On the Farrell Cohomology of Mapping Class Groups H. H. Glover, G. Mislin and Y. Xia Introduction. For any group of finite virtual cohomological dimension and a prime p, we * *say that is p-periodic, if there exists a positive integer k such that the Farrell coho* *mology groups H^i(; M) and ^Hi+k(; M) have naturally isomorphic p-primary components for all * *i 2 Z and Z-modules M. The p-period of is defined as the least value of k (cf. [B]* *). For instance, if is p-torsion free, then is p-periodic of period one. The mapping class group, g, is defined to be the group of path components o* *f the group of orientation preserving homeomorphisms of the oriented closed surface S* *g of genus g. For instance, 1 ~=SL(2; Z) and the cohomology is well known and easy to comp* *ute in this case. By writing SL(2; Z) as an amalgamated product of Z=4 and Z=6 over Z=* *2, one finds H^*(1; Z) ~=(Z=12)[x; x-1] with x of degree two. Thus 1 is 2- and 3-periodic, with periods equal to two. It is well known that g is of finite virtual cohomological dimension and, i* *f g > 1, vcd(g) = 4g - 5 (cf. [H]). In the sequel we will always assume that g > 1. Reca* *ll from [B] that a group of finite vcd is p-periodic if and only if it does not contain* * a subgroup isomorphic to Z=p x Z=p. Because for g > 1 the mapping class group g contains a* *lways a subgroup isomorphic to Z=2 x Z=2, g is never 2-periodic. However, for an odd * *prime p, g is p-periodic for almost all values of g. This corresponds to the intuitively* * obvious fact that it is hard to find two "different" homeomorphisms of order p on Sg, which * *commute with each other. The third author determined in [X1] all the genera g for whic* *h g is p-periodic. In particular, g is 3-periodic if and only if g 6 1 mod(3). For an * *odd prime p - 1 - and genus g 6 1 mod(p), g is always p-periodic. Moreover, there are only finite* *ly many "exceptional values" of g with g 1 mod(p) for which g is p-periodic. Recall that for a finite p-periodic group G and p an odd prime, the Sylow p* *-subgroup Gp of G is cyclic and, if Gp 6= 1, the p-period of G equals 2|N(Gp)=C(Gp)|, whe* *re N(Gp) (respectively C(Gp)) denotes the normalizer (respectively centralizer) of Gp in* * G; in par- ticular, the p-period of G divides 2(p - 1). Unlike the case of finite groups ,* * the p-period of a p-periodic infinite group, p a fixed prime, may be arbitrarily large. A si* *mple example is given by the group Z=pn o Z =< a; b|apn = 1; bab-1 = ap+1 >. For an odd prim* *e p, the p-period of Z=pn o Z equals 2pn-1. In this paper, however, we will show the surprising result that for a p-per* *iodic mapping class group g, the p-period is bounded by 2(p - 1). The precise theorem reads a* *s follows. THEOREM 1. Let p be an odd prime and assume that g is p-periodic. Then the p-period of g is given by lcm{2[N(ss) : C(ss)]|ss 2 S} where ss ranges over S, a set of representatives of conjugacy classes of subgro* *ups of order p of g, and N(ss) (respectively C(ss)) denotes the normalizer (respectively cen* *tralizer) of ss in g. In particular, the p-period of g divides 2(p - 1). We use the convention that lcm{2[N(ss) : C(ss)]|ss 2 S} = 1 in case S is em* *pty (the p-period of g equals one in that case). In case of the prime 3, one can find suitable subgroups of g to get the fol* *lowing even simpler result. THEOREM 2. Let g > 1 and assume that g is 3-periodic. Then the 3-period of* * g equals 4. Indeed, as we will see, it is also possible to give a more explicit descrip* *tion of the p-period of the mapping class group in the general case. In particular, for g * *1 mod(p) one finds the following. - 2 - THEOREM 3. Let p be an odd prime and g 1 mod(p). If g is p-periodic, then the p-period of g is 2(p - 1). The basic idea is to study for each subgroup ss of order p of g the action * *of the normalizer N(ss) of ss in g on a spherical space of the form Rd x (Rk - {0}), w* *hich comes up as a subspace of the normal bundle of the fixed point set of the action of s* *s on the Teichm"uller space of Sg. The numbers d and k in Rd x (Rk - {0}) turn out to de* *pend only on the orbifold Sg=ss. We use then the fact that the p-periods of the subgroups* * of the form N(ss) of g determine the p-period of g. The rest of the paper is organized as follows. In Section 1 we provide some* * background material on groups acting on spherical spaces. In Section 2 we will establish a* *n upper bound for the p-period of g. In Section 3 we finish the proof of Theorem 1 in the cas* *e that g 6 1 mod(p), and we also establish Theorem 2. Section 4 is devoted to the case g 1 * *mod(p), and in the last Section 5 we discuss an explicit formula for the p-period; in p* *articular, we will prove Theorem 3. We would like to thank the referee for valuable suggestions. 1. Groups acting on spherical spaces We call a (not necessarily compact) manifold X a spherical space, if X is h* *omotopy equivalent to a sphere. A classical result on finite groups G states that if G* * acts freely on an odd dimensional sphere Sk-1, then H^*(G; Z) is periodic, of period dividi* *ng k. For the convenience of the reader we provide below an adaption of the classical pro* *of to the setting of Farrell cohomology of groups of finite virtual cohomological dimensi* *on acting on spherical spaces. (1.1) LEMMA. Let denote a group of finite virtual cohomological dimension* * acting properly discontinuously on a spherical space X homotopy equivalent to an odd d* *imen- sional sphere Sk-1. Assume that the stabilizer of any point x 2 X is a finite p* *-torsion free group, p a fixed prime. Suppose furthermore that acts trivially on Hk-1 (X; Z)* *. Then is p-periodic of p-period dividing k. - 3 - Proof. For every finite p-subgroup P of , P acts freely on X and there is a* *n orientable spherical fibration (up to homotopy) Sk-1 ' X -! X=P -! BP : The Gysin sequence of this spherical fibration shows that the Euler class e(P )* * 2 Hk(P ; Z) induces, for n > dimX, an isomorphism - [ e(P ) : Hn (P ; Z) -! Hn+k (P ; Z) : It follows that ^H*(P ; Z) is periodic, and e(P ) maps under the canonical map H*(P ; Z) -! ^H*(P ; Z) to a unit ^e(P ) 2 ^Hk(P ; Z). If we write e() for the Euler class of Sk-1 ' X -! E x X -! B we conclude that ^e() 2 H^k(; Z) restricts to the unit ^e(P ) for every finite * *p-subgroup P < . This implies (cf. [B]) that ^e()(p)2 ^Hk(; Z)(p)is a unit. As a result, t* *he p-period of divides k. 2. An upper bound for the p-period of g Let Sg be a closed oriented surface of genus g > 1. It is classical that th* *e 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. If a subgroup* * G < g stabilizes a point of Tg, then G is necessarily finite and there exists a compl* *ex structure on Sg such that G lifts to a group of holomorphic automorphisms of (Sg; ) (cf.* * [K]). When no confusion can arise, we will in our notation not distinguish between G * *and the lifted group (isomorphic to G) acting on Sg. According to [M-H], the fixed point set of a finite group G < g acting on T* *g is a submanifold homeomorphic to the Teichm"uller space T (M) where M is a Fuchsian * *group containing ss1(Sg) with G ~=M=ss1(Sg). If the Fuchsian group M admits a present* *ation of the form Yh Yr M =< x1; y1; : :;:xh; yh; e1; : :;:er| eimi= 1 ; [xj; yj] ek = 1* * > ; j=1 k=1 where all mi 's are greater than one, then the Teichm"uller space T (M) is then* * homeomorphic to R6h-6+2r. Applied to the case of G ~=Z=p, we obtain the following. - 4 - (2.1) PROPOSITION. Let p be a prime and ss < g a subgroup of order p. Then the fixed point set (Tg)ssis homeomorphic to R6h-6+2n, where h is the genus of * *Sg=ss and n the number of fixed points of the ss-action on Sg. Proof. Consider the branched covering Sg -! (Sg=ss) =: Sh and define M to b* *e the orbifold fundamental group of Sg=ss, that is, Y Y M =< x1; y1; : :;:xh; yh; e1; : :;:en|e1p = : :=:enp= 1; [xj; yj] ek * *= 1 > : Let p1; : :;:pn 2 Sg be the fixed points of the ss-action on Sg. Then, writing * *p1; : :;:pnfor the images of the pi's in Sh, one obtains a regular p-sheeted covering space Sg - {p1; : :;:pn} -! Sh - {p1; : :;:pn} : The induced map of fundamental groups ' : ss1(Sg - {p1; : :;:pn}) -! ss1(Sh - {p1; : :;:pn}) gives rise to an injective map ' : ss1(Sg) -! M with image a normal subgroup of index p. It follows that T (M) ~=R6h-6+2n. Let Sg, g > 1, be a closed oriented surface of genus g and ss < g a subgro* *up of order p, p a fixed prime. We will write n(ss) for the number of fixed points of the s* *s-action on Sg, and h(ss) for the genus of Sg=ss. We will establish the following upper bou* *nd for the p-period of g. (2.2) THEOREM. Let g > 1 and p a prime. Assume that g is p-periodic. Then * *the p-period of g divides lcm{6(g - h(ss)) - 2n(ss)|ss 2 S} where S denotes a set of representatives of conjugacy classes of subgroups of o* *rder p of g. - 5 - Proof. It is well-known that there is a differentiable structure on Tg suc* *h that g acts smoothly. Let ss < g be a subgroup of order p. The normalizer N(ss) acts* * on (Tg)ss~=R6h(ss)-6+2n(ss)as well as the normal bundle E of (Tg)ssin Tg; of cours* *e, E is homeomorphic to R6h(ss)-6+2n(ss)x R6(g-h(ss))-2n(ss)since Tg ~=R6g-6. We now c* *onsider the N(ss)-action on the spherical space E - E0, where E0 denotes the zero secti* *on of the bundle E - ! (Tg)ss. First we check that the stablizer N(ss)e < N(ss) of every* * point e 2 E - E0 is a finite group of order prime to p. The projection E -! (Tg)ssis* * N(ss)- equivariant so that N(ss)e is mapped injectively to N(ss)e, which is contained * *in the finite stabilizer (g)e of e2 (Tg)ss Tg, where edenotes the image of e in (Tg)ss. Let x* * 2 N(ss)e. Assume that xp = 1. Since x 2 N(ss), it normalizes ss and, as xp = 1, it follo* *ws that x even centralizes ss. Thus, the subgroup generated by x and ss, < x; ss >, is is* *omorphic to Z=p x Z=p, or it is equal to ss. Because N(ss) is a subgroup of g, it is p-per* *iodic too, and it follows that < x; ss >6~= Z=p x Z=p. Thus x 2 ss and if x 6= 1, we have * *< x >= ss. Therefore the (linear) x-action on Ve - {0} must be free; here, Ve denotes the * *fibre over e of the bundle projection E ! (Tg)ss. Since xe = e, we conclude thus that x 6= 1* * implies that e 2 E0, the 0-section of the bundle E - ! (Tg)ss, and it follows that the * *action of N(ss) on (E - E0) ~=Rd x (Rk - {0}) has finite stabilizers of order prime to* * p, with d = 6h(ss) - 6 + 2n(ss), k = 6(g - h(ss)) - 2n(ss) . The action is properly dis* *continuous on E and therefore on E - E0, since it is properly discontinuous on (Tg)ss Tg, and* * the fibers of E -! (Tg)ssare locally compact. Also, N(ss) acts trivially on H*(E -E0; Z), * *as one can see as follows. The N(ss) action on Tg and (Tg)ssis by complex automorphisms wi* *th respect to some fixed complex structure on Tg. The normal bundle of (Tg)ssin Tg is the * *quotient bundle of o(Tg)|(Tg)ssby o(Tgss), where o(Tg) (respectively o(Tgss)) denotes th* *e tangent bundle of Tg (respectively (Tg)ss). All these bundles have a natural orientatio* *n, induced by the complex structure of Tg, and these orientations are preserved by the N(ss)-* *action. It is then obvious that N(ss) acts trivially on H*(E-E0; Z). We are therefore in the * *situation of (1.1) and conclude that the p-period of N(ss) divides k = 6(g-h(ss))-2n(ss). By* * a result of Q Brown [B], the fact that g is p-periodic implies that ^H*(g; Z)(p)~= ss2S^H*(N* *(ss); Z)(p), where S denotes a set of representatives for the conjugacy classes of subgroups* * of order p of g. The conclusion of the theorem now follows. In view of the application in Section 3, it is useful to rewrite our result* * on the p- periodicity of g as follows. - 6 - (2.3) COROLLARY. Let g > 1 and ss < g a subgroup of order p, p a fixed pri* *me. Assume that g is p-periodic . Then the p-period of N(ss) divides (3m + k)p - 3(m + k) - 2i where the integers k 0; m > 0 and i with 0 i p - 1 are uniquely determined b* *y the equations 2g - 2 = mp - i, and n(ss) = kp + i. Proof. From the proof of (2.2) we infer that the p-period of N(ss) divides * *6(g -h(ss))- 2n(ss), where h(ss) denotes the genus of Sg=ss and n(ss) the number of fixed po* *ints of the ss-action on Sg. By the Riemann-Hurwitz formula applied to the branched coverin* *g space Sg -! Sg=ss one has 2 - 2g = p(2 - 2h(ss)) - n(ss)(p - 1). Also, if one writes * *2g - 2 in the form mp - i with m > 0 and 0 i p - 1, then, the Riemann-Hurwitz formula shows* * that n(ss) = kp+i for some k 0. It follows that 2g = mp-i+2 and 2h(ss) = 2+m-k(p-1)* *-i so that 6(g - h(ss)) - 2n(ss) = (3m + k)p - 3(m + k) - 2i. 3. The p-period of g for g 6 1 mod(p) We will make repeated use of the following Lemma. (3.1) LEMMA. Let p be a prime and N a group of finite virtual cohomological dimension which is p-periodic. Suppose N contains a normal subgroup ss < N of o* *rder p. Then the following holds. (a)If x 2 ^H*(N; Z) restricts to a unit in ^H*(ss; Z) and degx 6= 0, then the * *p-period of N divides |degx|. (b)The p-period of N has the form 2[N : C(ss)]pff, where C(ss) < N denotes the* * centralizer of ss in N and ff 0 an integer. - 7 - Proof. We can write x 2 H^*(N; Z) uniquely as a sum of p-primary elements x* *(p)2 H^*(N; Z)(p). If the reduction x 2 H^*(N; Z=p) is a unit, then so is x(p)2 H^** *(N; Z)(p). This follows from the fact (cf. [B; Chapter X, 6.6]) that the reduction map : ^H*(N; Z)(p)-! ^H*(N; Z=p) has the property that ker() is nilpotent and for every u 2 ^H*(N; Z=p) there is* * an integer k such that upk2 Im(). Namely, if y is an inverse for xand ypk= (z), then (xpkz* *-1)(p) is nilpotent and thus x(p)is a unit. Thus, if x is a unit and degx 6= 0 , then * *the period of N divides |degx|. Because N is p-periodic and ss < N is normal, ss is the only * *subgroup of order p of N. Quillen's F-isomorphism theorem then implies (cf. [B]) that the r* *estriction map ' : ^H*(N; Z=p) -! ^H*(ss; Z=p)N(ss)=C(ss)=: H* has the property that ker(') is nilpotent and that for every v 2 H* there is an* * integer s such that vps 2 Im('). As before, we conclude that if '(x) is a unit, then so i* *s x. But '(x) 2 H* is invertible if and only if it is invertible as an element in ^H*(ss* *; Z=p), and the invertible elements of ^H*(ss; Z) ~=Z=p[w; w-1]; degw = 2, are precisely those,* * which map to invertible elements in ^H*(ss; Z=p), proving (a). For (b) we observe that the p* *-period of N is the smallest positive integer k for which ^Hk(N; Z)(p)contains a unit. By th* *e discussion above, this is, up to a pth-power, the smallest positive degree for which ^H*(N* *; Z=p) contains a unit and, using the map ', this is up to a pth-power the smallest positive de* *gree for which H* contains a unit. But H* is periodic with period 1 if p = 2, and period 2[N(s* *s) : C(ss)] if p is odd. Therefore, the p-period of N has the form 2[N(ss) : C(ss)]pff. W* *e will be interested in the case where ss < g and N = N(ss), the normalizer of ss in g. T* *o prove our main theorem stated in the introduction, it suffices to show that ff = 0 in* * (3.1, (b)) for N = N(ss) < g. In this section, we settle the case g 6 1 mod(p). - 8 - (3.2) PROPOSITION. Let g > 1 and assume that p is an odd prime. If g 6 1 mod(p) and ss < g a subgroup of order p, then N(ss) is p-periodic with p-period* * equal to 2[N(ss) : C(ss)]. Proof. As observed in the introduction, if p is an odd prime and g 6 1 mod(* *p), then g is p-periodic. Thus N(ss) is p-periodic and to prove the proposition, we need* *, because of (3.1, (b)), only show that the p-period of N(ss) is not divisible by p. Ther* *e will be two cases to consider. In the first case, the upper bound (2.3) for the p-period is* * prime to p and we are done. For the other case, we will construct an element cp-1(ae) 2 H2(p-1* *)(N(ss); Z) with the property that it restricts to a unit in H^2(p-1)(ss; Z); this implies * *by (3.1, (a)) that the p -period of N(ss) divides 2(p - 1), which is prime to p. To this end,* * consider the natural action of g on H1(Sg; Z) and write ae : g -! GL(2g; Q) for the corresponding representation over Q. Since ss ~=Z=p admits only two irr* *educible Q-representations, the trivial one, which we denote by o, and the reduced regul* *ar repre- sentation ffi : ss -! GL(p - 1; Q), the character of = ae|ss satisfies O = a(ss)Oo + b(ss)Offi (3:* *3) for some natural numbers a(ss) and b(ss). It is easy to check (cf. [G-M]) that * *the Chern class c(p-1)() 2 H2(p-1)(ss; Z) is non-zero if and only if b(ss) is relatively * *prime to p. The number b(ss) depends on the number of fixed point of the ss action on Sg an* *d can be determined as follows. Let x 2 ss be a generator and 1 2 ss the neutral element* *. Then, for the reduced regular representation ffi one has Offi(x) = -1 and therefore (3.3)* * yields Oae(x) = a(ss) - b(ss); Oae(1) = 2g = a(ss) + (p - 1)b(ss) (3:* *4) On the other hand, the Lefschetz Trace Formula shows that 2 - Oae(x) = n(ss); (3:* *5) - 9 - where n(ss) denotes the number of fixed points of the x-action on Sg. It is con* *venient to write 2g - 2 in the form mp - i with 0 i p - 1. Then, from the Riemann-Hurwi* *tz Formula, one has 2g - 2 -n(ss) mod(p) and thus n(ss) = kp + i for some k 0. S* *olving (3.4) and (3.5) for b(ss), yields b(ss) = m + k (3:* *6) As 2g -2 = mp-i and g 6 1 mod(p), we have i 6 0 mod(p). By (2.2) the p-period o* *f N(ss) divides (3m+k)p-3(m+k)-2i. As observed, if p does not divide (3m+k)p-3(m+k)-2i, we are done, because then necessarily ff = 0 in (3.1). On the other hand, if p* * divides (3m + k)p - 3(m + k) - 2i, then 3(m + k) -2i 6 0 mod(p) since p is odd and i 6* * 0 mod(p); of course, p 5 in that case. It then follows that m + k 6 0 mod(p) and* * by (3.6), that b(ss) is not divisible by p. We conclude then that cp-1(ae) 2 H2(p-1)(ss; * *Z) is nonzero, and it is clear that then cp-1() 2 ^H2(p-1)(ss; Z) is necessarily a unit , conc* *luding the proof of the proposition. (3.7) COROLLARY. The 3-period of every 3-periodic mapping class group g is* * 4. Proof. According to [X1], g is 3-periodic if and only if g 6 1 mod(3). Th* *us, by Proposition (3.2), the 3-period of a 3-periodic g is 2 or 4. To rule out the v* *alue 2, it suffices to find a subgroup of g whose 3-period is 4. We claim that for g 6 1 m* *od(3), g contains the Dihedral group D6 of order 6. Namely, by [X3], 2; 3; 5 and 6 conta* *in D6, and, as shown in [X1], if g contains a finite subgroup G of order |G| then so d* *oes g+|G|. Thus, every g, g 6 1 mod(3), contains D6. - 10 - 4. The p-period of g for g 1 mod(p) Suppose that g is p-periodic and 1 < g 1 mod(p). As discussed in Section 3* *, to show that the p-period of g is given by the formula of Theorem 1 in the introdu* *ction amounts to showing that for ss < g any subgroup of order p, the p-period of N(s* *s) is relatively prime to p. This will be done by constructing for each such ss a sy* *mplectic characterstic class dk(ss)(ae) 2 H2k(ss)(g; Z) where k(ss) satisfies 1 k(ss) p - 1, such that dk(ss)(ae) restricts to a unit* * in H^*(ss; Z). These characteristic classes arise as follows. We look at the natural represen* *tation of ae : g -! Sp(2g; R), by letting g act on H1(Sg; R), preserving the symplectic f* *orm given by the cup-product. 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); 1 i g : For our application, the following algebraic Lemma will be useful. (4.1) LEMMA. Let p be a prime and f(x) 2 Fp[x] a polynomial satisfying f(x* *) 1 mod(xp) and which factors completely over Fp into linear factors. Then each roo* *t of f has multiplicity divisible by p. - 11 - Proof. Let 1; : :;:n be the roots of f with multiplicities. The assumptio* *n on f implies that the first (p - 1) elementary symmetric functions in the 's vanish* *. As a result Xn i k = 0; 1 k p - 1 : i=1 If n 0 denotes the multiplicity of 2 Fp as a root of f, then we infer X n k = 0; 1 k p - 1 : 2Fp-{0} Since the Van der Monde matrix [k] with 2 Fp - {0} and 1 k p - 1 is regular,* * we conclude that n 0 mod(p) for all 2 Fp - {0}. The following useful Lemma on Chern classes of representations of Z=p is no* *w an easy consequence. (4.2) LEMMA. Let ' : Z=p -! U(n) be a representation of Z=p, p a fixed pri* *me. Assume that ci(') 2 H2i(Z=p; Z) is zero for 1 i p - 1. Then ' is of the form * *p o, with o a trivial representation. Proof. Decompose ' into one-dimensional representations, ' = p-1k=0nk!k, wi* *th ! a faithful one-dimensional representation of Z=p. We put o = n0!0 and need to sho* *w that each ni; 1 i p - 1, is divisible by p. Consider the injective ring homomorphi* *sm Fp[x] -! H*(Z=p; Z=p) given by mapping x to c1(!) 2 H2(Z=p; Z=p), the reduction mod(p) of c1(!). Note* * that Q p-1 the polynomial f(x) 2 Fp[x] defined by k=0(1+kx)nk is mapped to the total Che* *rn class P c(') = ci('). Since ci(') = 0 for 1 i p - 1, we conclude that f(x) 1 mod(x* *p). By the previous Lemma, we conclude that nk is divisible by p for 1 k p - 1: We now return to the study of the representation ae : g -! Sp(2g; R) and complete the proof of Theorem 1. - 12 - (4.3) PROPOSITION. Let p be an odd prime and g 1 mod(p). Assume that g is p-periodic. Then, for every subgroup ss < g of order p there exists an integer * *k(ss) with 1 k(ss) p - 1 such that dk(ss)(ae|ss) 2 ^H2i(ss; Z) is a unit, where i = k(ss* *). Proof. The representation ae : g -! Sp(2g; R) induces a representation ae|s* *s : ss -! Sp(2g; R) which factors uniquely (up to a conjugation) through a maximal compac* *t sub- group U(g) < Sp(2g; R). We can therefore think of the classes di(ae|ss) as Che* *rn classes ci(ae) of the representation ae: ss -! U(g) obtained in this way. We want to sh* *ow that if we had di(ae|ss) = 0 for 1 i p - 1, then gwould contain a subgroup of the form Z* *=p x Z=p, contradicting the assumption that g be p-periodic. The representation ae: ss -!* * U(g) can be realized by choosing a complex structure and a Hermitian metric on H1(Sg; R)* * compat- ible with the ss-action and symplectic structure. This can be done by choosing * *a complex structure on Sg such that ss acts by holomorphic automorphisms on Sg; the induc* *ed action on the space W 1of holomorphic 1-forms of Sg is then a model for the representa* *tion ae. If we decompose aeas p-1i=0ni!i, ! a faithful irreducible one-dimensional represen* *tation then, assuming di(ae|ss) = ci(ae) = 0 for 1 i p - 1, we infer ni 0 mod(p) for 1 i* * p - 1 (cf. (4.2)). Let x 2 ss be a generator and denote by (fi1; : :;:fin(ss)) the fi* *xed point datum of the x-action on Sg; thus 1 fii p - 1, and for some numbering of the fixed * *points of x and in a suitable local coordinate system about the j'th fixed point, the act* *ion of x-1 is p ___ given by z 7! exp(2ss -1 fij=p). We can think of the numbers ni in the decompo* *sition of aeas dimensions of eigenspaces of the x-action on W 1, the space of holomorphic* * 1-forms. According to [F-K; Chapter V, 2.2.3 and 2.5.4] one has then n0 = h(ss) where h(ss) denotes the genus of Sg=ss, and for 1 j p - 1, n(ss)X nj = h(ss) - 1 + n(ss) - < ffi(j)fii=p > (4:* *4) i=1 for a suitable permutation ffi of {1; 2; : :;:p - 1}; we use the notation < q >* * to denote the fractional part of a rational number q. By renumbering the fixed points of ss s* *uitably, we - 13 - may assume that ffi is the identity permutation. The equations (4.4) imply then* *, because nj 0 mod(p) for 1 j p - 1, that X X X np-1 - n1 = fii=p - < (p - 1)fii=p >= 2 fii=p - n(ss) 0 mod(p)(;4* *:5) and np-1 + n1 = 2((h(ss) - 1 + n(ss)) - n(ss) 0 mod(p) : (4:* *6) Moreover, the Riemann-Hurwitz equation 2 - 2g = p(2 - 2h(ss)) - n(ss)(p - 1) sh* *ows that, because g 1 mod(p), one has n(ss) 0 mod(p) so that (4.5) and (4.6) imply h(ss) 1 mod(p) ; (4:* *7) and X fii 0 mod(p2) : (4:* *8) With g = kp + 1; h(ss) = tp + 1 and n(ss) = sp, the Riemann-Hurwitz equation ca* *n now be written in the form (2 - 2g)=p2 = (2 - 2(t + 1)) - s(1 - 1=p) : (4:* *9) Because of (4.8) we see that either n(ss) = 0, or else n(ss) > p, so that n(ss)* * = sp with s = 0 or s > 1. We wish to prove that this implies that Z=pxZ=p < g. For this, it suf* *fices in view of (4.9) to show that Z=pxZ=p admits a generating set (a1; : :;:at+1; b1; : :;:* *bt+1; c1; : :;:cs) Q Q satisfying the relations [ai; bi] cj = 1 with each cj of order p (see for i* *nstance [T]). By choosing a generating set {a; b} for Z=p x Z=p, we can proceed as follows. In c* *ase s = 0 (that is, there are no cj's), we choose ai = a and bi = b for 1 i t + 1. In * *case s 2, if s 1 mod(p), choose ai = a and bi = b (1 i t + 1), and c1 = c2 = : :* *=: cs-2 = a; cs-1 = a2; cs = a-1; if s 6 1 mod(p), choose ai = a; bi = b (1 i t * *+ 1) and c1 = c2 = : :=:cs-1 = a; cs = a1-s. We conclude that the assumption that di(ae|* *ss) = 0 for 1 i p - 1 yields a contradiction. Thus, for some k(ss) with 1 k(ss) p -* * 1, we infer that dk(ss)(ae|ss) 2 H2k(ss)(ss; Z) is a non-zero element. But then dk(ss* *)(ae|ss) 2 ^H*(ss; Z) is necessarily a unit, because ^H*(ss; Z) ~=Z=p[w; w-1] with degw = 2, finishing the proof of the Proposition. - 14 - 5. An explicit formula for the p-period of g We first give a proof of Theorem 3 of the introduction. It is clear from co* *vering space theory that if g 1 mod(p), say g = kp + 1, then the surface Sh of genus h = k * *+ 1 will have Sg as p-fold regular covering space, and therefore there exists a subgroup* * ss < g of order p, which acts freely on Sg. By a classical result of Nielsen [N], all* * fixed point free homeomorphisms of Sg, having the same finite order, are conjugate in g so * *that we conclude that N(ss)=C(ss) ~= Z=(p - 1). Theorem 1 then implies immediately tha* *t the p-period of g equals 2(p - 1), which proves Theorem 3. For the general case, we use of the following formula. (5.1) LEMMA [X2; 3.1]. Let g > 1 and write S for the set of conjugacy clas* *ses of subgroups ss < g of order p. Then lcm{[N(ss) : C(ss)] |ss 2 S} = lcm{gcd(p - 1; n(ss)) |ss 2 S} where n(ss) denotes the number of fixed points of the ss-action of Sg. It is therefore possible to compute the p-period of a p-periodic g, if one * *knows the possible fixed point numbers n(ss) for homeomorphisms of order p on Sg. These n* *umbers were determined in [X1] and look as follows. If we write 2g - 2 in the form mp * *- i with 0 i p - 1, define for an odd prime p ae Bg;p= {i;{i1++p;p:;:;:i:+:[2g=(p:-;1)1-+m]p};if[i26g1=mod(p);(pi-* *f1)i- m]p};1 mod(p). As usual, we use here the notation [x] to denote the integral part of the ratio* *nal number x. According to [X1], Bg;pconsists precisely of the set of all those numbers, w* *hich occur as cardinalities of the fixed point set for homeomorphisms of order p on Sg; (i* *f, in case i 6 1 mod(p), one has 2g=(p - 1) < m, then Bg;pis empty and g contains no eleme* *nt of order p, similarly in case i 1 mod(p)). Combining Theorem 1 with (5.1), one o* *btains the following explicit formula for the period of a p-periodic mapping class gro* *up. (5.2) THEOREM. Suppose p is a prime, g > 1 and that g is p-periodic. Then * *the p-period of g is given by lcm{gcd(2(p - 1); 2n)|n 2 Bg;p} : For instance, (5.2) implies that for large values of g (e.g., 2g > p3), the* * p-period of a p-periodic g is precisely 2(p - 1). - 15 - References. [B] K. S. Brown, Cohomology of Groups, Graduate Texts in Math. Vol. 87, Sprin* *ger Verlag, 1982. [F-K] H. Farkas and I. Kra, Riemann Surfaces, Graduate Texts in Math. Vol. 71, * *Springer Verlag, 1982. [G-M] H. Glover and G. Mislin, Torsion in the mapping class group and its cohom* *ology, J. Pure and Applied Alg. 44 (1987), 177-189. [H] J. L. Harer, The virtual cohomological dimension of the mapping class gro* *ups of orientable surfaces, Ann. of Math. 121 (1985), 215-249. [K] S. Kerckhoff, The Nielsen realization problem, Ann. of Math. 117 (1983), * *235-265. [M-H] C. MacLachlan and W. J. Harvey, On mapping-class groups and Teichm"uller * *spaces, Proc. London Math. Soc. (3) 30 (1975), 496-512. [N] J. Nielsen, Die Struktur periodischer Transformationen von Fl"achen, Dans* *ke Vid. Selsk. Mat.-Fys. Medd. 15 (1937), 1-77. [T] T. Tucker, Finite groups acting on surfaces and the genus of a group, J. * *Comb. Theory B 34 (1983), 82-98. [X1] Y. Xia, The p-periodicity of the mapping class group and the estimate of * *its p-period, to appear in Proc. Amer. Math. Soc. [X2] Y. Xia, The p-period of an infinite group, to appear in Publicacions Mate* *matiques. [X3] Y. Xia, Farrell-Tate cohomology of the mapping class group, Thesis, The O* *hio State University, 1990. Ohio State University ETH-Zentrum Ohio State University Columbus, Ohio Z"urich, Switzerland Columbus, Ohio and Ohio State University Columbus, Ohio September 1991, revised February 1* *992 - 16 -