On the Farrell Cohomology of Mapping Class Groups
H. H. Glover, G. Mislin and Y. Xia
Introduction.
For any group of finitevirtual cohomological dimension and a prime p, wesa*
*y that
is p-periodic,if there exists a positive integer ksuch that the Farrell cohomo*
*logy 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 definedto b ethe group of path components of*
* the
group of orientation preserving homeomorphisms of the oriented closed surface S*
*gof genus
g. For instance,1 = SL(2;Z) and the cohomology is well known and easy to comput*
*e 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; x1 ]
with x of degree two. Thus 1 is 2- and 3-periodic, with periods equal to two.
It is well known that gis of finite virtual cohomological dimension and, if*
* g >1,
vcd(g) = 4g 5 (cf. [H]). In the sequel we will always assume that g > 1. Recall*
* 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 Z=p. Because for g> 1 the mapping class group gcontains alwa*
*ys
a subgroup isomorphic to Z=2 Z=2,g is never 2-periodic.However, for an odd pri*
*me 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 gfor which *
*g is
p-periodic. In particular,g is 3-periodic if and only if g6j 1 mod(3). For an o*
*dd prime p
and genus g 6j1 mod(p), gis always p-periodic. Moreover, there are only finitel*
*y many
"exceptional values" of gwith g j1 mod(p) for which g is p-periodic.
Recall that for a finite p-periodic group G and p ano ddprime, the Sylow p-*
*subgroup
Gp of G iscyclic and, if Gp 6=1, the p-period of G equals 2jN(Gp)=C(Gp)j, where*
* N(Gp)
(respectively C(Gp)) denotes the normalizer (resp ectively centralizer) of Gp i*
*n G;in par-
ticular, the p-period of G divides 2(p 1). Unlike the case of finite groups , *
*thep-perio d
of a p-periodic infinite group, p a fixed prime,may be arbitrarily large. A sim*
*ple example
is given by the group Z=pno Z =. For an odd prime p*
*,the
p-period of Z=pno Z equals 2pn1 .
In this paper, however, we will show the surprising result that for a p-per*
*iodicmapping
class group g,the p-period is bounded by 2(p 1). The precise theorem reads as *
*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
lcmf2[N(ss) : C(ss)]jss 2 Sg
where ss ranges over S, a set of representatives of conjugacy classes of subgro*
*ups oforder
p of g, andN (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 lcmf2[N (ss ) :C (ss)]jss2 Sg = 1 in case S is e*
*mpty (the
p-period of g equalsone in that case).
In case of the prime 3,one can find suitable subgroups of gto get the follo*
*wing 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 possibleto give a more explicit descript*
*ion of the
p-period of the mapping class group in the general case. In particular, for g j*
*1 mod(p)
one finds the following.