ON THE YAGITA INVARIANT OF MAPPING CLASS GROUPS H. H. Glover, G. Mislin and Y. Xia March 1993 Abstract. Let denote a group of finite virtual cohomological dimension an* *d p a prime. If the cohomology ring H*(; Fp) has Krull dimension one, the p-period of * *is defined; it measures the periodicity of H*(; Fp) in degrees above the virtual cohomolo* *gical dimension of . The Yagita invariant p() of is a natural generalization of the p-per* *iod to groups with H*(; Fp) of Krull dimension larger than one. We compute the Yagita invaria* *nt p(g) for the mapping class group g with respect to an arbitrary odd regular prime p. Introduction The mapping class group g is defined to be the group of path components of t* *he group of orientation preserving diffeomorphisms of an oriented closed surface Sg of g* *enus g. In this paper, we study for a given prime p the Yagita invariant p(g), which is de* *fined as follows (see [Y] for the case of finite groups and [T] for more general groups). Let be a group of finite virtual cohomological dimension and ss any subgr* *oup of prime order p. It is well known that the image Im (Hk(; Z) ! Hk(ss; Z)) of * *the restriction map in cohomology is non-zero for some degree k > 0. Because the n* *atural map H*(ss; Z) ! H*(ss; Fp) maps onto Fp[u] H*(ss; Fp) with u a generator in H2* *(ss; Fp), there exists a maximum value m = m(ss) such that Im((H*(; Z) -! H*(ss; Fp)) Fp[um ] H*(ss; Fp) : It is easy to see (cf. Lemma 1.1) that the possible values m(ss) are bounded by* * a number depending on only. The Yagita invariant p() of with respect to the prime p is* * then defined to be the least common multiple of values 2m(ss), where ss ranges over * *all subgroups of order p of . We use the convention that p() = 1 if is p-torsion free. The i* *nvariant p() agrees with the p-period of a p-periodic group (i.e., a group with p-period* *ic Farrell cohomology groups, see [X1] and [X2] for a discussion of that concept) and, as * *it is the case for the p-period, the Yagita invariant p() divides 2(p - 1)pk, for some k * * 0 (see Section 1). The interest in p() stems from the fact that it provides a lower bound for t* *he dimension of a complex, which admits a certain type of action of (see [Y] for the case o* *f finite groups). For instance, one checks easily that if acts properly discontinuously* * on Rn x (Sm )k and trivially on H*(Rn x (Sm )k; Z), in a way that the stabilizer of any* * point Typeset by AM S-* *TEX 1 2 H. H. GLOVER, G. MISLIN AND Y. XIA x 2 Rn x (Sm )k is a p-torsion free group, then m + 1 is a multiple of the Yagi* *ta invariant p(). We will discuss a variation of this in Section 4. The mapping class group g is never 2-periodic for g > 1. For an odd prime p* * and p-periodic g we completely determined the p-period in [GMX]. We recall that for* * an odd prime p and genus g 6 1 mod p, g is always p-periodic; thus we will only need * *to be concerned with the case g 1 mod p in the sequel. We have a complete result in* * case p is an odd regular prime, and partial results for general primes (recall that a pri* *mepp_is called regular if p does not divide the class number of the cyclotomic field Q(exp(2ss* * -1 =p)); the smallest irregular prime is 37). It is convenient for what follows to employ th* *e following terminology. Definition. Let p be a prime. We say that an integer g satisfies the (p)-condit* *ion if and only if g is of the form lpff+ 1 with l prime to p, ff > 0, and 2l = p(2h - 2) * *+ k(p - 1) for some integers h > 0, k 0 with k 6= 1. Our main result is the following. Theorem 1. Let p be an odd regular prime and assume that g = lpff+ 1 with l pri* *me to p and ff > 0. Then the Yagita invariant p(g) is determined as follows. (i)If g does not satisfy the (p)-condition, then p(g) equals 2(p - 1)pff-1. (ii)If g satisfies the (p)-condition, then p(g) equals 2(p - 1)pff. For the case of a general odd prime, we have the following partial results, * *which underline the role of the (p)-condition. Theorem 2. Let p be an odd prime and g = lpff+ 1 with l prime to p and ff > 0. * *Then the following holds. (i)p(g) has the form 2(p - 1)pffor 2(p - 1)pff-1. (ii)If g satisfies the (p)-condition, then p(g) = 2(p - 1)pff. (iii)If 1 < 2l < p - 1 then p(g) = 2(p - 1)pff-1. Remark 1. For a fixed prime p and ff > 0 there are obviously only finitely many* * genera g of the form lpff+ 1 with l prime to p which do not satisfy the (p)-condition. T* *hus, we can think of (ii) in Theorem 2 as the generic case. Remark 2. If the Krull dimension of H*(g; Fp) equals one so that g is p-periodi* *c, and if we assume g of the form lpff+ 1 with l prime to p and ff > 0, then the (p)-c* *ondition does not hold for g. This can be seen by comparing the (p)-condition with the * *formula for the Krull dimension for g as stated in ([B]). Moreover, ff is then necessar* *ily equal to 1 (otherwise the Krull dimension is larger than 1 by [B]), and we recover th* *e formula p(g) = 2(p - 1) of [GMX] for that case. The finite set of values l prime to p f* *or which g = lp + 1 gives rise to a p-periodic g was determined in an explicit way in [X* *3], where it is proved that lp+1with l prime to p is p-periodic if and only if l + 1 is p* *rime to p and the interval [(2l + 3)=p; (2l + 2)=(p - 1)] does not contain an integer. The re* *ader can then easily check that the smallest genus g for which there is an odd prime such tha* *t the Yagita invariant is not given either by [GMX], Theorem 1 or Theorem 2 is g = 1296 = 35* * . 37 + 1. ON THE YAGITA INVARIANT OF MAPPING CLASS GROUPS 3 Remark 3. From Lemma 2.2 it will become obvious that the (p)-condition has a ge* *ometric interpretation as follows. If p denotes a prime and g = lpff+ 1 with l prime to* * p and ff > 0 then g satisfies the p-condition if and only if g contains a cyclic subgroup C * *of order pff+1 such that C lifts to an action on Sg with all stabilizers of order p; note als* *o that the C action on Sg cannot be free, since g - 1 = lpffis not divisible by pff+1. The remainder of this paper is organized as follows. In Section 1 we provide* * some basic facts about the Yagita invariant and establish rather crude lower and upper bou* *nds. In Section 2 we recall the definition of the fixed point data of an element of fin* *ite order in the mapping class group g and develop a technique of moves, which requires the * *prime in question to be regular. The moves are used in Section 3 to study representa* *tions of subgroups of order p of g on the space of holomorphic differentials of a Rieman* *n surface of genus g. As a result, we will obtain precise lower bounds for g. Using the a* *ction of g on a suitable submanifold of the Teichm"uller space of surfaces of genus g we e* *stablish in Section 4 an upper bound for g and use it to determine sharp upper bounds in Se* *ction 5, completing the proofs of the Theorems 1 and 2. We would like to thank R. Swan for providing the proof of Proposition 2.3. Section 1 : Some basic facts concerning the Yagita invariant Our first lemma provides an upper bound for the Yagita invariant p() . Lemma 1.1. Let p be a prime and a group of finite virtual cohomological dimens* *ion, which has p-torsion. Let G denote a finite factor group of such that the kerne* *l of the projection ! G is p-torsion free. Then the Yagita invariant p() divides 2(p - * *1)pk-1, where pk denotes the largest power of p which divides the order of G. Proof. We consider the regular representation ae : G ! Gl|G|(C) of the finite g* *roup G. If we restrict ae to a subgroup ss of order p in G, then ae|ss is of the form spk-* *1oe, where s is prime to p and oe denotes the regular representation of ss. Since the total Che* *rn class c(oe) in H*(ss; Z) has the form 1 + cp-1(oe) with cp-1(oe) 6= 0, we see that c(p-1)pk* *-1(ae) restricts to scp-1(oe)pk-1, which is non-trivial in the integral cohomology of ss. Theref* *ore, the Yagita invariant of the factor group G must divide 2(p - 1)pk-1. But every subgroup of* * order p in g injects into G via the projection, and therefore viewing ae as a represent* *ation "aeof , its Chern class c(p-1)pk-1("ae)will restrict non-trivially to any subgroup of* * order p in , showing that p() divides 2(p - 1)pk-1 too. It is often possible to improve on the power of p in the upper bound of p() * *as follows. Lemma 1.2. Let p be a prime, a group of finite virtual cohomological dimension* *, and assume that ae : ! Gld(C) is a representation of degree d such that ae does no* *t have any element of order p in the kernel. (i)If d < pm then p() divides 2(p - 1)pm-1 . (ii)If ae() Gld(Q) and d < (p - 1)pm then p() divides 2(p - 1)pm-1 . Proof. We know already that p() divides 2(p - 1)pn for some n. If we were not a* *ble to choose n = m - 1, then there would exist a subgroup ss such that the restrict* *ion map H*(; Z) ! H*(ss; Z) is zero for 0 < * < 2pm . But, assuming d < pm , one would* * infer 4 H. H. GLOVER, G. MISLIN AND Y. XIA for the total Chern class c(ae|ss) equals 1. This is a contradiction, since ae * *is faithful when restricted to ss . In case ae() Gld(Q) , all non-zero Chern classes of ae|ss l* *ie in degrees of the form 2(p - 1)pl, see ([EM]), and thus it suffices to assume d < (p - 1)pm * *to conclude that some Chern class of ae|ss is non-zero in the range 0 < * < 2pm . Applying this to the case of the mapping class group g we obtain the followi* *ng upper bound for the Yagita invariant. Lemma 1.3. Let p be an arbitrary prime and 0 < 2g < (p - 1)pm . Then p(g) divid* *es 2(p - 1)pm-1 . Proof. We consider the natural action of g on H1(Sg; Q) which defines a represe* *ntation ae : g ! Gl2g(Q) with torsion-free kernel. The result then follows from (ii) of* * Lemma 1.2 . It is plain from the definition that the Yagita invariant for a subgroup of * * divides p(). We can therefore find lower bounds for p() by looking at suitable subgroups of * *. This leads to the following useful lemma. Lemma 1.4. Let ss be a cyclic subgroup of p-power order of , with p an odd prim* *e. If we denote by C(ss) (respectively N(ss)) the centralizer (respectively normalize* *r) of ss in then p(g) is a multiple of 2[N(ss) : C(ss)]. Proof. Let W be the cyclic group N(ss)=C(ss). The image of the restriction map * *in coho- mology H*(; Z) ! H*(ss; Z) maps into the subring of W -invariant elements, whic* *h is of the form Z[x]=(pkx; xm ), where x 2 H2(ss; Z) denotes a generator, m = |W | the* * order of W , and pk the order of ss. The result then follows readily. As an application of Lemma 1.4, we deduce the lower bound (i) of Theorem 2 o* *f the introduction. Lemma 1.5. Let p be an odd prime and assume that g is of the form lpff+ 1 with * *l prime to p and ff > 0. Then p(g) equals 2(p - 1)pfifor some fi ff - 1. Proof. We know from Lemma 1.1 that p(g) divides 2(p-1)pn for some n 0. If g = * *lpff+1 with l prime to p, the surface Sg is a pff-fold regular (unbranched) covering s* *pace of Sh where h = l+1, with cyclic covering transformation group generated by a map f :* * Sg ! Sg. Since f acts freely, it is conjugate in Diffeo+(Sg) to fj for any j prime to p * *([N], see also Lemma 2.1 below). The subgroup g generated by the image of f in g has order pffand N()=C() is isomorphic to the full automorphism group of ~=Z=pffwhich has order (p-1)pff-1(see also Lemma 2.1 below). Therefore, using Lemma 1.1 and Lemm* *a 1.4, the result follows. Comparing the lower and upper bounds for the Yagita invariant of the mapping* * class group, we obtain part (iii) of Theorem 2 of the Introduction. Corollary 1.6. Let p be an odd prime and g = lpff+ 1 with 1 < 2l < p - 1 and ff* * > 0. Then the Yagita invariant p(g) equals 2(p - 1)pff-1. Proof. Since 2g = 2lpff+ 2 (p - 2)pff+ 2 < (p - 1)pffwe infer from Lemma 1.3 t* *hat p(g) divides 2(p - 1)pff-1. On the other hand Lemma 1.5 shows that p(g) is a mu* *ltiple of 2(p - 1)pff-1. ON THE YAGITA INVARIANT OF MAPPING CLASS GROUPS 5 Section 2 : Fixed point data and moves A basic invariant of an orientation preserving diffeomorphism f of Sg of per* *iod n > 1 is its fixed point data. It is defined as follows. Since f preserves orientation, * *the singular set of f is necessarily discrete in Sg. Let {xi} be a set of representatives of the* * singular orbits of f and write ffi for the order of stabf(xi), the stabilizer of f at xi 2 Sg. * * Then fn=ffi generates stabf(xi) and, with respect to a fixed Riemannian structure, the diff* *erential of f acts faithfully by rotation on the tangent space at xi. Let fii be an intege* *r such that ffiin=ffiacts by rotation through 2ss=ffi. The number fii is well defined modul* *o ffi, and fii is prime to ffi. The fixed point data of f, denoted ffi(f), is then the collect* *ion ffi(f) =< g; n | fi1=ff1; : :;:fiq=ffq > where g is the genus of the surface Sg, n the order of f, and q the number of s* *ingular orbits of the f-action; the numbers fi1=ff1; : :;:fiq=ffq are unique up to order, if w* *e consider them as elements in Q=Z. A classical theorem of Nielsen [N] states that two diffeomorphisms of finite* * order are conjugate in Diffeo+(Sg) if and only if they have the same fixed point data. Sy* *monds [Sy] proved that the fixed point data of a diffeomorphism of finite order depends on* *ly upon its isotopy class, and thus is well defined for an element of finite order of the m* *apping class group g; we will thus write ffi(x) for the fixed point data of an element of fi* *nite order x 2 g. He also shows that the Nielsen Theorem is still true for the mapping cla* *ss group g, that is, two elements of finite order in g are conjugate if and only if they* * have the same fixed point data. As an immediate consequence, one can deduce the followin* *g. Lemma 2.1. Suppose that f 2 Diffeo+(Sg) has finite order n and acts freely on S* *g. Denote by x the image of f in g. Then the index [N(x) : C(x)] of the centralize* *r of x in its normalizer equals OE(n), OE the Euler function. Proof. Indeed, all the generators of the cyclic group < x > generated by x have* * the same fixed point data ffi =< g; n| > and are therefore conjugate in g. Thus N(x) map* *s onto the automorphism group of < x >, with kernel C(x), and the result follows. The well-known techniques on realizing fixed point data lead to the followin* *g result, of which we sketch the proof for the convenience of the reader. Lemma 2.2. Let p be a prime and fi1; : :;:fiq integers prime to p. Then the col* *lection < g; pt| fi1=pt1; : :;:fiq=ptq> can be realized as the fixed point data of an element of order pt in g if and o* *nly if the following three conditions are satisfied. P (i) fii=ptiis an integer . P (ii)The Riemann-Hurwitz formula 2g - 2 = pt(2h - 2) + pt (1 - 1=pti)holds * *for some h 0. (iii)If h = 0 in (ii), then t = max(t1; : :;:tq). 6 H. H. GLOVER, G. MISLIN AND Y. XIA Proof. Suppose the given collection is the fixed point data of some x 2 g, repr* *esented by the diffeomorphism f of Sg of order pt. Then there is a branched covering Sg ! * *Sh with set of branch points {y1; : :;:yq} Sh and group Z=pt, giving rise to a regular* * covering Sg \ ss-1 {y1; : :;:yq} -! Sh \ {y1; : :;:yq}; which induces a short exact sequence ss1(Sg \ ss-1 {y1; : :;:yq}) -! ss1(Sh \ {y1; : :;:yq}) -@!< f >~=Z=* *pt: Note that Yh ss1(Sh \ {y1; : :;:yq}) =< a1; b1; : :;:ah; bh; x1; : :;:xq| [ai; bi]x* *1 : :x:q= 1 > i=1 and @(xi) = ffiipt-tiforQ1 i q. Therefore (i) is true since the map @ preser* *ves the relation hi=1[ai; bi]x1 : :x:q= 1. By calculating the Euler characterist* *ics of Sg \ ss-1 {y1; : :;:yq} and Sh \ {y1; : :;:yq} one gets (ii), and (iii) follows from* * the surjectivity of the map @. Conversely, given (i), (ii) and (iii) we begin by constructing a* * surjective homomorphism @ : ss1(Sh \ {y1; : :;:yq}) -! Z=pt such that @(xi) is a suitable element of order ptifor 1 i q. This can be done* *, if h > 0, by putting @(a1) = @(b1) = 1, @(ai) = @(bi) = 0 for 2 i h, and @(xj) = fijpt-* *tjfor 1 j q. In case h = 0, we still put @(xj) = fijpt-tjfor 1 j q. The condition* * (iii) then guarantees that the map @ is surjective. The kernel of @ defines a pt-shee* *ted regular covering Sg\{z1; : :;:zr} -! Sh\{y1; : :;:yq}, giving rise to a branched coveri* *ng Sg -! Sh, with covering transformation group generated by a diffeomorphism with the desir* *ed fixed point data. Note that given an element x 2 g of order pt, one can determine the fixed po* *int data ffi(xk) from ffi(x) =< g; pt|fi1=pt1; : :f:i1=ptq> as follows. When k is prime * *to p then ffi(xk) =< g; pt| lfi1=pt1; : :;:lfiq=ptq> where l is a multiplicative inverse of k mod pt . When k is a multiple of p, s* *ay k = mps, with m prime to p, the subgroup generated by k in Z=pt is naturally isomorphic * *to Z=pt-s by mapping k = mps to m in Z=pt-s. If we write n for a multiplicative inverse * *of m mod pt-s then s t-s ffi(xmp ) =< g; p | A1;1; : :;:A1;m1; : :;:Aq;1; : :;:Aq;mq> ; 1 i q * *; 1 j mi; where the Ai;jcorrespond to fii in ffi(x), mi = pmin(s;t-ti), and Ai;j= nfii=pm* *in(t-s;ti)for 1 j mi. ON THE YAGITA INVARIANT OF MAPPING CLASS GROUPS 7 For example, if ffi(x) =< 183; 81 | 11=81; 14=81; 1=27; 1=9; 1=3 > then ffi(x41) =< 183; 81 | 22=81; 28=81; 2=27; 2=9; 2=3 > ffi(x3) =< 183; 27 | 11=27; 14=27; 1=27; 1=27; 1=27; 1=9; 1=9; 1=9; 1=3; * *1=3; 1=3 > ffi(x9) =< 183; 9 | 2=9; 5=9; 1=9; 1=9; 1=9; 1=9; 1=9; 1=9; 1=9; 1=9; 1=9; 1=9; 1=9; 1=9; 1=3; 1=3; 1=3; 1=3; 1=3; 1=3; 1=3; 1=3; 1=* *3 > and so on. The following proposition, which will be used repeatedly later on, was sugge* *sted to us by an explicit computation with the help of a computer program. The actual * *proof as presented below was communicated to us by R. Swan ([Sw]). For an integer n a* *nd a fixed prime number p we will use the notation "nto denote the unique integer sa* *tisfying 0 "n< p and n "nmod p. We will also write < r > for the fractional part of a* * number r 2 Q, so that r = [r]+ < r >, with [r] the integral part of r. Note that for a* *ny n 2 Z one has "n= p < n=p >= n - p[n=p]. Proposition 2.3. Let p be an odd prime and consider the integral (p - 1)=2 x (p* * - 1)=2 matrix A = (aij) whose (i; j) entry aijis defined to be p if i = 1, and eijif 2* * i (p-1)=2. Then the matrix A is non-singular. Moreover, the entries of the matrix pA-1 are* * rational numbers which have, in reduced form, no p in the denominators if and only if p * *is a regular prime. Proof [Sw]. For p 5 the claim is easily checked directly. If p > 5 we proceed * *as follows. By subtracting each column of A from the next one we get a matrix B which, after s* *ubtracting the second column of B from the first one, takes the form 0 p 0 0 : : : 01 BB0 2 2 : : : 2CC C = BB0 3 CC: @ ... ... ai;j- ai;j-1 A 0 (p - 1)=2 Now ai;j- ai;j-1= ij - p[ij=p] - i(j - 1) + p[i(j - 1)=p] = i - pei;j, where ei;j= [ij=p] - [i(j - 1)=p] ; 3 i; j (p - 1)=2 are the entries of a (p - 5)=2 x (p - 5)=2 matrix E. Subtracting the second col* *umn of C from the rest reduces C to 0 p 0 0 : : : 01 BB0 2 0 : : : 0CC D = BB0 3 CC @ ... ... -pE A 0 (p - 1)=2 8 H. H. GLOVER, G. MISLIN AND Y. XIA with 0 1 p-1 0 0 : : : 0 BB 0 2-1 0 : : : 0 C B 0 f3;2 CC D-1 = BBB0 f4;2 CC: B@ .. . CC . .. -_1pE-1 A 0 f(p-1)=2;2 The last (p - 5)=2 entries of the second column of D-1 are given by 0 f3;2 1 0 3 1 BB f4;2 CC _1_ -1 BB 4 CC @ ... A = -2p E @ ... A f(p-1)=2;2 (p - 1)=2 and pD-1 has no p in the denominators if and only if detE is prime to p.p By_[C* *O], detE = h1, where h1 is the first factor of the class number of Q(exp 2ss -1 =p* *). But Kummer showed p is regular if and only if p does not divide h1. The following lemma is immediate and we state it without proof. Lemma 2.4. Suppose we are given a prime p and an integral linear system (L) Ax = b such that (i)A 2 Mn(Z) has non-zero determinant, (ii)b 0 mod p2, (iii)pA-1 in reduced form has the property that the matrix entries have no p* * in their denominators. If x denotes an integral solution of (L), then x 0 mod p. As a consequence we will prove the following proposition, which plays a cruc* *ial role in the sequel. Proposition 2.5. Let p be a regular prime and suppose we are given an integer n* * > 0 divisible by p and n integers fii with 0 < fii< p, where 1 i n. Write ffj for* * the number of fii's which equal j for 1 j < p. If the fractional parts < jfii=p > satisfy X (1) < jfii=p > 0 mod p; 1 j < p i then ffj ffp-j mod p for all j. Proof. The case p = 2 is trivial, so we will assume p 3. One can then rewrite * *each sum in (1) in terms of the ffi's as follows: X X e1 = < fii=p >= ffi< i=p > 0 mod p i i ON THE YAGITA INVARIANT OF MAPPING CLASS GROUPS 9 X X e2 = < 2fii=p >= ffi< 2i=p > 0 mod p i i : :::::::::: : X X ep-1 = < (p - 1)fii=p >= ffi< (p - 1)i=p > 0 mod p : i i For (p + 1)=2 i p - 1 we have 2 < i=p > - < 2i=p >= 1, and the first two equa* *tions yield p-1X ffi= 2e1 - e2 : i=(p+1)=2 P p-1 Noting that i=1 ffi = n = kp ; we obtain an equation which we consider as the* * first equation of a linear system (L) of the type Ax = b considered in Lemma 2.4, wit* *h the transpose of x the vector (x1; : :;:x(p-1)=2) = (ff1 - ffp-1; : :;:ff(p-1)=2- ff(p+1)=2) namely the equation px1 + px2 + . .+.px(p-1)=2= p(n - 4e1 + 2e2) : Observe that the right hand side is 0 mod p2. By using the fact that < (p - j* *)i=p >= 1- < ji=p >, we obtain (p - 3)=2 additional equations for our system (L), which* * take the form p-1X p < 2=p > x1 + p < 4=p > x2 . .+.p < (p - 1)=p > x(p-1)=2= p(e2 - f* *fi) i=(p+1)=2 p-1X p < 3=p > x1 + p < 6=p > x2 + . .+.p < 3(p - 1)=2p > x(p-1)=2= p(e3 - * * ffi) i=(p+1)* *=2 : ::::::: : p-1X p < (p - 1)=2p > x1 + . .+.p < (p - 1)2=4p > x(p-1)=2= p(e(p-1)=2- f* *fi) i=(p+1)=2 It is easy to see that the right hand sides ofPall equations are 0 mod p2 sinc* *e by assumption all ei's are 0 mod p and, as we have seen, p-1i=(p+1)=2ffi 0 mod p. Observi* *ng that for any integer k one has k p < k=p > mod p, we see that the matrix A of the lin* *ear system (L) is precisely the matrix A of Proposition 2.3. Thus, by applying Lemm* *a 2.4, we infer that ffj - ffp-j 0 mod p for 1 j p - 1, completing the proof. The following theorem can be viewed as a purely algebraic version of our "mo* *ves" concerning fixed point data as considered in the next section. 10 H. H. GLOVER, G. MISLIN AND Y. XIA Theorem 2.6. Suppose p is an odd regular prime and n > 0 an integer which is di* *visible by p. Suppose the system of p - 1 equations Xn < jxi=p > 0 mod p; 1 j p - 1 i=1 has an integral solution x = (x1; : :;:xn) with 0 < xi < p for all i. Then it * *also has an integral solution z = (z1; : :;:zn) with 0 < zi< p for allPi such thatPfor any * *given integer j the number of zi's which equal j is divisible by p, and i< jxi=p >= i< jzi=* *p > for all j. Proof. Let x = (x1; : :;:xn) be a solution as above, and denote by ffj(x) the n* *umber of xi's which are equal to j, where 0 < j < p. Then we can rewrite our system of e* *quations as p-1X ffi(x) < ji=p > 0 mod p ; 1 j p - 1 i=1 so that for all j one has ffj(x) ffp-j(x) mod p by Proposition 2.5. We can no* *w alter the solution x = (x1; : :;:xn) to y = (y1; : :;:yn) by applying a move of type * *(s; t), where 1 s; t p - 1 and s 6= t, by which we mean the following : (i)replace ffs(x) by ffs(y) = ffs(x) + 1 (ii)replace ffp-s(x) by ffp-s(y) = ffp-s(x) + 1 (iii)replace fft(x) by fft(y) = fft(x) - 1 (iv)replace ffp-t(x) by ffp-t(y) = ffp-t(x) - 1 . Note that for such a move to be possible we need to have a t such that fft(x) >* * 0. If we alter x accordingly into y, we obtain a new system of equations satisfying p-1X p-1X ffi(y) < ij=p >= ffi(x) < ij=p > i=1 i=1 for 0 < j < p, because < ij=p > + < (p - i)j=p >= 1. Suppose now that not all f* *fi(x)'s are already divisible by p. Then, using Proposition 2.5 and writing n as kp, we* * have p-1X (p-1)=2X n = kp = ffi(x) 2 ffi(x) 0 mod p; i=1 i=1 and we conclude that there must exist a pair (s; t) with 1 s; t (p - 1)=2 and* * s 6= t such that ffs(x) and fft(x) are both not divisible by p. Performing a move of type * *(s; t) will provide a new solution y. If ffs(y) and fft(y) are both still not divisible by * *p, we repeat the move of type (s; t), and eventually ffs or fft will be a multiple of p. By * *continuing in this manner, we will end up with the required solution z. ON THE YAGITA INVARIANT OF MAPPING CLASS GROUPS 11 Section 3 : Chern classes of representations on holomorphic differentials The action of g on the symplectic space H1(Sg; R) with its intersection pair* *ing gives rise to a representation ae : g ! Sp(2g; R), which we call the canonical repres* *entation of g in the sequel. Recall that H*(BSp(2g; R); Z) ~=Z[d1; : :;:dg] where the di's are such that they restrict to the universal Chern classes of a * *maximal compact subgroup U(g) Sp(2g; R). The images of the di's under the induced map ae* : H*(BSp(2g; R); Z) -! H*(g; Z) gives rise to symplectic characteristic classes di(ae) := ae*(di) 2 H2i(g; Z); 0 i g : We are going to study the behavior of the restriction of these symplectic chara* *cteristic classes to cyclic subgroups of order pn in g. Note that ae| factors, up to con* *jugation, through U(g) Sp(2g; R) and we can view the classes di(ae|) as the Chern classe* *s ci(ae) of a representation ae: ! U(g). This representation can be thought of in the f* *ollowing way. One chooses a lift of a generator of to an element of order pn in Diffeo+* *(Sg) and chooses a complex structure on Sg compatible with the orientation of Sg such th* *at f acts by a holomorphic automorphism. Then f acts on the associated space of holomorp* *hic differentials, which is a complex vector space of complex dimension g. This act* *ion defines a representation of , whose dual is equivalent to the representation ae: ! U(G) introduced above. We will sometimes, by abuse of language, refer to aeas the re* *presentation of on holomorphic differentials of Sg. We will make use of the following basic property of Chern classes of represe* *ntations. Lemma 3.1. Let ae : Z=pn ! U(g)nbe a representation of Z=pn, p a fixed prime. A* *ssume that ae can be decomposed as pi-1=0ni!i where ! denotes a faithful one dimensio* *nal repre- sentation and where ni= nj for all (i; j) which satisfy 0 < i; j < pn and i j * *6 0 mod p. Then the following are equivalent. (i)The Chern classes ci(ae) 2 H2i(Z=pn; Z) are all 0 mod p for 1 i pn - * *1, (ii)ni 0 mod p for 0 < i < p. Proof. We first prove that (i) implies (ii). It is convenient for this proof to* * assume that ae satisfies in addition n0 = np = . .=.npn-p; this will not change the mod p Ch* *ern classes of ae. Consider the ring homomorphism OE : H*(Z=pn; Z) ! (Z=pn)[x] Q* * pn-1 given by mapping c1(!) 2 H2(Z=pn;QZ) to x. Let f(x) 2 (Z=pn)[x] be defined by * * i=0 (1+ ix)ni andQF (x) 2 (Z=p)[x] byQ p-1i=0(1 + ix)ni. Notice that OE(c(ae)) = f(x).* * We then have f(x) p-1k=0(1 + kx)nkpn-1 p-1k=0(1 + kxpn-1)nk mod p, thus f(x) F (xpn-1* *) mod p. 12 H. H. GLOVER, G. MISLIN AND Y. XIA Now, thenassumption that ci(ae) 0 mod(p) for 1 i pn-1 implies that f(x) 1+p* *g(x) mod xp , and therefore, F (x) 1 mod xp. So, by Lemma 4.1 of [GMX], we concl* *ude that nk 0 mod p for all k satisfying 1 k p - 1. It remains to check that (* *ii) implies (i). If (ii) holds, we can write ae in the form pkoe + o, where oe deno* *tes the reduced regular representation of Z=pn, and o a representation which factors through Z=* *pn-1 so that c(o) 1 mod p. Since c(oe) = 1 + c(p-1)pn-1(oe) we see that c(ae) = c(oe)pk . c(o) (1 + c(p-1)pn-1(oe)p)k mod p ; so that ci(ae) 0 mod p for 0 < i < (p - 1)pn. Our theorem on algebraic moves leads to the following result, obtained by a * *geometric version of "moves" of fixed point data. It will provide a proof of case (i) of * *Theorem 2 of the introduction (see Corollary 3.3), and it will also provide us with an impro* *ved lower bound for g, needed for the proof of case (ii) of Theorem 2. Proposition 3.2. Let p denote an odd regular prime and ae the canonical represe* *ntation of g in Sp(2g; R). Assume g = lpff+ 1 with l prime to p and ff > 0, and suppose* * g does not satisfy the (p)-condition. Then for every subgroup ss g of order p, at le* *ast one characteristic class di(ae|ss) 2 H2i(ss; Z) is non-zero in the range 1 i pff-* * 1. Proof. Suppose there is a subgroup ss of order p in g such that di(ae|ss) = 0 f* *or 1 i pff-1. We will show that this leads to a cyclic subgroup of g of order pff+1, w* *hich acts on Sg with stabilizers of order p. The Riemann-Hurwitz equation associated with t* *his action yields then an equation contradicting the assumtion concerning the (p)-conditio* *n. The construction of such a subgroup is by induction on its order. First, we begin b* *y modifying ss slightly, not changing the Chern classes, using our technique of "moves" as * *follows. Let ae: ss ! U(g) be the representation of ss on the space of holomorphic different* *ials on Sg (with respect to a suitable complex structure) and write aeas p-1i=0ni!i, ! * *a faithful irreducible one dimensional representation of ss. Thus we have di(ae|ss) = ci(* *ae) = 0 for 1 i p-1. Applying Lemma 3.1, we infer ni 0 mod p for 1 i p-1. Let x 2 ss be a generator and denote by < g; p | fi1=p; : :;:fin=p > the fixed point data of * *x, normalized such that 0 < fii < p. As we discussed in Proposition (4.3) of [GMX], see also * *[FK], one has then Xn (2) nj = h - 1 + n - < jfii=p > ; 0 < j < p ; i=1 where h denotes the genus of Sg=ss. Note that the Riemann-Hurwitz equation 2g -* * 2 = p(2h - 2) + n(pP- 1) shows that n 0 mod p and, as argued in [GMX], by conside* *ring nj- np-j = 2 i< jfii=p > +n it follows, because for 0 < j < p each nj is divi* *sible by p, that n X < jfii=p > 0 mod p ; 0 < j < p : i=1 Equation (2) then implies that h - 1 0 mod p, which in particular shows that * *the orbit genus h is greater than 0. By applying Theorem 2.6 and Lemma 2.2, there * *are n ON THE YAGITA INVARIANT OF MAPPING CLASS GROUPS 13 integers fli with 0 < fli< p, forming the fixed point data < g; p|fl1; : :;:fln* * > of a generator of a subgroup "ssof order p in g, which gives rise to a representation "ss! U(g* *) with multiplicities ni of the one dimensional components still given by (2), but suc* *h that the number of fli's equal to a fixed j is always divisible by p. By reordering the * *fl's, we may assume that fls flt for s < t, and we can then consider (3) ffi =< g; p2|fl1=p; flp+1=p; : :;:fl(k-1)p+1=p > where n = kp. Note that k-1X Xn flip+1=p= fii=p2 2 Z : i=0 i=1 Also, with h - 1 = p(s - 1) and n = pk, we get (4) 2g - 2 = p(2h - 2) + n(p - 1) = p2(2s - 2) + kp(p - 1) so that ffi is the fixed point data of an elementPof order p2 in g by Lemma 2.2* *. Fur- thermore, s > 0 since h > 0, and k 6= 1, because k-1i=0flip+1=p is an integer* *. Thus, if ff = 1 equation (4) implies that g satisfies the (p)-condition, a contradiction* *, and we are done. If ff 2, we repeat our construction in the following manner. Let ss(2) d* *enote the subgroup of g generated by an element whose fixed point data is given by equati* *on (3). By construction, the associated representation ae(2) : ss(2) ! U(g) on holomorp* *hic differ- entials (with, perhaps, a different complex structure on Sg), is such that agai* *n ci(ae(2)) 0 2-1 mod p for 0 < i < pff. Decomposing ae(2) as pi=0ni!i, with ! faithful one dime* *nsional, yields equations of the form k(2)X (5) nj = h(2) - 1 + k(2) - < jfii(2)=p > 0 mod p ; 1 < j < p ; i=1 with h(2) the genus of Sg=ss(2), fii(2) = fl(i-1)p+1, and k(2) = k as before. * *Also (see [FK])), one has nj = nj+p if 0 < j < j + p < p2 and j prime to p, so that by Le* *mma 3.1 ni 0 mod p for 0 < i < p. The Riemann-Hurwitz equation for the ss(2) action * *is 2g - 2 = p2(2h(2) - 2) + k(2)p(p - 1) and, as ffP 2, k(2) 0 mod p; in particu* *lar, k(2) 6= 1. Again, by considering nj - np-j = 2 k(2)i=1< jfii(2)=p > +k(2) 0 * * mod p, we see that k(2)X < jfii(2)=p > 0 mod p ; 0 < j < p ; i=1 and thus h(2) 1 mod p from equation (5). This permits us to find k(2)=p = k(3* *) integers ffi1; : :;:ffik(3); 1 < ffii< p, so that k(3)X k(2)X ffii=p = fii(2)=p2 2 Z i=1 i=1 14 H. H. GLOVER, G. MISLIN AND Y. XIA as well as (6) 2g - 2 = p3(2s(2) - 2) + k(3)p2(p - 1) P k(3) As h(2) 6= 0 we have s(2) > 0, and k(3) 6= 1 because i=1 ffii=p 2 Z. If ff = * *2, equation (6) shows that g must satisfy the (p)-condition, a contradiction. If ff > 2 we* * continue our construction. If g does not satisfies the (p)-condition, we will eventuall* *y arrive at a contradiction, finishing the proof. But we also note that if g satisfies the (* *p)-condition, we end up with a cyclic subgroup ss(ff + 1) g of order pff+1, which is generat* *ed by an element with fixed point data < g; pff+1|ffl1=p; : :;:fflw =p > and associated Riemann-Hurwitz equation of the form 2g - 2 = pff+1(2s(ff) - 2) + k(ff + 1)pff(p - 1) with the property that on pffss(ff+1), the subgroup of order p of ss(ff+1), the* * characteristic classes di(ae|pffss(ff + 1))agree with those associated with the original group* * ss; in particular, they vanish in the range 0 < i < pff. As an application, we can now prove part (i) of Theorem 1. Corollary 3.3. Let p be an odd regular prime and assume g = lpff+ 1 with l prim* *e to p and ff > 0. If g does not satisfy the (p)-condition then p(g) = 2(p - 1)pff-1. Proof. We know from Lemma 1.5 that p(g) has the form 2(p - 1)pfifor some fi ff* * - 1. By Proposition 3.2. we can find for every subgroup ss g of order p a non-zero * *element di 2 H2i(ss; Z) for some i > 0 of the form mpflwith m prime to p and fl < ff, w* *hich lies in the image of the restriction map H*(g; Z) ! H*(ss; Z). Thus the largest powe* *r of p dividing p(g) must be less than ff and the result follows. This proof of Proposition 3.2 reveals, as a by-product, the following. Corollary 3.4. Let p be an odd regular prime and g = lpff+ 1 with l prime to p * *and ff > 0. If g contains a subgroup ss order p with di(ae|ss) = 0 for 0 < i < pffa* *nd if g satisfies the (p)-condition, then g contains a cyclic subgroup of order pff+1with associa* *ted action on Sg having stabilizers of order p. The converse of that Corollary is also true, by a simple construction; it do* *es not need any regularity condition on the prime involved. Corollary 3.5. Let p be an arbitrary odd prime and assume g = lpff+ 1 with l pr* *ime to p and ff > 0. If g satisfies the (p)-condition then g contains a cyclic subgrou* *p ss(ff + 1) of order pff+1generated by an element with fixed point data of the form < g; pff+1|fi1=p; : :;:fik=p > : ON THE YAGITA INVARIANT OF MAPPING CLASS GROUPS 15 Moreover, the subgroup ss = pffss(ff + 1) of order p satisfies di(ae|ss) = 0 fo* *r 0 < i < pff. Proof. If g satisfies the (p)-condition, we have integers h > 0 and k 0 with k* * 6= 1 satisfying 2g - 2 = pff+1(2h - 2) + pffk(p - 1). Therefore, by Lemma 2.2, Sg a* *dmits a diffeomorphism of order pff+1with exactly k singular orbits, and fixed point da* *ta < g; pff+1|fi1=p; : :;:fik=p > P where the fii's satisfy 0 < fii< p and are chosen in such a way that ifii=p i* *s an integer; this is possible since k 6= 1. If ss(ff + 1) denotes the corresponding subgroup* * in g then its action on holomorphic 1-forms will be given by a representation of the form ff+1-1 i pi=0 ni! ; with ! one dimensional and faithful, such that ni = ni+p for any i prime to p s* *uch that 0 < i < i + p < pff+1; this follows easily from the general formula concerning * *the ni's, as presented for instance in [FK], by noting that the only nontrivial stabilizer o* *ccurring for the action on Sg is of order p. We can now apply Lemma 3.1 to the classes di(ae* *|ss(ff + 1)) and infer that di(ae|ss) = 0 for 0 < i < pff. This Corollary leads to the following lower bound for p(g), which will be us* *ed in the proof of (ii) of Theorems 1 and 2, presented in Section 5. Corollary 3.6. Let p be an odd prime and suppose g = lpff+ 1 with l prime to p * *and ff > 0. If g satisfies the (p)-condition then p(g) is a multiple of 2(p - 1)pff. Proof. By Corollary 3.5 we can find a cyclic subgroup ss(ff + 1) in g of order * *pff+1given by an element x with fixed point data ffi(x) =< g; pff+1|fi1=p; : :;:fik=p > for some fii with 0 < fii < p. From our discussion on fixed point data we see * *that ffi(xj) = ffi(x) if j 1 mod p. But this implies that pffdivides [N(x) : C(x)]* *. Applying Lemma 1.3 we see that p(g) is a multiple of pffand, using Lemma 1.5, it follows* * that p(g) is actually a multiple of 2(p - 1)pff. Section 4 : The action on Teichm"uller space Let Sg be a closed oriented surface of genus g > 1. It is classical that the* * Teichm"uller space Tg of Sg is homeomorphic to R6g-6, and Tg admits a complex structure such* * that g acts on Tg properly discontinuously by holomorphic automorphisms. For a fixed s* *ubgroup ss g of order p, let {ssi}i2J be the set of all order p subgroups which are co* *njugate to ss in g. This is a countable set (in general infinite) so that we may assume 0 2 J* * N with ss0 = ss. The fixed point sets Fi = (Tg)ssi Tg are closed submanifolds homeomor* *phic to R6h-6+2n where h denotes the genus of the surface Sg=ss and n the number of fix* *ed points of the ss-action on Sg, see Proposition 2.1 of [GMX] (strictly speaking, it is * *not ss which acts on Sg, but some lift of ss to Diffeo+(Sg)). We may choose a triangulation * *of Tg, by triangulating the algebraic variety Tg=g, such that g acts simplicially. Thus e* *ach Fi is 16 H. H. GLOVER, G. MISLIN AND Y. XIA a subcomplex of Tg. We now choose a subset I J containingT0 such that for i; * *j 2 I with iT6= j one has Fi 6= Fj.S Put Xi = Tg \ Fi and X1 = Xi = X0 \ X>0, where X>0 = i>0Xi. Note that F = Fi is closed in Tg, since it is a subcomplex, an* *d X1 is therefore a (connected) open submanifold of Tg, on which g acts, with stabil* *izers not containing any subgroup conjugate to ss. One checks easily using the Riemann-H* *urwitz equation that the only case for which F = Tg (and thus X1 an empty space) is t* *he case of ss being generated by the hyperelliptic involution, acting on a genus 2 surf* *ace; in our applications we are only interested in actions of groups of odd order and thus * *X1 will not be empty. We will keep the notation introduced here through the entire section. S Lemma 4.1. Let ss g be a subgoup of prime order p and F = Fi Tg the subspace of points fixed by some conjugate of ss. In case g = 2 assume that ss is not ge* *nerated by the hyperelliptic involution. Denote by h be the genus of Sg=ss and n the numbe* *r of fixed points of the ss-action on Sg. Then the singular cohomology with compact suppor* *ts of F satisfies H6h-6+2ncpt(F ; Z) ~=i2IZ ; and Hkcpt(F ; Z) = 0 for k > 6h - 6 + 2n. ` Proof. Consider the natural map : Fi! F . Since the action of g on Tg is pro* *per, no f 2 F lies in infinitely many Fi's, and therefore the map is proper and induce* *s a map in cohomology with compact supports. Let Sing(F ) F be the subcomplex of points w* *hich lie in more than one Fi. Since for i 6= j in I the complex submanifold Fi\Fj is* * empty or has at least (real) codimension 2 in Fi, Sing(F)is empty, or is a subcomplex of F o* *f codimension at least 2, and it follows that is a H*cpt-isomorphism for * 6h - 6 + 2n = di* *m(F ). Since each Fi is homeomorphic to R6h-6+2n the result follows. Lemma 4.2. Let F Tg be as in Lemma 4.1 and put X1 = Tg \ F . Then X1 is a connected manifold of dimension 6g - 6 with singular cohomology satisfying Y H6(g-h)-2n-1(X1 ; Z) ~= Z ; i2I and Hk(X1 ; Z) = 0 for 0 < k < 6(g - h) - 2n - 1. Furthermore, the natural map : X1 ! X0 x X>0 induces an isomorphism Hk(X0 x X>0; Z) ! Hk(X1 ; Z) for 0 < k 6(g - h) - 2n - 1. Proof. Since F is closed in Tg we get by Alexander Duality Hk(X1 ; Z) ~=H(6g-6)-kcpt(Tg; F ; Z) and, since Tg ~=R6g-6 and using Lemma 4.1, H6g-6-kcpt(Tg; F ; Z) = 0 for 0 < k * *< 6(g - h) - 2n - 1. Moreover, for k = 6(g - h) - 2n - 1 we have H6(g-h)-2n-1(X1 ; Z) ~=H6h-6+2n+1cpt(Tg; F ; Z) ~=H6h-6+2ncpt(F ; Z) ~=i2* *IZ : ON THE YAGITA INVARIANT OF MAPPING CLASS GROUPS 17 Noting that by the universal coefficient theorem Hk(X1 ; Z) ~=Hom (Hk(X1 ; Z)) for 0 k 6(g - h) - 2n - 1, the result on the cohomology of X1 follows. For t* *he map : X1 ! X0 x X>0 one observes that the same type of argument applied to X0 and* * X>0 shows that they are homologically 6(g - h) - 2n - 2 connected too, and satisfy H6(g-h)-2n-1(X0; Z) ~=Z ; and H6(g-h)-2n-1(X>0; Z) ~=i>0Z : Using the K"unneth Formula and the universal coefficient theorem the isomorphis* *m result follows readily. Proposition 4.3. Let g > 1 and p an odd prime. Let ss be a subgroup of order p,* * with associated Riemann-Hurwitz formula 2g - 2 = p(2h - 2) + n(p - 1). Then there e* *xists a cohomology element e 2 H6(g-h)-2n(g; Z) whose restriction to H6(g-h)-2n(ss; Z* *) is non-trivial. Proof. Since p is odd, 6(g - h) - 2n > 0 (it is the codimension of F0 in Tg) so* * that, in the notation used above, X1 = Tg \ F is non-empty. We first study the Serre s* *pectral sequences with Z-coefficients, associated with the fibrations (A) X1 ! Eg xg X1 ! Bg and (B) X1 ! Ess xssX1 ! Bss : Let us put ! = 6(g - h) - 2n - 1. Then, in the obvious notation, one has E!;02(A) ~=H! (X1 ; Z)g =< a >~=Z ; Q since g acts on H! (X1 ; Z) ~= i2IZ by permuting the factors transitively (the* * action is induced by the action on the set I). Because Hk(X1 ; Z) = 0 for 0 < k < !, one * *has A E!;02(A) = E!;0!(A) -d!-!E0;!+1!(A) = H!+1 (g; Z) : We will show that dA!(a) 2 H!+1 (g; Z) restricts non-trivially to H!+1 (ss; Z).* * For this it suffices to show that the element a considered as an element in E!;0!(B) = H! (* *X1 ; Z)ss satisfies dB!(a) 6= 0 in H!+1 (ss; Z). We will analyze dB!(a) using the ss-map * *X1 ! X0xX>0 considered in Lemma 4.2 . and the spectral sequences associated with (C) X0 x X>0 ! Ess xss(X0 x X>0) ! Bss (D) X0 ! Ess xssX0 ! Bss 18 H. H. GLOVER, G. MISLIN AND Y. XIA and (E) X>0 ! Ess xssX>0 ! Bss : Since the ss-action on X0 is free, Ess xssX0 is homotopy equivalent to X0=ss, a* * finite dimensional complex, so that all of E!;02(D) = H! (X0; Z)ss= H! (X0; Z) is mapped isomorphically by dD!onto E0;!+1!(D) ~=H!+1 (ss; Z); otherwise, as H** *(X0; Z) ~= H*(S! ; Z), one would get a contradiction to the finite dimensionality of X0* *=ss. The fibration (E) has a section, because the ss-action on X>0 has a fixed point; th* *is follows from the fact that F0 \ ([i>0Fi) is either empty or a subcomplex of F0 codimens* *ion 2, and F0 is non-empty. Consequently, all elements in E0;!+12(E) = E0;!+1!(E)* * = H!+1 (ss; Z) are permanent dE -cycles. But this implies that dE!: E!;0!(E) ! E* *0;!+1!(E) is the zero map. The ss-map : X1 ! X0 x X>0 induces a map of spectral sequen* *ces E*;**(C) ) E*;**(B). As * : H*(X0 x X>0; Z) ! H*(X1 ; Z) is an isomorphism for 0 < * !, and also H! (X0 x X>0; Z) ~= H! (X0; Z) x H! (X>0; Z), we see that for x = (x0; x>0) 2 H! (X0 x X>0; Z)ssone has dC!(x) = dB!(*(x)) = dD!(x0) 2 H!+1 (ss; Z) : Thus, if we decompose the generator a 2 H! (X1 ; Z)g H! (X0; Z) x H! (X>0; Z)ss as a = (a0; a>0), we have resgss(dA!(a)) = dD!(a0) 2 H!+1 (ss; Z) ; and we need only show that a0 2 H! (X0; Z) is non-zero, to ensure that dD!(a0) * *6= 0. For this we view a as a g invariant function on H!(X1 ; Z) = H!(X0; Z) H!(X>0;* * Z) But every g-invariant function on H!(X1 ; Z), which is constant on H!(X0; Z) ~=* * Z, must be trivial, because H!(X1 ; Z) is isomorphic to i2IZ, with g acting by per* *muting transitively the summands of i2IZ. It follows that a0 6= 0 and we are done. Corollary 4.4. Let p be an odd prime and g = lpff+ 1 with l prime to p and ff >* * 0. If ss denotes a subgroup of order p of g such that the Riemann-Hurwitz equation of* * ss has the form 2g - 2 = p(2h - 2) + n(p - 1) with h - 1 = pff(s - 1) and n = (kp + i)pffwhere 0 < i < p, then 2l has necessa* *rily the form mp - i, and the restriction map ff [(3m+k)p-3(m+k)-2i]pff H[(3m+k)p-3(m+k)-2i]p(g; Z) ! H (ss; Z) is non-trivial. Proof. Since 2g - 2 = 2lpff, the Riemann-Hurwitz equation shows that 2l -i mo* *d p so that we can write 2l in the form mp - i for a unique m > 0. Thus 2g - 2 = (mp -* * i)pff and 2h - 2 = [m - k(p - 1) - i]pffso that 6(g - h) - 2n = [(3m + k)p - 3(m + k) - 2i]pff: Our claim then follows from Proposition 4.3 . ON THE YAGITA INVARIANT OF MAPPING CLASS GROUPS 19 Section 5 : The sharp upper bounds for g We have already established that if g = lpff+ 1 with l prime to p and ff > 0* *, then, p(g) = 2(p - 1)pfifor some fi ff - 1, see Lemma 1.5. To show that necessarily * *fi ff, we need only verify that for every subgroup ss g there exists a number j(ss) p* *rime to p such that the restriction map ff 2j(ss)pff (7) H2j(ss)p(g; Z) ! H (ss; Z) is non-trivial. This will be established in the next theorem, which proves (i) * *of Theorem 2 of the introduction. Theorem 5.1. Let p be an odd prime and g = lpff+ 1 with l prime to p and ff > 0* *. Then the Yagita invariant g equals 2(p - 1)pffor 2(p - 1)pff-1. Proof. As explained above, we need to study restriction maps of the type (7). L* *et ss be a subgroup of g of order p and consider the associated Riemann-Hurwitz equation 2g - 2 = p(2h - 2) + n(p - 1) ; where n 0 denotes the number of fixed points of the ss action on Sg. Write :* * g ! GL 2g(C) for the representation given by the action of g on H1(Sg; C). Note tha* *t factors through Gl2g(Q) so that |ss = ao + boe, with o the trivial one-dimensional repr* *esentation and oe the reduced regular one of ss. Thus dim CH1(Sg; C) = 2g = a + (p - 1)b = 2lpff+ 2 and, for a generator OE of the ss-action on H1(Sg; C), trace(OE : H1(Sg; C) ! H1(Sg; C)) = a - b = 2 - n : Thus pb = 2g - a + b = 2g - 2 + n = 2lpff+ n so that the following possibilitie* *s arise. (a) n 6 0 mod pff. Then b = b0pflwith fl < fff-l1 and b0 prime to p. But th* *en the Chern class c(p-1)pfl(|ss) = b0cp-1(oe)p is non-zero, providing a j(ss)* * for which the restriction map (7) is non-zero. (b) n = (kp + i)pffwith 0 i < p. Since then pb = 2lpff+ n = (2l + kp + i)pf* *f, we see that in case 2l+i 6 0 mod p, b is not zero mod pff, so that the same * *argument as before shows that c(p-1)pff-1() has a non-trivial restriction under the * *map (7). It remains to consider the case where 2l + i = mp. This yields pb = (m + k)* *pff+1and we find again two sub-cases. Firstly, if m + k is prime to p, then we co* *nclude again that the restriction map (7) is non-zero, with j(ss) = p - 1. Secondly, * *if m + k 0 mod p, we observe that in the notation above 2h = a = 2g -(p-1)b, which* * implies that 2h - 2 is divisible by pff. Also, we cannot have i = 0 in that cas* *e, because otherwise 2g - 2 were divisible by pff+1. Thus 0 < i < p and we are prec* *isely in the situation of Corollary 4.4, which shows that the restriction map (7) is * *non-trivial if one chooses 2j(ss) = [(3m + k)p - 3(m + k) - 2i] ; and j(ss) 6 0 mod p since we assume that m + k is divisible by p. 20 H. H. GLOVER, G. MISLIN AND Y. XIA This completes the proof of the theorem. As a corollary, we obtain part (ii) of Theorem 2, by combining Theorem 5.1 w* *ith Corol- lary 3.6. Of course, this also implies (ii) of Theorem 1, and therefore we have* * completed all proofs of the theorems stated in the introduction. References [B] S.A. Broughton, The equisymmetric stratification of the moduli space and * *the Krull dimension of mapping class groups, Topology and its Applications 37 (1990), 101-113. [CO] L.Carlitz and F.R. Olson, Maillet's determinant, Proc. Amer. Math. Soc. 6* * (1955), 265-269. [EM] B. Eckmann and G. Mislin, Chern classes of group representations over a n* *umber field, Math. Ann. 271 (1985), 349-358. [FK] H. Farkas and I. Kra, Riemann Surfaces, Graduate Texts in Math., vol. 71,* * Springer Verlag, 1982. [GMX] H.H. Glover, G. Mislin and Y. Xia, On the Farrell Cohomology of Mapping C* *lass Groups, Invent. Math. 109 (1992), 535-545. [N] J. 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Glover, Ohio State University, Columbus, Ohio E-mail address: glover@math.ohio-state.edu G. Mislin, ETH Z"urich, Switzerland; and Ohio State University, Columbus, Oh* *io E-mail address: mislin@math.ethz.ch and mislin@math.ohio-state.edu Y. Xia, Northwestern University, Evanston, Illinois E-mail address: xia@math.nwu.edu