposes itself into the double cover Φ and the fiberwise Veronese embeddin of P into PΛ. We define a complex line bundle L on P by restrictin the dual of the fiberwise Hopf bundle on PΛ to the Veronese imae of P.Thenwehave Φ L = T X /B. 3.1 From the naturality of Gysin maps we obtain e m X = π! c 1 T X /B m+1 = 2 ϖ! c 1 L m+1, and so e m X = 1 m+1 2 ϖ! c1 L m+1 H 2m B ; Z. 3.2 Especially we have ϖ! c1 L = 1 H 0 B ; Z. 3.3 Now we consider the locus W = W = W X /B of Weierstrass points in the family of hyperelliptic curves X = X. It coincides with the ramified locus of the coverin Φ. The projection π : X B ives a fold unramified coverin map π W : W B. If we denote the fundamental roup of the locus W by,then the subroup of is of index 2 + 2, and the space W is an aspherical space: W = B. The subroup is equal to the inverse imae of the 2 + 1st symmetric roup S 2+1 S 2+2 by the surjective homomorphism of onto the 2 + 2nd symmetric roup S 2+2 defined by permutation of the Weierstrass points. In our case the Weierstrass locus W admits a fiberwise tubular neihborhood so that we have natural isomorphisms H X, X W; Z = H T X/B W,T X/B zero section W ; Z = H 2 W; Z. 3.4 The riht one is the Thom isomorphism associated to the complex line bundle T X/B W. We denote by U H 2 X; Z the imae of the Thom class of the bundle T X/B W.Thenwe obtain a lon exact sequence H q 2 W; Z U H q X; Z H q X W; Z H q 1 W; Z exact. 3.5 Since U W = e W H 2 W; Z,wehave π! e m k U k = cor e W m 1 H B ; Z = H ; Z 3.6

12 374 N. Kawazumi / Topoloy and its Applications for any 0 <k m. Herecor : H ; Z H ; Z is the transfer map. For eneralities of transfer maps see [5]. See also [14, 2]. The double branched cover Φ induces an isomorphism of complex line bundles T X/B X W = Φ T P/B X W. Hence, if we denote e P := Φ c 1 T P/B H 2 X ; Z, then we have e e P = au for some a Z. Applyin the Gysin map to this relation, we have = a2 + 2 H 0 B = Z,sothata = 1. Hence we obtain e + U = e P H 2 X ; Z Morita Mumford classes In this section we ive two kinds of descriptions of the Morita Mumford classes on the hyperelliptic mappin class roup. One description follows computations of Miller [15] and Morita [16]. Proposition 4.1. For any n 0, we have e 2n = 4p n n+1 cor e W 2n H 4n ; Z, e 2n 1 = 1 2 2n cor e W 2n 1 H 4n 2 ; Z. Here p 1 = p 1 P H 4 ; Z is the Pontrjain class of the S 2 -bundle P. Proof. From 3.7 and the naturality of the Gysin maps 2e m P = π! ep m+1 = π! e + U m+1 = e m + 2 m+1 1 cor e W m. As was shown in Corollary 2.2, e 2n 1 P = 0ande 2n P = 2p 1 P n. The other involves with the line bundle L over P introduced in Section 3. Theorem 4.2. For any m 0 the mth Morita Mumford class e m H 2m ; Z is iven by [m/2] j=0 [m/2] j=0 m + 1 2j 1/2 2s1 m 2j p j 1, if is odd, 2j + 1 m j 2s 1 m 2j j ĉ 2, if is even. 2j + 1 Here s 1 = s 1 Λ = c 1 Λ H 2 ; Z is the first Newton class, and, in the case is even, ĉ 2 H 4 ; Z is the second Chern class of the S 2 -bundle P of even type. In both cases 2 2 divides any of the Morita Mumford classes on. All the computations in the rest of this paper are based on

13 N. Kawazumi / Topoloy and its Applications Lemma 4.3. { H 2 Z/22 + 1, if is even, ; Z = Z/42 + 1, if is odd. This lemma follows from the presentation of the roup iven by Birman Hilden [3] toether with the fact H 2 ; Q = 0 [6,12]. Now suppose the enus is odd.define h := 1/2. Recall the Gysin sequence of the oriented S 2 -bundle ϖ : P B 0 H 2 B ; Z ϖ H 2 P ; Z ϖ! H 0 B ; Z. Then 3.3 implies that there exists a unique cohomoloy class u H 2 ; Z such that c 1 L = hc 1 T P /B ϖ u H 2 P ; Z. 4.1 Lemma s 1 = u H 2 ; Z s 1 and u 2s 1 are both nontrivial 4-torsion elements in H 2 ; Z. Proof. Let V be a 2-dimensional complex vector space, on which SU2 acts as a maximal compact roup of SLV.Themthsymmetric product S m V is of complex dimension m+1 for any non-neative inteer m.ifm = 2n is even, then the action of SU2 on S 2n V factors throuh the rotation roup SO3 = SU2/{±1}, and so we may consider a complex vector bundle over BSO3 with fiber S 2n V, which we denote also by S 2n V. We may reard the universal S 2 -bundle ϖ : P SO3 BSO3 as a continuous family of complex projective lines by makin use of the standard metric on S 2. So the holomorphic n-tensors on each fiber ϖ 1 t forms a 2n + 1-dimensional complex vector bundle over BSO3: π TPSO3 /BSO3 n := H 0 ϖ 1 t; O ϖ t 1 Tϖ 1 t n, t BSO3 which is naturally isomorphic to the vector bundle S 2n V. Therefore we have an isomorphism of vector bundles over B Λ = ϖ T X /B = ϖ L = S 2h V L u. Here L u is a continuous vector bundle over B correspondin to u H 2 B ; Z. Clearly we have c 1 S 2n V= 0 H 2 BSO3; Z = 0. Hence we obtain s 1 = c 1 Λ = c 1 S 2n V u = u, as was to be shown.

14 376 N. Kawazumi / Topoloy and its Applications From Lemma 4.3 the cohomoloy class 2 + 1s 1 and u 2s 1 = 2 + 1u are 4-torsion. To prove s 1 and 2u 2s 1 0, it suffices to construct a roup homomorphism ϕ 0 : Z/2 satisfyin ϕ 0 s 1 0 H 2 Z/2; Z = Z/2. In fact, ϕ 0 u 0 because is odd, and so ϕ 0 u 2s 1 0, while the unique nontrivial homomorphism of Z/ onto Z/2 maps all the 2-torsions to zero. Consider the hyperelliptic involution ι. This induces a roup homomorphism ϕ 0 : Z/2.Asisknown,ι θ = θ for any holomorphic 1-form θ on any hyperelliptic Riemann surface. If we denote by u 0 the unique enerator of H 2 Z/2; Z = Z/2, we obtain ϕ 0 s 1 = u 0 = u 0 0. This completes the proof. Proof of Theorem 4.2 in the odd enus case. From 3.2, Lemma 2.1 and Corollary 2.2 we have e m = e m X = 1 m+1 2 ϖ! c1 L m+1 = 2 ϖ! ϖ u hc 1 T P /B m+1 = 2 = 4 [m/2] j=0 [m/2] j=0 = 2 2 m + 1 ϖ! ϖ u m 2j h 2j+1 c 1 T P /B 2j + 1 2j+1 m + 1 u m 2j h 2j+1 p 1 P j 2j + 1 [m/2] j=0 m + 1 u m 2j h 2j p 1 P j. 2j + 1 Now 4 divides 2 2. Therefore we may replace u by 2s 1 from Lemma 4.4. This completes the proof for the case is odd. Next we consider the case where the enus is even. Thenϖ! L = 1 is odd, so that there exists a bundle map ψ : P P SU2 by Lemma 2.3. We introduce a cohomoloy class y on P defined by y := ψ y 0 H 2 P ; Z. Here y 0 H 2 P SU2 ; Z = Z is the enerator satisfyin ϖ! y 0 = 1 H 0 BSU2; Z as in Section 2. From the Gysin sequence of ϖ : P B there exists a unique cohomoloy class u H 2 ; Z such that c 1 L = 1y ϖ u H 2 P ; Z. 4.2 Lemma s 1 = u H 2 ; Z s 1 = 2u 2s 1 = 0 H 2 ; Z. Proof. Let V be a 2-dimensional complex vector space, on which SU2 acts as a maximal compact roup of SLV. We may consider a complex vector bundle over BSU2 with

15 N. Kawazumi / Topoloy and its Applications fiber the symmetric product S m V, which we denote also by S m V. In a similar way to Lemma 4.4 we obtain s 1 = c 1 Λ = c 1 S 1 V u = u. From Lemma 4.3 we have 2u 2s 1 = u = 0. Now is even. So divides Hence 2 + 1s 1 = 0. This completes the proof. Proof of Theorem 4.2 in the even enus case. From 2.3 follows y 2 = ϖ ĉ 2. By 3.2 we have e m = e m X = 1 m+1 2 ϖ! c1 L m+1 = 2 ϖ! ϖ u 1y m+1 = 2 = 2 [m/2] j=0 [m/2] j=0 = 2 2 m + 1 ϖ 2j + 1! ϖ u m 2j 1 2j+1 y 2j+1 m + 1 u m 2j 1 2j+1 j ĉ 2 2j + 1 [m/2] j=0 m + 1 u m 2j 1 2j j ĉ 2. 2j + 1 Here we may replace u by 2s 1 from Lemma 4.5. This completes the proof for the case is even. 5. Akita s conjectures In this section we compute the cohomoloy class 4s 2n 1 Λ H 4n 2 ; Z explicitly to prove Theorem B stated in Introduction: numb 2n e 2n 1 = 1 n 1 2n denb 2n s 2n 1 H 4n 2 ; Z. Our proof is based on Voronoi s conruence cf., e.., [20, Chapter IX] or [10]: β 2n 1 m 1 [ kβ numb 2n 1 n 1 2nβ 2n 1 denb 2n k 2n 1 m Here m and β Z are coprime positive inteers. In order to start the computation we need Lemma cor e W 2n 1 = 0 H ; Z. ] mod m. 5.1 Proof. There exist cohomoloy classes u H 2 B ; Z and v H 4 B ; Z such that U + u 2 = v H 4 X ; Z.

16 378 N. Kawazumi / Topoloy and its Applications In fact, in the case the enus is odd, wehavee = he P + u H 2 X ; Z for some u H 2 B ; Z from 4.1. Here h := 1/2. This toether with 3.7 implies U +u = h + 1e P. From Lemma 2.1 U + u 2 = h e P 2 = h p 1 H 4 X ; Z. In the case the enus is even, wehavee = 1Φ y + u H 2 X ; Z for some u H 2 B ; Z from 4.2. This toether with e + U = e P = 2Φ y implies U + u = + 1Φ y. From 2.3 we have U + u 2 = Φ y 2 = ĉ 2 π H 4 B ; Z. We have u = 0 by Lemma 4.3. So we obtain cor e W = 2 + 1π! U 2 = π! Uu = u = 0 H 2 ; Z. Moreover 2 + 1U 4 = u 2 U u 4 v 2, and 2 + 1U 2n = u 2 U 2n u 4 v 2 U 2n 4 for any n 3. Hence cor e W 3 = u 2 cor e W = 0and cor e W 2n 1 = u 2 cor e W 2n u 4 v 2 cor e W 2n 5. Consequently the lemma is deduced inductively on n. This completes the proof. Theorem 5.2. For any n 1, 2 [ ] 2k 4s 2n 1 Λ = 2 2n+1 k 2n 1 cor e W 2n 1 H ; Z. Proof. We write simply A,n := k2n 1 and C,n := 1 k 2 1,k: odd k2n 1.Then we have 2 2n A,n 2 1 k 2,k: even k2n 1 2C,n mod 2 + 1,since 2 2 k 2n 1 Moreover we have 2 k 2n 1 + k 2n 1 0 mod n A,n 2C,n 2 2n+1 A,n 2 2n+1 A,n 2 2n n+1 2 k 2n 1 [ ] 2k k 2n 1 mod Now, from 1.5 and 3.4, we have π s 2n 1 Λ + A,n e 2n 1 = t n U H 4n 2 X ; Z for some t n H 4n 4 W; Z.SinceU 2 = eu H 4 X ; Z, we obtain 0 = t n UU e = U eπ s 2n 1 Λ + A,n e 2n 1 U A,n e 2n.

17 N. Kawazumi / Topoloy and its Applications This toether with Proposition 4.1 implies 4s 2n 1 Λ = π! U es 2n 1 Λ = A,n cor e W 2n 1 e 2n 1 = 2 2n A,n cor e W 2n 1. As is known, the ap sequence of each Weierstrass point on hyperelliptic Riemann surfaces is 1, 3, 5,...,2 1. See, for example, [9, 6.6, p. 219]. In terms of Section 1 this means W = W {1,3,5,...,2 1}. Hence we have π s 2n 1 Λ W = C,n e W 2n 1 H 4n 2 ; Z, and so 2 + 2s 2n 1 Λ = C,n cor e W 2n 1 H 4n 2 ; Z. Consequently we have 4s 2n 1 Λ = s 2n 1 Λ = 2 2n A,n 2C,n cor e W 2n 1 2 [ ] 2k = 2 2n+1 k 2n 1 cor e W 2n 1. The last equality follows from Lemma 5.1. This completes the proof. Proof of Theorem B. From Theorem 5.2 and Voronoi s conruence 5.1 for m = and β = 2 we obtain 1 n 1 2n denb 2n s 2n 1 = 1 n 1 n denb 2n /2 4s 2n 1 Λ = 1 n 1 n denb 2n /2 2 [ 2k 2 2n+1 k 2n [ 2k = 1 n 1 2n2 2n 1 denb 2n k 2n = 2 2n 1 numb 2n cor e W 2n 1 = numb 2n e 2n 1. This completes the proof of Theorem B. ] cor e W 2n 1 ] cor e W 2n 1 6. modp Morita Mumford classes This section is devoted to the proof of Theorem A stated in Introduction. Theorem 4.2 implies that 2 2 divides e m H 2m ; Z for any m 0. So clearly the assertion 1 of Theorem A holds true. The odd Morita Mumford classes are much simpler than the even ones.

18 380 N. Kawazumi / Topoloy and its Applications Theorem 6.1. The order of the odd Morita Mumford class e 2n 1 in the cohomoloy roup H 4n 2 ; Z is equal to that of 2 2n 1 in the roup Z/ In order to prove the theorem we need: Lemma 6.2. There exists a roup homomorphism ϕ 1 : Z/2 + 1 such that the order of ϕ 1 cor e W m H 2m Z/2 + 1; Z = Z/2 + 1 is just for any m 1. Proof. Consider the compactification C of the complex plane curve { z, w C 2 ; w 2 = z z }. The compact Riemann surface C is a hyperelliptic Riemann surface of enus.the2 + 1st root of unity ζ := exp 2π defines an automorphism of C by z, w ζ 2 z, ζ w. It induces a roup homomorphism ϕ 1 : Z/2 + 1.DefineG = Z/2 + 1 and let G act on C by this action. The Borel construction π : C G := EG G C BG is a continuous family of hyperelliptic Riemann surfaces. If we denote by S the set of the Weierstrass points on C, the locus of Weierstrass points W CG /BG is equal to S G = EG G S C G.Wehave ϕ 1 cor e W m = π SG! e SG m. The set S decomposes itself into one Z/2 + 1-free orbit and a sinle fixed point z, w = 0, 0. Hence the Gysin homomorphism π SG! : H S G ; Z H BG; Z is an isomorphism for > 0. On the other hand, the first Chern class of the complex line bundle over BG induced by the tanent space at 0, 0, T 0,0 C, is a enerator of the rin H G; Z. Consequently ϕ 1 cor e W m is a enerator of H 2m G; Z, and so is of order 2 + 1, as was to be shown. Proof of Theorem 6.1. From Lemma 5.1 and Lemma 6.2, the order of the class cor e W 2n 1 in H 4n 2 ; Z is just On the other hand, we have e 2n 1 = 1 2 2n cor e W 2n 1 H 4n 2 ; Z by Proposition 4.1. Consequently the order of e 2n 1 in H 4n 2 ; Z is equal to that of 2 2n 1 in the roup Z/ This completes the proof. The order of the even Morita Mumford classes are estimated as follows. Lemma 6.3. For any n 1, we have e 2n = 0, e 2n = 0, if is even, if is odd, in H 4n ; Z.

19 N. Kawazumi / Topoloy and its Applications Proof. Let be the inverse imae of S 2 1 S 2+2 by the surjective homomorphism S 2+2 defined by permutation of the Weierstrass points. Since a linear fractional transformation of CP 1 fixin 3 distinct points is trivial, an oriented S 2 -bundle which admits 3 mutually disjoint sinle-valued sections is trivial. Thus the pullback of the S 2 -bundle P over B by the inclusion is a trivial S 2 -bundle, so that p 1 = 0 H 4 ; Z. Since the index [ : ] is equal to , we obtain p 1 = 0 H 4 ; Z. 6.1 Suppose is even. Lemma 4.3 implies s 1 = 0. Hence, from Theorem 4.2, we have 2 + 1e 2n = n ĉ 2 n. On the other hand, we have p 1 = 4ĉ 2 in H 4 BSU2; Z. Hence e 2n = n 4 ĉ 2 n = 0. In the case is odd. Similarly we have e 2n = /2 2n p1 2n = /2 2n+1 p1 2n = 0 from Theorem 4.2 and Lemma 4.3. This completes the proof. As a corollary of Lemma 6.3, we obtain the assertion 2 of Theorem A. If p does not divide 2 + 1, then e p 2n 1 = 0 from Theorem 6.1. Therefore, in order to prove the assertion 3 of Theorem A, it suffices to show Lemma There exists a roup homomorphism ϕ 2 : Z/4 such that, for any n 1, ϕ 2 e 2n H 4n Z/4; Z = Z/4 is of order 2 respectively, if is even respectively odd. 2 There exists a roup homomorphism ϕ 3 : Z/2 + 2 such that, for any n 1, ϕ 3 e 2n H 4n Z/2 + 2; Z = Z/2 + 2 is of order + 1respectively + 1/2,if is even respectively odd. Proof. 1 In [19, Example 5.1], Uemura studied the hyperelliptic Riemann surface C defined as the compactification of the complex plane curve { z, w C 2 ; w 2 = z z 2 1 }. The 4th root of unity ζ := exp2π 1/4 defines an automorphism of C by z, w ζ 2 z, ζ w. It induces a roup homomorphism ϕ 2 : Z/4. As was proved by Uemura [19] loc. cit., ϕ 2 e 2n = 2 + 2u 2n H 4n Z/4; Z = Z/4.

20 382 N. Kawazumi / Topoloy and its Applications Here u H 2 Z/4; Z is the first Chern class of the complex line bundle over BZ/4 induced by the 1-dimensional Z/4-module defined by multiplication by ζ.the cohomoloy class u is a enerator of the rin H Z/4; Z. Therefore ϕ 2 e 2n is of order 2 respectively, if is even respectively odd, as was to be shown. 2 In order to prove 2 we consider the compactification C of the complex plane curve { z, w C 2 ; w 2 = z }, on which the cyclic roup Z/2 + 2 acts by z, w ζ z, w.hereζ is the primitive 2 + 2nd root of unity ζ := exp2π 1/ This action defines a roup homomorphism ϕ 3 : Z/ In terms of the first Chern class u of the line bundle over BZ/2 + 2 induced by the 1-dimensional Z/2 + 2-module defined by multiplication by ζ, the even Morita Mumford class e 2n is iven by ϕ 3 e 2n = 4u 2n H 4n Z/2 + 2; Z = Z/2 + 2 from a fixed-point formula established by Uemura and the author [14]. See also [19]. The order of this cohomoloy class is equal to + 1 respectively + 1/2, if is even respectively odd. This completes the proof. Finally we consider the case where p divides Recall [ : ]= So we have 2 + 2e 2n P = cor res e2n P = cor 2eP W 2n = 2 2n+1 cor e W 2n H 4n ; Z from Lemma 2.1 and Corollary 2.2. This toether with Proposition 4.1 implies 2 + 2e 2n = 2 2n n+1 cor e W 2n H 4n ; Z 6.2 for any n 1. If a prime number p divides 2 + 1, we have e m p = 1 2 m+1 cor e W m H 2m ; Z/p for any m 1. The cohomoloy class cor e W m is not nilpotent in H 2m ; Z/p from Lemma 6.2. Thus we obtain Theorem A4. This completes the proof of the whole of Theorem A. Acknowledement The author thanks Shieyuki Morita, Toshiyuki Akita and Yuki Tadokoro for inspirin discussions.

21 N. Kawazumi / Topoloy and its Applications References [1] T. Akita, Nilpotency and triviality of mod p Morita Mumford classes of mappin class roups of surfaces, Naoya Math. J., to appear. [2] D.J. Benson, F.R. Cohen, Mappin Class Groups of Low Genus and Their Cohomoloy, in: Mem. Amer. Math. Soc., Vol. 443, [3] J. Birman, H. Hilden, On the mappin class roups of closed surfaces as coverin spaces, in: Advances in the Theory of Riemann Surfaces, in: Ann. of Math. Stud., Vol. 66, 1971, pp [4] C.F. Bödiheimer, F.R. Cohen, M.D. Peim, Mappin class roups and function spaces, Preprint. [5] K.S. Brown, Cohomoloy of Groups, Spriner, Berlin, [6] F.R. Cohen, Homoloy of mappin class roups for surfaces of low enus, Contemp. Math [7] F.R. Cohen, On the mappin class roups for punctured spheres, the hyperelliptic mappin class roups, SO3, Spin c 3, Amer. J. Math [8] C.J. Earle, J. Eells, A fiber bundle description of Teichmüller theory, J. Differential Geom [9] S. Iitaka, Alebraic Geometry, Spriner, Berlin, [10] K. Ireland, M. Rosen, A Classical Introduction to Modern Number Theory, Spriner, Berlin, [11] N. Kawazumi, Foldin surface bundles of enus 2, J. Fac. Sci. Univ. Tokyo Sect. IA [12] N. Kawazumi, On the homotopy type of the moduli space of n-point sets of P 1,J.Fac.Sci. Univ. Tokyo, Sect. IA [13] N. Kawazumi, Homoloy of hyperelliptic mappin class roups for surfaces, Topoloy Appl [14] N. Kawazumi, T. Uemura, Riemann Hurwitz formula for Morita Mumford classes and surface symmetries, Kodai Math. J [15] E.Y. Miller, The homoloy of the mappin class roup, J. Differential Geom [16] S. Morita, Characteristic classes of surface bundles, Invent. Math [17] D. Mumford, Towards an enumerative eometry of the moduli space of curves, in: Arithmetic and Geometry, in: Pror. Math., Vol. 36, Birkhäuser, Boston, 1983, pp [18] S. Smale, Diffeomorphism of the 2-sphere, Proc. Amer. Math. Soc [19] T. Uemura, Morita Mumford classes on finite cyclic subroups of the mappin class roup of closed surfaces, Hokkaido Math. J [20] J.V. Uspensky, M.A. Heaslet, Elementary Number Theory, McGraw-Hill, New York, 1939.