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 -
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[B] K. S. Brown, Cohomology of Groups, Graduate Texts in Math. Vol. 87, Sprin*
*ger
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[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*
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[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), *
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[M-H] C. MacLachlan and W. J. Harvey, On mapping-class groups and Teichm"uller *
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[X1] Y. Xia, The p-periodicity of the mapping class group and the estimate of *
*its p-period,
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[X2] Y. Xia, The p-period of an infinite group, to appear in Publicacions Mate*
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[X3] Y. Xia, Farrell-Tate cohomology of the mapping class group, Thesis, The O*
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and
Ohio State University
Columbus, Ohio
September 1991, revised February 1*
*992
- 16 -