Mapping Class Groups, Characteristic
Classes and Bernoulli Numbers
GUIDO MISLIN
ETH Z"urich, Switzerland; and Ohio-State University, Columbus, Ohio
Introduction
The mapping class group g of a closed, connected and oriented surface Sg of
genus g is defined as the group of connected components of the group of orienta*
*tion
preserving diffeomorphisms of Sg. This group has been the object of many recent
studies. Of particular interest are its finite subgroups; these are precisely t*
*he finite
groups which occur as groups of symmetries of the surface Sg equipped with a
complex structure (a Riemann surface). The interplay of algebra, topology and
analysis in the study of g make it one of the most fascinating groups. As it is*
* the
case for classical arithmetic groups, the finite subgroups of g are related to *
*certain
concepts in number theory. We shall discuss in this essay invariants of g which*
* are
related to number theory via Bernoulli numbers. The invariants we have in mind
are firstly certain characteristic classes, associated with a natural flat vect*
*or bundle
over Bg, secondly, the orbifold Euler characteristic of the group g and thirdly
its Yagita invariant. The characteristic classes are related to the denominator*
*s of
Bernoulli numbers, the Euler characteristic involves the whole Bernoulli number*
*s,
and our theorems concerning the Yagita invariant have to do with the notion of
regular primes, which is expressible in terms of numerators of Bernoulli number*
*s.
Although the three concepts which we study seem rather unrelated from the point
of view of their definitions, the fact that they all are tightly linked to prop*
*erties of
finite subgroups and their normalizers and centralizers in g renders it plausib*
*le,
that the resulting invariants must be somehow linked. The precise relationship,
however, remains for the time being a mystery.
We have tried to make these notes easy to read for non-experts. We therefore
recall in Sections 1 through 3 many facts and definitions, and state the releva*
*nt
properties of the mapping class group without proofs, in form of a survey. We a*
*lso
have included a substantial bibliography, helping the reader to find the proofs*
* of the
basic theorems of the subject, which are scattered through the literature and w*
*hich
are crucial for analyzing homological properties of the mapping class group. F*
*or
the classical part, not dealing with the cohomology of the mapping class group,*
* the
reader should consult Birman's book [Bi]. Section 4 contains a short introducti*
*on
to the theory of characteristic classes for group representations, and in Secti*
*on 5 we
1
2 MAPPING CLASS GROUPS, CHARACTERISTIC CLASSES AND BERNOULLI NUMBERS
compute the order of the Euler class e2g(g), associated with the flat bundle ov*
*er
Bg induced by the action of g on the homology group H1(Sg; R). In Section 6
this Euler class is related to the Euler characteristic O(g) of the group g, an*
*d in
Section 7 we discuss periodicity phenomena of g as well as the Yagita invariant.
1. The Definition of the Mapping Class Group
Let Sg denote a closed, connected and oriented topological surface of genus*
* g.
It is well-known that Sg admits a unique smooth structure; we shall also write *
*Sg for
the corresponding smooth (oriented) manifold. There are four basic ways of view*
*ing
the mapping class group g of the surface Sg, one being purely topological, the
second more geometric in nature, the third homotopical and the fourth algebraic,
involving the fundamental group of the surface in question only. The definition*
*s we
have in mind have the following form :
(I) g = Homeo+ (Sg)=Homeo0(Sg)
(II) g = Diffeo+ (Sg)=Diffeo0(Sg)
(III) g = Hoequ+ (Sg)=Hoequ0(Sg)
(IV) g = Out+ (ss1(Sg; s0)) .
We shall first give some background information and comments concerning these
equivalent definitions. Let s0 2 Sg denote a basepoint. The fundamental group of
Sg has a presentation
Y
ss1(Sg; s0) =< a1; b1; : :;:ag; bg| [ai; bi]>
and thus ss1(Sg; s0)ab ~=H1(Sg; Z) ~=Z2g. Since Sg is assumed to be orientable *
*one
has H2(Sg; Z) ~=Z. A map f : Sg ! Sg is said to be orientation preserving, if t*
*he
induced map
H2(f) : H2(Sg; Z) ! H2(Sg; Z)
is the identity map. It is useful to notice that this is equivalent to the requ*
*irement
that the determinant of
H1(f) : H1(Sg; Z) ! H1(Sg; Z)
equals one (the multiplicative structure of the cohomology ring H*(Sg; Z) revea*
*ls
that H2(f) is multiplication by detH1(f) ). Let Homeo(Sg) denote the topological
group of homeomorphisms of Sg, with the compact-open topology. We shall write
Homeo+ for the subgroup of orientation preserving homeomorphisms, and Homeo0
for the connected component of the identity (we use the "+" here as a subscript
rather than as a superscript, to avoid confusion with Quillen's plus-constructi*
*on).
The mapping class group g of the surface Sg is then defined as the discrete gro*
*up
of connected components
(I) g = Homeo+ (Sg)=Homeo0(Sg) :
We will consider (I) as our basic definition for g, and want to compare it with
(II),(III) and (IV). Consider now Sg as a smooth oriented manifold. In accor-
dance to the notation used above, we write Diffeo+ (Sg) for the group of orien-
tation preserving diffeomorphisms of Sg with the C1 -topology, and Diffeo0(Sg)
for the connected component of the identity. It was proved by Dehn [De] that
1. THE DEFINITION OF THE MAPPING CLASS GROUP 3
Homeo+ (Sg)=Homeo0(Sg) is generated by "Dehn twists", which are diffeomor-
phisms obtained by splitting Sg along a simple closed smooth curve, rotating one
part by 2ss, and gluing the surface back together. It follows that the natural *
*map
Diffeo+ (Sg) ! Homeo+ (Sg)=Homeo0(Sg)
is surjective. The kernel is precisely Diffeo0(Sg); namely, if f :Sg ! Sg is a
diffeomorphism isotopic to the identity (i.e. f 2 Homeo0(Sg)), then f is a fort*
*iori
homotopic to the identity, and therefore, according to Earle and Eells [Ea-Ee],*
* the
map f can be connected by a path in Diffeo+ (Sg) to the identity map. We have
thus established that
(II) g = Diffeo+ (Sg)=Diffeo0(Sg) :
In case g = 0, that is S0 = S2 the 2-sphere, Diffeo+ (S2) is connected; the inc*
*lusion
of SO(3) in Diffeo+ (S2) is actually a homotopy equivalence by Smale's result [*
*Sm].
Thus 0 = {e}. For g > 0 however, the mapping class groups g turn out to be
all non-trivial. The group 1 can be most easily understood using the definitions
(III) and (IV) respectively, which we shall discuss now. Let Hoequ+ (Sg) be the
topological group of orientation preserving homotopy equivalences of Sg with the
compact-open topology, and Hoequ0(Sg) the connected component of the identity.
By a result due to Nielsen ([Ni1]), the natural map
Homeo+ (Sg) ! Hoequ+ (Sg)=Hoequ0(Sg)
is surjective, and Baer proved [Ba] that any homeomorphism which is homotopic to
the identity, is actually isotopic to the identity, showing that the kernel is *
*precisely
Homeo0(Sg), (compare also Mangler [Ma]). Therefore, we conclude:
(III) g = Hoequ+ (Sg)=Hoequ0(Sg) :
Denote the set of free homotopy classes of maps between the spaces X and Y by
[X; Y ]. We have then a natural map
: Hoequ+ (Sg) ! [Sg; Sg] :
The homotopy set [Sg; Sg] may be identified with the set of orbits of the usual
ss1(Sg; s0)-action on the pointed homotopy set [(Sg; s0); (Sg; s0)]* of pointed*
* homo-
topy classes of pointed maps. Since for g > 0 the surface Sg has a contractible
universal covering space, there is a natural bijection
[(Sg; s0); (Sg; s0)]* ~=Hom(ss1(Sg; s0); ss1(Sg; s0)) ; (g > 0):
Passing to orbit spaces with respect to the ss1(Sg; s0)-action, we obtain
[Sg; Sg] ~=Rep(ss; ss) ;
where Rep(ss; ss) stands for the set of conjugacy class of homomorphisms ss ! s*
*s,
with ss = ss1(Sg; s0). Homotopy equivalences correspond then to automorphisms
modulo inner automorphisms of ss. If we denote by Out(ss) the group of outer
automorphisms of ss, we can view this group as a subset of Rep(ss; ss), and the*
* map
defined above yields a surjective homomorphism
Hoequ(Sg) ! Out(ss1(Sg; s0)) [Sg; Sg] :
4 MAPPING CLASS GROUPS, CHARACTERISTIC CLASSES AND BERNOULLI NUMBERS
The kernel consists of course of all homotopy equivalences homotopic to the ide*
*n-
tity. If we write Out+ for the "orientation-preserving" outer automorphisms, th*
*at
is, the subgroup of Out(ss1(Sg; s0)) consisting of those elements which act on *
*the
abelianized fundamental group ss1(Sg; s0)ab by a homomorphism of determinant
one, then we infer that
Hoequ+ (Sg)=Hoequ0(Sg) ~=Out+ (Sg) :
Note that the formula is also correct in case g = 0. From (III) we conclude th*
*en
that
(IV) g ~=Out+ (ss1(Sg; s0)):
For the case of g = 1 one has S1 = S1 x S1 a torus, and therefore 1 is isomorph*
*ic
to Out+ (Z Z) ~=Sl2(Z) :
2. Some Algebraic Properties of the Mapping Class Group
As mentioned earlier, g is generated by Dehn twists, associated with (isoto*
*py
classes) of simple closed curves on Sg and it follows that g is finitely genera*
*ted,
see Dehn [De]. An explicit finite set of generators is described in Lickorish *
*[Li].
Actually, g is finitely presented. This is obvious for 1. Indeed, 1 ~=Sl2(Z), so
that one obtains a finite presentation for 1 from the well-known decomposition *
*of
Sl2(Z) as an amalgamated free product :
Sl2(Z) ~=Z=4Z * Z=6Z :
Z=2Z
Note also that this implies
(1)ab ~=Z=12Z :
Birman and Hilden described in [Bi-Hi1] an explicit finite presentation for 2,
which can be used to show that
(2)ab ~=Z=10Z :
It was proved by Powell [Po] that, for higher genus the mapping class group is
always perfect :
(g)ab = 0 for g > 2 :
That g is finitely presented for general g was first proved by McCool in [McCo],
(see also Hatcher-Thurston [Ha-Th], as well as Wajnryb [Wa] for explicit presen-
tations).
A lot is known about finite subgroups of g. It follows from Harvey [Harv2]
that the number of conjugacy classes of finite subgroups of g is finite. The in*
*di-
vidual finite subgroups of g can be described as follows. Let o denote a complex
structure on Sg compatible with the smooth structure. Then the group of holomor-
phic automorphisms Aut(Sg; o ) is a subgroup of Diffeo(Sg). It is a classical r*
*esult
that for g > 1 the group Aut(Sg; o ) is finite and the induced map
o : Aut(Sg; o ) ! g (g > 1);
is injective (cf. Farkas-Kra [Fa-Kr]). According to Kerckhoff [Ke], the finite
subgroups of g are precisely the subgroups of the form o(F ) for F a finite gro*
*up of
holomorphic automorphisms of (Sg; o ) for some o , amounting to a positive solu*
*tion
of the Nielsen realization problem; for an account on the long history of this *
*problem
2. SOME ALGEBRAIC PROPERTIES OF THE MAPPING CLASS GROUP 5
as well as the partial results proved earlier, the reader should consult Ziesch*
*ang's
book [Zi]. Another classical result, due to Hurwitz [Hu], states that
|Aut(Sg; o )| 84(g - 1) ; (g > 1):
As a consequence, all finite subgroups F g satisfy the Hurwitz bound
|F | 84(g - 1) ; (g > 1):
For a finite cyclic subgroup F g this bound can be improved to
|F | 4g + 2 ;
and this bound is sharp, that is, g always contains a cyclic subgroup of order *
*4g+2,
see Wiman [Wi]. Note also that every finite group F admits an embedding into
some g since, as is well-known, every finite group occurs as a group of symmetr*
*ies
of some Riemann surface (see Broughton [Br] for an explicit condition, in terms*
* of a
presentation of F , for the existence of an embedding of F in g). It is even th*
*e case
that every finite group is isomorphic to the full automorphism group Aut(Sg; o )
of some closed Riemann surface, a result due to Greenberg [Gre]. The minimal
g = g(F ) such that F admits an embedding into g is called the genus of F . It
is an interesting problem to determine g(F ) for certain families of finite gro*
*ups F .
For the case of a finite cyclic group F of prime power order |F | > 2 the probl*
*em
is quite elementary and one finds g(F ) = 1_2OE(|F |), where OE denotes the Eu*
*ler-
function, see Glover-Mislin [Gl-Mi2]; the case of general finite cyclic groups *
*was
settled by Harvey [Harv1]. For a more involved example, the reader should consu*
*lt
Glover-Sjerve [Gl-Sj], where the genus of the group P Sl2(Fq) is computed for Fq
an arbitrary finite field.
Maps from g to finite groups were studied by Grossman in [Gro]. She proved
that g is residually finite (which means that g admits an embedding into a prod-
uct of finite groups). In particular, the elements of g may therefore be separa*
*ted
by means of finite dimensional linear representations. However, it is still an *
*open
question whether g admits a faithful finite-dimensional linear representation. *
*But
it is known that for g > 1, g is not isomorphic to an arithmetic group; for a
discussion of this fact see Ivanov [Iv2] or Harer [Ha5].
Another important result is that g contains a torsion-free subgroup of fini*
*te
index. This can be seen in an elementary way as follows. From our definition (I*
*V)
for g, there is a natural map ae : g Out(ss) ! Aut((ss)ab), where ss denotes t*
*he
fundamental group of Sg, and thus ssab ~=Z2g the abelianized fundamental group.
Hence there is a left exact sequence
Tg = kerae ! g ! Gl2g(Z) ;
where T g denotes the Torelli group, which is easily seen to be torsion-free (e*
*very
element of finite order in g can be realized as a holomorphic automorphism on s*
*ome
Riemann surface (Sg; o ), and these act non-trivially on H1(Sg; Z)). Since Gl2g*
*(Z)
contains a torsion-free subgroup of finite index, it follows that g too possess*
*es one.
To conclude this Section, we want to mention two additional basic results,
concerning infinite subgroups of g. The first one is an analogue of a Theorem
of Tits on linear groups. According to McCarthy [McC] and Ivanov [Iv1] the
following Tits-alternative holds for the mapping class group: every subgroup of*
* g
either contains a free subgroup on two generators, or a solvable subgroup of fi*
*nite
6 MAPPING CLASS GROUPS, CHARACTERISTIC CLASSES AND BERNOULLI NUMBERS
index. Solvable subgroups of g were analyzed in Birman-Lubotzky-McCarthy
[B-L-M]. They proved that every solvable subgroup of g is virtually abelian, and
that the maximal rank of a free abelian subgroup in g is, for g > 1, equal to 3*
*g -3.
3. Some Cohomological Results on g
Since Diffeo0(Sg) is a contractible space for g > 1 (see Earle-Eells [Ea-Ee*
*]),
the natural map of classifying spaces BDiffeo+ (Sg) ! Bg is a homotopy equiv-
alence, and thus
H*(g; Z) ~=H*(BDiffeo+ (Sg); Z) ; g > 1 :
One can therefore think of the cohomology elements of g as universal characteri*
*stic
classes for smooth orientable Sg-bundles. The most important tool for studying
the cohomology of g is its action on Teichm"uller space Tg, which is for g > 1
a smooth manifold homeomorphic to R6g-6. Teichm"uller space is a parameter
space for complex structures on the oriented closed smooth surface Sg, where two
complex structures on Sg are considered as equivalent if and only if there exis*
*ts
a diffeomorphism f : Sg ! Sg diffeotopic to the identity (i.e. f 2 Diffeo0(Sg)*
*),
carrying one complex structure to the other. According to Earle-Eells [Ea-Ee],
one can describe Tg as follows. Consider the space CS(Sg) of complex structures
on Sg compatible with the smooth structure and orientation; it carries a natural
topology and an obvious action of Diffeo+ (Sg). Then one has
Tg = CS(Sg)=Diffeo0(Sg) :
There remains a natural action of g = Diffeo+ (Sg)=Diffeo0(Sg) on Tg, which is
known to be properly discontinuous. The orbit space
Mg = Tg=g = CS(Sg)=Diffeo+ (Sg)
is called the moduli space of Sg. It has the structure of a complex variety and
its points correspond to conformal equivalence classes of complex structures on*
* Sg.
Because the action of g on Tg is properly discontinuous, the stabilizers of poi*
*nts
of Tg are finite. The natural projection
Tg ! Mg
is a branched covering space, and it can be thought of as a resolution of the s*
*ingu-
larities for the variety Mg. In particular, Tg inherits a natural complex stru*
*cture
such that g acts by complex automorphisms. By a result due to Royden [Roy],
g is actually the full automorphism group of Teichm"uller space with this compl*
*ex
structure. If t 2 Tg is fixed by the (finite) subgroup F g, then we can think
of F as a group of symmetries for a complex structure on Sg. On the other hand,
by the positive solution of the Nielsen realization problem mentioned earlier, *
*every
finite subgroup F g is a group of symmetries of some complex structure on
Sg and thus has a fixed point when acting on_Tg. Harvey proved in [Harv2] that
there is a contractible simplicial complex_T gcontaining Tg such that_the g-act*
*ion
extends to a proper simplicial action on T g, with compact quotient T g=g. From
general principles (see Brown's book [Brow1]), this immediately implies a wealth
of finiteness properties for g, the first of which has also been discussed in t*
*he
previous Section:
(1) g is finitely presented.
3. SOME COHOMOLOGICAL RESULTS ON g 7
(2) g is of finite virtual cohomological dimension (vcd); actually it follo*
*ws that
vcd(g) 6g - 6 = dim Tg, (g > 1). Note also that vcd(g) 3g - 3, since
g possesses a free abelian subgroup of rank 3g - 3. It has been proved *
*by
Harer in [Ha3] that, more precisely, one has
vcd(g) = 4g - 5 ; g > 1 :
Of course, vcd(1) = 1, since 1 ~=Sl2(Z) is an amalgamated free product
of finite groups.
(3) g is of type W F L (i.e. all torsion-free subgroups of finite index ad*
*mit
finitely generated free resolutions of finite length).
(4) For every subgroup H g of finite index, the homology groups Hi(H; Z)
are finitely generated, and they are finite if i > vcd(g). In particula*
*r, the
naive Euler characteristic
X
Oe(H) = (-1)irkHi(H; Z)
is well defined.
Remark. There are of course many other ways to define Teichm"uller space Tg.
We just want to mention one different definitions, which is discussed in Goldma*
*n's
paper [Go], and which is particularly attractive. By choosing a complex structu*
*re
on Sg one obtains an embedding in ss1(Sg) into P Sl2(R), which is the group
of isometries of the upper half-plane. Teichm"uller space can then be identifi*
*ed
with the component of in Rep(ss1(Sg); P Sl2(R)), the space of conjugacy classes
of homomorphisms from ss1(Sg) to P Sl2(R). Since ss1(Sg) is finitely presented,*
* this
space admits a natural embedding as a real algebraic subvariety in a quotient o*
*f a
product of copies of P Sl2(R). The action of g on Tg is just the one induced by
the action of the group of orientation preserving automorphisms of ss1(Sg) on t*
*he
space Hom(ss1(Sg); P Sl2(R)).
3.1 Rational cohomology. Next, we shall describe some results concerning
the rational cohomology of g. We first recall that, because the stabilizers of *
*the
g-action on the contractible space Tg are all finite, the rational cohomology o*
*f the
moduli space Mg satisfies
H*(Mg; Q) ~=H*(g; Q) ;
which explains the great interest in the rational cohomology of the mapping cla*
*ss
group. Obviously, 1 is Q-acyclic, being an amalgamated free product of finite
groups. By a result due to Igusa [Ig], 2 is Q-acyclic too, but this is not the *
*case
for g 3. Of course, for g 3, H1(g; Q) is trivial, since g is perfect. However,
Harer proved in [Ha4] that
H2(g; Q) ~=Q for g 3 ;
and he also showed that
H3(g; Q) = 0 for g 6 :
For all values of g one can describe as follows a part of the rational cohomolo*
*gy of
g whose dimension increases rapidly with g. According to Miller [Mil] and Morita
[Mo] there are classes yj in H2j(g; Q), j 1, such that the induced map of the
polynomial ring
: Q[y1; y2; y3 : :]:! H*(g; Q)
8 MAPPING CLASS GROUPS, CHARACTERISTIC CLASSES AND BERNOULLI NUMBERS
is injective in dimensions less than g=3. The classes yj for odd j can be descr*
*ibed
as symplectic characteristic classes in the following way, an interpretation wh*
*ich is
useful in many contexts. Consider the natural map
Bg ! BSp2g(R)
induced by the action of g on H1(Sg; R), viewed as a symplectic space using the
cup product. Since a maximal compact subgroup in Sp2g(R) is isomorphic to U(g),
one has
H*(BSp2g(R); Z) = Z[d1; d2; : :;:dg] ;
with dj 2 H2j(BSp2g(R); Z) the universal symplectic characteristic classes, cha*
*r-
acterized by the property that dj restricts to the universal Chern class cj of *
*the
maximal compact subgroup U(g) Sp2g(R). Each even indexed class d2j restrict
in H4j(g; Q) to the image of a polynomial in di's involving only odd i's; this *
*can
easily be seen using the fact that the rational Pontrjagin classes of a flat re*
*al vector
bundle vanish, and that the Pontrjagin class pi 2 H4i(BGl2g(R); Z) restricts wi*
*th
respect to the inclusion of Sp2g(R) in Gl2g(R) according to the formula
im (1 - p1 + p2 - : :):= (1 + d1 + d2 + : :):(1 - d1 + d2 - : :):2 H*(BSp2g(R);*
* Z) :
From a rational point of view, the only part of the map
H*(BSp2g(R); Z) ! H*(g; Z)
which is of any relevance, is therefore the induced map
H*(BSp2g(R); Q) Q[d1; d3; d5; : :]:-! H*(g; Q) :
It is shown in Miller [Mi] and Morita [Mo] that the image of this map agrees
with the image of the restriction of the map to Q[y1; y3; y5; : :]:, and in pa*
*rticular
is therefore injective in dimensions less than g/3 too.
3.2 Mod-p cohomology in the stable range. Again, we can look at the
representation of g obtained by letting g act on H1(Sg; R), yielding a homomor-
phisms
g ! Gl2g(R) ! Gl2g(C) :
Recall that
H*(BGl2g(R); F2) = F2[w1; w2; : :;:w2g] ;
a polynomial algebra in the universal Stiefel-Whitney classes. Kaufmann showed
in [Kau] that the odd Stiefel-Whitney classes w2j+1 restrict to zero in H*(g; F*
*2),
and that the induced map
H*(BGl2g(R); F2) F2[w2; w4; w6; : :]:! H*(g; F2)
is injective in dimensions less than g=3. He also proved (loc. cit.) a correspo*
*nding
result for the the case of an odd prime p, by considering
H*(BGl2g(C); Fp) = Fp[c1; c2; : :;:c2g] ;
where cistands for the mod-p reduction of the universal Chern class ci. The res*
*ult
is that the mod-p Chern classes ci restrict to zero in H*(g; Fp) in case i 6 0
mod (p - 1), whereas the induced map
Fp[cp-1 ; c2(p-1); : :]:! H*(g; Fp)
3. SOME COHOMOLOGICAL RESULTS ON g 9
is injective in dimensions less than g=3 (the dimension condition should be tho*
*ught
of as a stability condition, cf. (STAB) in Section 3.4.) Since [Kau] is not so *
*readily
available, we give a short outline of his proof. First, the vanishing condition*
*s follow
from standard results on characteristic classes of classical groups. Namely, t*
*he
natural inclusion of Sp2g(R) in Gl2g(R) induces a map
H*(BGl2g(R); F2) ! H*(BSp2g(R); F2)
for which the odd universal Stiefel-Whitney classes w2i+1 restrict to zero; the*
* cor-
responding result for g then follows, since the Gl2g(R)-representation of g con*
*sid-
ered above factors through Sp2g(R). For the case of the Chern classes one consi*
*ders
the restriction map
H*(BGl2g(C); Z) ! H*(BGl2g(Z); Z)
which maps the universal Chern class cj to cj(Z) 2 H2j(BGl2g(Z); Z), a torsion
class of order prime to p if j 6 0 mod (p - 1) (see [Eck-Mi2]). The mod p C*
*hern
classes cjrestrict therefore already in H*(BGl2g(Z); Fp) to zero, if j is not d*
*ivisible
by (p - 1); thus cj restricts to zero in H2j(g; Fp) too, if j 6 0 mod (p - 1)*
*. For
the part of Kaufmann's result which deals with injectivity, one proceeds as fol*
*lows.
One makes use of the mapping class groups g;1 of "oriented surfaces of genus g,
with one boundary component", which can be arranged to form a natural increasing
sequence
g;1,! g+1;1
and one defines the stable mapping class group by putting
1 := [gg;1:
The point is that both natural maps
g g;1! 1
induce integral cohomology isomorphisms in dimensions less than g=3, see Harer
[Ha2], so that 1 is suitable for computing the cohomology of g in that range.
Note also that g;1acts on the homology of the surface Sg;1= Sg \ D, with D the
interior of a closed disk. Because the inclusion Sg;1 Sg induces an isomorphism
in H1, one obtains natural representations g;1! Sl2g(Z), which are compatible
with the corresponding representations of g. Moreover, there are natural pairin*
*gs
g;1x h;1! g+h;1
compatible with the usual pairings
Sl2g(Z) x Sl2h(Z) ! Sl2g+2h(Z) ;
inducing H-space structures on B+1 and BSl(Z)+ respectively, where the plus
stands for Quillen's plus-construction. By naturality, the induced map
B+1 ! BSl(Z)+
is an H-map, thus inducing morphisms of Hopf-algebras
H*(BGl(C); Fp) ! H*(BGl(R); Fp) ! H*(B1 ; Fp) ;
10 MAPPING CLASS GROUPS, CHARACTERISTIC CLASSES AND BERNOULLI NUMBERS
with the Hopf-algebra structure on H*(BGl(C); Fp) and H*(BGl(R); Fp) induced
by the Whitney sum construction. We assume now that p is an odd prime; in case
p = 2 one argues similarly. The Hopf-algebra structure on
A* = H*(BGl(C); Fp)
is given by
Xk
(ck) = ck-i ci:
i=0
Now consider B* = Fp[cp-1 ; c2(p-1); : :]:, with a Hopf-algebra structure defin*
*ed by
Xk
ck(p-1) = c(k-i)(p-1) ci(p-1):
i=0
Although the inclusion B* A* is not a morphism of Hopf-algebras, any Hopf-
algebra map A* ! H*(1 ; Fp) which maps cj to zero for j not divisible by
(p - 1), will restrict to a morphism of Hopf-algebras on B*. Note also that a
morphism of (graded) Hopf-algebras with domain B* is injective, if it is inject*
*ive
when restricted to the subspace P B* of primitive elements of B*. One checks th*
*at
the graded vector space P B* has a basis consisting of the Newton polynomials
Nk = Nk(cp-1 ; c2(p-1); : :):, given by the usual recursion formula:
N1 = cp-1
and, for k > 1
Nk = cp-1Nk-1 - c2(p-1)Nk-2 + : :+:(-1)k-2 c(k-1)(p-1)N1 + (-1)k-1 kck(p-1):
Kaufmann proves then that these classes Nk 2 H2k(p-1)(BGl(C); Fp) do not restri*
*ct
to zero in H2k(p-1)(1 ; Fp), by evaluating them on a suitable subgroup of order*
* p
in (pn-1)(p-1)=2, with n >> k. His result then follows.
3.3 The case of genus less than three. Since 1 is isomorphic to an
amalgamated free product of a cyclic group of order four and a cyclic group of
order six over a cyclic group of order two, the Mayer-Vietoris sequence reveals*
* that
H*(1; Z) = Z[x]=(12x) ;
with x 2 H2(1; Z). An explicit description of x can be given as follows. Consid*
*er
the flat real vector bundle over B1, induced by the inclusion of 1 = Sl2(Z) into
Sl2(R). One checks that the universal Euler class e2 in H2(BSl2(R); Z) restrict*
*s to a
generator of H2(C; Z) for any cyclic subgroup C Sl2(R). It therefore restricts*
* to a
generator e2(1) = x in H2(1; Z). Also, it follows that the projection 1 ! (1)ab,
a cyclic group of order 12, induces in integral cohomology an isomorphism.
As mentioned earlier, 2 is Q-acyclic and therefore, because the integral co*
*ho-
mology groups of g are finitely generated, Hk (2; Z) is a finite group for k > *
*0.
Moreover, Lee-Weintraub proved in [Le-We] that 2 is Fp-acyclic for all primes
p > 5. For the remaining primes p = 2; 3 and 5 it is useful to study the short *
*exact
sequence
0 ! Z=2Z ! 2 ! 60! 0 ;
where 60is the mapping class group of the 2-sphere with 6 punctures, that is
60= ss0(Diffeo+ (S2 \ {x1; : :;:x6})) :
3. SOME COHOMOLOGICAL RESULTS ON g 11
This short exact sequence is discussed in Birman-Hilden [Bi-Hi1]. It is obtained
by considering the genus two surface S2 as a branched covering space of the 2-
sphere S2 with six branch points, by forming the quotient surface S2= < o >, o *
*the
hyperelliptic involution, which is known to generate the center of 2. There is *
*an
obvious map 60! 6, the symmetric group on six letters, with kernel denoted by
K6. An analysis of K6 led Cohen and Benson [Co4][Be][Be-Co] to the following
results concerning the mod-p cohomology of 2.
(2 mod p)
a) There is a subgroup Z=5Z 2 such that the restriction map induces an
isomorphism
H*(2; F5) ~=H*(Z=5Z; F5) :
b) There are elements x, y, z and w of degree 3, 4, 4 and 5, such that
H*(2; F3) ~=F3[x; y; z; w]= < x2; xz; z2; zw; w2; yz - xw > :
c) The Poincare series of H*(2; F2) is given by
1_+_t2_+_2t3_+_t4_+_t5_= 1 + t + 2t2 + 4t3 + 6t4 + . .:.
(1 - t)(1 - t4)
A discussion of the integral cohomology of 2 is presented in Cohen [Co3], see a*
*lso
[Co1-2]; we want to mention in particular the following result, which we will u*
*se
later on:
120 . Hi(2; Z) = 0 for i > 0:
Away from the prime 2 the cohomology is particularly easy to describe, because *
*it
is "periodic with period 4 from dimension four on". It can be expressed as foll*
*ows:
H*(2; Z[1=2]) = Z[1=2][x; y; z]= < 5x; 3y; 3z; z2 > ;
with x 2 H2, y 2 H4 and z 2 H5. The groups in low dimension are
8
>>>0; for i = 1
>> Z=2Z; for i = 3
>>> 2
>: Z=120Z (Z=2Z) ; for i = 4
Z=6Z (Z=2Z)2; for i = 5:
3.4 Torsion in the cohomology of the mapping class group. The
Bernoulli numbers Bn are rational numbers defined recursively by the formula
(B + 1)#n - Bn = 0 ; n 2 ;
where the exponent "# n" means that after evaluating the n'th power of the mono-
mial, one replaces the power Bk by Bk. For n = 2 this yields B2+2B1+1-B2 = 0,
hence B1 = -1=2. It turns out that for odd n > 1 one always has Bn = 0 and,
as already observed by Euler, the B2k's are related to the Taylor series of tan*
*(x),
which is given by
X1 22k(22k - 1)B
x + 1_3x3 + _2_15x5 + _17_315x7 . .=. (-1)k-1 _____________2k_x2k-1 :
k=1 (2k)!
12 MAPPING CLASS GROUPS, CHARACTERISTIC CLASSES AND BERNOULLI NUMBERS
Thus B2 = 1_6, B4 = - 1_30, B6 = _1_42, . .,.B12 = - __691___2.3.5.7.13, . . .a*
*nd so on. There are
several conflicting notations in use concerning Bernoulli numbers. The one we u*
*se
here differs by a sign from the one used in [Gl-Mi1] and [Gl-Mi2], but agrees w*
*ith
the one used in [Brow] and [Ha-Za]. The Bernoulli numbers turn up in number
theory in several places. Our convention is such that for any integer k > 0, t*
*he
Riemann zeta function satisfies the equation
i(1 - 2k) = -B2k=2k :
In Glover-Mislin [Gl-Mi2] it is proved that for g > (8m)2 the cohomology
group H4m (g; Z) contains an element of order
E2m = den(B2m =2m) ;
the denominator of B2m =2m, expressed as a fraction in lowest terms (E2 = 12,
E4 = 120, E6 = 252; : :):. On the other hand, Harer proved in [Ha2] a stability
result for g :
(STAB) Hk (g; Z) is for g > 3k independent of g.
It follows then that :
(TOR) H4m (g; Z) contains for g > 12m an element of order E2m .
This result will be improved in Section 5. In [Gl-Mi2] the torsion result (TOR)
was used to construct strange torsion in the cohomology of g (torsion, which is
not present in the group g itself). It is shown that for p a prime larger than*
* 13
and g = g(p) = (p2 - 4p + 1)=2, the mapping class group g(p)is p-torsion-free, *
*but
H2(p-1)(g(p); Z) contains an element of order p.
Remark. An alternative way to construct torsion in H*(g; Z) is to use a
result due to Charney-Cohen [Ch-Co], which states that, in the stable range, the
cohomology of g contains a direct summand isomorphic to the cohomology of
Im J1=2, where Im J1=2 is a space which is a factor of BGl(Z)+ , usually referr*
*ed to
as "the image of the J-homomorphism localized away from 2".
3.5 Periodicity and Krull dimension. It is well-known that for a finite
group F the cohomology ring H*(F ; Fp) is noetherian. More generally, Quillen's
Proposition 14.5 of [Qu] implies that if denotes a group of finite virtual coh*
*omo-
logical dimension acting simplicially on a finite dimensional contractible simp*
*licial
complex, with compact quotient and finite stabilizers, then H*(; Fp)_is_noether*
*ian.
In particular, this implies that, by considering the action of g on T g:
(NOETH) H*(g; Fp) is a noetherian ring:
To get an idea of the growth rate of Hn as n ! 1, one considers the Krull
dimension. Recall that the Krull dimension of a commutative ring R with 1 is
defined as the supremum of the lengths n of chains of distinct prime ideals
p0 p1 : : :pn :
For an arbitrary (discrete) group and prime p, the Krull dimension of at p,
(; p), is per definition the Krull dimension of the commutative ring Hev(; Fp) *
*of
even dimensional cohomology classes. If H*(; Fp) is noetherian, standard results
3. SOME COHOMOLOGICAL RESULTS ON g 13
from commutative algebra imply that (; p) is the smallest integer 0 such
that there is a constant C > 0 satisfying for all n 0
X
dim Fp Hi(; Fp) C . n :
in
Note that in this last formula we did not restrict to the even cohomology; inde*
*ed
one easily checks that for a finitely generatedPgraded-commutative Fp-algebra H*
satisfying for all n 0 the conditionP 2i2n dim H2i C(2n) with C > 0, one
can find a constant D > 0 such that in dim Hi Dn , and conversely. In
particular, if H*(; Fp) is noetherian then
(; p) = 0 () dim FpH*(; Fp) < 1 :
For groups of finite virtual cohomological dimension, the prime ideals of the
ring Hev(; Fp) are intimately linked to the elementary abelian subgroups of ; in
case has only finitely many conjugacy classes of elementary abelian p-subgroup*
*s,
the minimal prime ideals are in one-one correspondence with the conjugacy class*
*es
of maximal elementary abelian p-subgroups (see Quillen [Qu]). It was moreover
proved in [Qu] that for a finite group F the Krull dimension (F; p) equals the
maximal rank of an elementary abelian p-subgroup of F . This result still holds*
* for
certain infinite groups, in particular for g, as was proved by Broughton in [Br*
*].
Note that from our description of the cohomology of 1 and 2 one readily sees
that:
o (1; 2) = (1; 3) = 1 ; and (1; p) = 0 for p > 3 ;
o (2; 2) = 2 ; (2; 3) = (2; 5) = 1 ; and (2; p) = 0 for p > 5 :
In [Br] Broughton established an explicit general formula for the Krull dimensi*
*on
of g, by determining the maximal rank of an elementary abelian p-subgroup con-
tained in the mapping class group:
(KRULL-DIM I)
Let g > 1. Then the Krull dimension (g; p) is the largest integer such that
there are nonnegative integers k 6= 1 and h satisfying:
1) 2g - 2 = p (2h - 2) + p-1 (p - 1)k, and
2) 2h if k = 0, and
3) < 2h + k if k > 1.
It will be useful to record the following immediate consequences.
(KRULL-DIM II)
Let g > 1. Then the Krull dimension of g satisfies the following:
1) Case p = 2 : (g; 2) 2.
1.1) if g is even, (g; 2) = 2
1.2) if g is odd, (g; 2) 3
2) Case p odd:
2.1) if g 6 1 mod p, then (g; p) 1
2.2) if g 1 mod p, then (g; p) 1 and if we write g in the form l.pf*
*f+1
with l prime to p and ff > 0, then (g; p) ff + 1 ; moreover, if l >> p
one has (g; p) = ff + 1 :
14 MAPPING CLASS GROUPS, CHARACTERISTIC CLASSES AND BERNOULLI NUMBERS
A particularly interesting case arises when (g; p) = 1. It was observed by Venk*
*ov
[Ve] that if is a group of finite virtual cohomological dimension with (; p) *
*1,
then H*(; Fp) is "periodic in sufficiently high dimensions", meaning that
9k > 0; 8i >> 0 : Hi(; Fp) ~=Hi+k (; Fp) :
Periodicity phenomena are much easier to handle using Farrell cohomology instead
of ordinary cohomology. We will write
H^i(; M) ; i 2 Z
for the i-th Farrell cohomology group [Fa] of the group of finite virtual coho*
*mo-
logical dimension, and M a -module. For an in depth discussion of the properties
of these Farrell cohomology groups, see Brown's book [Brow1]. The interested
reader might also consult [Mis2], where "Tate cohomology groups" H^i(; M) are
defined for arbitrary groups , in a way that for groups of finite virtual cohom*
*o-
logical dimension one obtains the Farrell cohomology groups, and thus for finite
groups the classical Tate cohomology groups. Here are some basic facts concerni*
*ng
these generalized Tate groups.
(TATE-GROUPS)
a) For an arbitrary group and projective -module P one has ^H*(; P ) = 0
b) H^0(; Z) = 0 () cd() < 1
c) Suppose vcd() < 1. Then
c1) H^i(; M) is a torsion group for all i and all -modules M
c2) if i > vcd(), the natural map Hi(; M) ! ^Hi(; M) is an isomorphism
for all -modules M.
c3) the following conditions are equivalent :
c3.1) every abelian p-subgroup of has rank 1
c3.2) there exists a d > 0 such that H^i(; M)(p)~= ^Hi+d(; M)(p)for a*
*ll
-modules M and all i 2 Z (the smallest such d is called the p-period *
*of
an is denoted by p(); for an abelian torsion group A and prime p we
write A(p)for its p-torsion subgroup)
c3.3) the ring H^*(; Z)(p)contains an invertible element of positive *
*de-
gree.
Property a) results from the definition of these general Tate groups,, which we
will not recall here; the interesting fact b) was proved by Kropholler [Kr1-2],*
* and
the list c) just recalls some of the basic results on Farrell cohomology. A gr*
*oup
satisfying one of the equivalent conditions listed under (c3) above will be cal*
*led
p-periodic (see Xia [Xi1] for a discussion of that concept). For a group like *
*g,
being p-periodic is therefore equivalent to the condition that its Krull dimens*
*ion
be less than or equal to one. A basic example of an infinite p-periodic group i*
*s the
semidirect product
S(n,p):= Z=pnZ o Z ;
with p-period equal 2(p - 1)pn-1 in case p is an odd prime, n 1, and Z is acti*
*ng
by means of a surjective map Z ! Aut(Z=pnZ). It was proved in [G-M-X1] that
in general, the p-period of a p-periodic group divides 2(p - 1)pn for some n *
*0.
For g one has a stronger result. It was shown in [G-M-X1] that the p-period of a
p-periodic g actually always divides 2(p - 1), and the exact value of the p-per*
*iod
3. SOME COHOMOLOGICAL RESULTS ON g 15
as a function of p and g was computed in an explicit way. Moreover, the values *
*of
g for which g is p-periodic were determined by Xia in [Xi2]. The results on the
p-period of g may be summarized as follows:
(p-PERIOD)
Let g > 1. Then
a) g is never 2-periodic
b) for an odd prime p, g is p-periodic if and only if one of the follow*
*ing
two condition holds:
b1) g 6 1 mod p
b2) g is of the form kp + 1 with k 6 0; -1 mod p and the interval
[(2k + 3)=p; (2k + 2)=(p - 1)] does not contain any integer
c) The p-period of p-periodic g, denoted by p(g), is given by
p(g) = lcm{2[N(ss) : C(ss)]|ss 2 P } ;
where ss ranges over the set P of subgroups of order p in g, and N(ss)
(respectively C(ss)) denotes the normalizer (respectively centralizer) *
*of ss
in g. In particular, the p-period p(g) divides 2(p - 1).
We use the convention that lcm{2[N(ss) : C(ss)]|ss 2 P } = 1 in case P is the e*
*mpty
set; in that case p(g) = 1 too, according to our definition. It is possible to *
*convert
the general formula for p(g) in an explicit formula in terms of g and p as foll*
*ows.
If ss g is a subgroup of order p then, as discussed earlier, one can lift ss to
a subgroup of Diffeo+ (Sg), and it is a classical result, that the number of fi*
*xed
points n(ss) of this ss-action on Sg does not depend on the lift chosen. It was*
* proved
in [G-M-X1] that for g > 1 one has
lcm{2[N(ss) : C(ss)] | ss 2 P } = lcm{gcd(2(p - 1); 2n(ss)) | ss 2 P }*
* :
According to Xia [Xi2], the numbers n(ss), which occur as cardinalities of such*
* fixed-
point sets, form a set denoted by Bg;p, which is given by the following formula*
*. We
may restrict to the case p odd, since g is never 2-periodic for g > 1. Write 2g*
* - 2
in the form mp - i with 0 i < p. Then
ae{i; i + p; : :;:i + [2g=(p - 1) - m]p};if i 6 1 mod p
Bg;p=
{1 + p; : :;:1 + [2g=(p - 1) - m]p}; if i 1 mod p:
As usual, the notation [x] stands for the integral part of the rational number *
*x. We
also use the natural convention that gcd(2(p - 1); 0) = 2(p - 1) and, as before*
*, the
lcm of an empty set of numbers is understood to equal 1. For example, one easily
checks that for p = 3 the sets Bg;p contain always an even number. We also note,
by using (KRULL-DIM I) or by checking the condition (b2) of (p-PERIOD), that
for g > 1 and g 1 mod 3, g is never 3-periodic. This shows that the following
holds.
(3-PERIOD)
Suppose g > 1. Then g is 3-periodic if and only if g 6 1 mod 3 and the
3-period of g is always 4.
It is easy to see from the definition of the p-period that a subgroup of a p-pe*
*riodic
group is also p-periodic, with p-period dividing the p-period of . As a resul*
*t,
one concludes for example:
16 MAPPING CLASS GROUPS, CHARACTERISTIC CLASSES AND BERNOULLI NUMBERS
o The groups S(2,3)= Z=9Z o Z cannot be embedded into any mapping class
group g with g 6 1 mod 3.
Indeed, using the same notation as earlier, S(2; 3) is 3-periodic with 3-period*
* 12,
which is larger than 4, the 3-period of a 3-periodic g. Returning to the case *
*of
an arbitrary prime p, we like to mention one more result, which follows from (p-
PERIOD). Consider g with g 1 mod p. As mentioned earlier, there can only
be a finite number of such g's which are p-periodic (p a fixed prime). On the o*
*ther
hand, because Sg can be considered as an unramified covering space of Sh with h
given by p(2 - 2h) = 2 - 2g, it follows that g contains a subgroup ss of order p
satisfying n(ss) = 0, corresponding to a fixed-point-free action. From our form*
*ula
for the p-period we infer thus:
o Suppose g is p-periodic and g 1 mod p. Then its p-period is 2(p - 1).
We would like to conclude this Section by mentioning computations, which involve
the mapping class groups (p-1)=2, respectively (p-1), and which demonstrate the
power of using p-periodicity and the p-period in the course of computing Farrell
cohomology. For an odd prime p, the smallest value g 1 such that g has p-
torsion, is g = (p - 1)=2, and g = p - 1 is the second smallest such value. The*
* Krull
dimension at p > 2 is 1 for (p-1)=2and p-1 , so that these groups are p-periodi*
*c.
The p-primary part of the Farrell cohomology has been completely computed for
these two families of examples by Xia in his papers [Xi3-4].
4. Characteristic Classes for Group Representations
In Eckmann-Mislin [Eck-Mi1-5] characteristic classes of group representatio*
*ns
where discussed in relationship with their field of definition. In our applica*
*tions
here we will mainly be concerned with representations defined over Q. But first
we shall recall some basic facts; a general reference on characteristic classes*
* of
representations is Thomas' book [Th1].
4.1 Chern classes. If ae : G ! Gln(C) denotes a complex representation of
the (discrete) group G, then the induced map of classifying spaces
(Bae)* : H*(BGln(C); Z) ! H*(G; Z)
maps the universal Chern classes ci 2 H2i(BGln(C); Z) to the Chern classes of a*
*e,
defined by
ci(ae) := (Bae)*ci 2 H2i(G; Z) :
To understand the Chern classes of representations of finite groups, it is usef*
*ul to
analyze first the case of cyclic groups, which we will quickly review. The boun*
*dary
homomorphism associated to the short exact sequence
0 ! Z ! C -exp-!Gl1(C) ! 1
induces an isomorphism
c1 : Hom (Z=nZ; Gl1(C)) ~=H2(Z=nZ; Z) ;
mapping a one dimensional representation to its first Chern class. Since every
complex representation ae of Z=nZ decomposes as a sum aei of one dimensional
representations, we can express the total Chern class
c(ae) = 1 + c1(ae) + c2(ae) + : : :
4. CHARACTERISTIC CLASSES FOR GROUP REPRESENTATIONS 17
as Y Y
c(ae) = c(aei) = (1 + c1(aei)) :
Let OE(n) denote the number of generators of the cyclic group Z=nZ (the Euler O*
*E-
function). Note that there are OE(n) faithful irreducible C-representations of *
*Z=nZ,
and their sum, which we denote by oen, is a representation of degree OE(n), whi*
*ch
is defined over Z; the representation oen is sometimes called the cyclotomic re*
*pre-
sentation, since as a Q-representation it is equivalent to the one obtained fro*
*m the
Galois action of Gal(Q(in)=Q) on the cyclotomic extension of Q gotten by adjoin-
ing a primitive n-th root of unity in. It is obvious that any faithful irreduc*
*ible
representation of Z=nZ over Q must involve oen and thus oen is characterized as
being the smallest faithful irreducible Q-representation of Z=nZ. Note that its*
* top
Chern class
cOE(n)(oen) 2 H2OE(n)(Z=nZ; Z)
has (maximal) order n, because the total Chern class c(oen) is given by
Y Y
(1 + jc1(oen)) = 1 + : :+: j . c1(oe)OE(n):
j;(j;n)=1 j;(j;n)=1
Recall that E2m denotes the denominator of B2m =2m. From the well-known di-
visibility properties of Bernoulli numbers one can infer that a prime power pk *
*> 1
divides E2m if and only if (p - 1)pk-1 divides 2mi (von Staudt's Theorem). Thus
E2m = lcm{n | 2m 0 mod OE(n)} :
Therefore, if ps > 1 denotes the highest power of a prime p dividing E2m , we c*
*an
write 2m in the form (p-1)ps-1 .t with t prime to p. If oe denotes the represen*
*tation
given by taking a t-fold sum of the cyclotomic representation of Z=psZ composed
with the projection of Z=E2m Z onto Z=psZ, then c2m (oe) will have order ps. Ta*
*king
a sum of representations of this type, one for each prime divisor of E2m , one *
*ends
up with a representation
m : Z=E2m Z ! Gl(Z)
with c2m (m ) 2 H4m (Z=E2m Z; Z) of (maximal) order E2m . The main result of
[Eck-Mi1] says that this order is optimal for Q-representations of arbitrary fi*
*nite
groups, in the following sense:
o If ae : F ! Gl(Q) denotes an arbitrary representation of a finite group*
* F
over Q, then the order of ci(ae) is at most two for i odd, and it divid*
*es Ei
for i > 0 even.
The bound is actually also best possible for Q-representations of infinite grou*
*ps, up
to possibly a factor 2. This follows from [Eck-Mi4] in conjunction with Arletta*
*z's
result [Ar] who proved that the universal Chern classes restrict to torsion cla*
*sses
in the cohomology of the group Gl(Q) considered as a discrete group (the compli-
cation to overcome is the fact that the integral homology of Gln(Q) is not fini*
*tely
generated); see also [Mis1] for a general torsion result on characteristic clas*
*ses.
4.2 The Euler class. It is well-known that the integral cohomology ring of
BSln(R) is generated by elements of order two, the Euler class
en 2 Hn (BSln(R); Z) ;
18 MAPPING CLASS GROUPS, CHARACTERISTIC CLASSES AND BERNOULLI NUMBERS
and the Pontrjagin classes
pj 2 H4j(BSln(R); Z); 2j n :
For n odd, the Euler class has order 2. If n is even, say n = 2m, one has a rel*
*ation
of the form
e22m= pm = (-1)m res(c2m )
with c2m 2 H4m (BGl(C); Z) the universal Chern class, and res(c2m ) the image
under the restriction map induced by the inclusion of Sl2m (R) in Gl(C). For any
subring R of R, we will write en(R) for the restriction res(en) 2 Hn (BSln(R)ff*
*i; Z),
where BSln(R)ffistands for the classifying space of the group Sln(R), considered
as a discrete group. In case that R is a discrete subring of R, we will omit t*
*he
superscript ffi; a similar convention is used for the case of the Chern classes*
*. The
Euler class
e2m (Q) 2 H2m (BSl2m (Q)ffi; Z)
has infinite order (see Milnor [Miln]), but, according to Sullivan [Su]
o e2m (Z) 2 H2m (BSl2m (Z); Z) is a torsion class.
The order of e2m (Z) is _ the same number is coming up again _ E2m or 2E2m .
For the restriction of e2m (Q) to the cohomology of a finite subgroup Sl2m (Q) *
*it was
proved in [Eck-Mi1] that the "universal bound" for the order is precisely E2m :
o If ae denotes a representation F ! Sl2m (Q) of a finite group F , then *
*the
order of e2m (ae) := (Bae)*e2m (Q) divides E2m , and E2m is the best un*
*iversal
bound for the order of the Euler class of such representations.
One should note that a finite subgroup of Sln(Q) is not necessarily conjugate t*
*o a
subgroup of Sln(Z)!
5. Torsion in g and the Homology Representation
Our goal in this Section is to improve the (TOR)-result concerning torsion *
*in
the integral cohomology of the mapping class group, stated in Section 3.3. The
techniques we use are essentially the ones used in Glover-Mislin [Gl-Mi2], just*
* a
little bit refined.
5.1 Fixed point data. A basic invariant of an orientation preserving diff*
*eo-
morphism of finite order n > 1, f 2 Diffeo+ (Sg) ; is its fixed point data. It*
* is
defined as follows. Because f preserves orientation, the singular set of f, th*
*at is,
the points x 2 Sg for which the orbit {fk (x)|k 2 Z} has fewer than n elements,*
* is
necessarily a finite set. Let {xi} be a set of representatives of the singular*
* orbits
of f and write ni for the order of stabf(xi), the stabilizer of < f > at xi 2 S*
*g,
where < f > denotes the subgroup generated by f. Then fn=ni generates stabf(xi)
and, with respect to a fixed Riemannian structure, the differential of fn=ni ac*
*ts by
rotation on the tangent space at xi. Let ki be an integer such that fkin=ni act*
*s by
rotation through 2ss=ni. The number ki is well defined modulo ni, and ki is pri*
*me
to ni. The fixed point data of f, denoted by ffi(f), is then the collection
ffi(f) =< g; n | k1=n1; : :;:kq=nq >
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 k1=n1; : :;:kq=nq are unique up to order, i*
*f we
choose ki so that 1 ki < ni.
5. TORSION IN g AND THE HOMOLOGY REPRESENTATION 19
A classical theorem of Nielsen [Ni2] states that two diffeomorphisms of fin*
*ite
order are conjugate in Diffeo + (Sg) if and only if they have the same fixed po*
*int
data. Symonds [Sy] proved that the fixed point data of a diffeomorphism of fini*
*te
order depends only upon its isotopy class, that is, its image in g. By the clas*
*sical
case of the "Nielsen Realization Theorem" (cf. Fenchel [Fe]), every element of *
*finite
order of g can be represented by a diffeomorphism of the same order. It follows
that one can define the fixed point data ffi(x) for a torsion element x 2 g by *
*putting
ffi(x) = ffi(f), where f denotes any lift of x to Diffeo+ (Sg) of the same orde*
*r. As a
consequence we infer:
o Two elements of finite order in g are conjugate if and only if they have
the same fixed point data.
If x 2 g has finite order and has fixed point data
ffi(x) =< g; n | k1=n1; : :;:kq=nq >
and if f denotes a diffeomorphism of order n representing x, then the orbit spa*
*ce
Sg= < f > is a surface Sh such that the natural projection ss : Sg ! Sh is an n-
sheeted branched covering, with q branch points in Sh corresponding to the sing*
*ular
orbits of the action of the group < f > generated by f. The order of a branch
point P 2 Sh is defined as n=|ss-1 (P )|, and these orders correspond therefore*
* to
the orders of stabilizers, denoted by ni above. From covering space theory one
sees that the genus h is determined by g and the fixed point data ffi(x) via the
Riemann-Hurwitz Relation:
Xq
(R-H) 2g - 2 = n((2h - 2) + (1 - 1=ni)) :
i=1
Remark. The branching orders can be computed by counting the numbers
of fixed points of the different powers of the map f. The number of such fixed
points can be computed from the representation of the group < f >~= Z=nZ on
H1(Sg; R), by the Lefschetz-Hopf trace formula. From character theory it is then
plain that the conjugacy class of the image of f in Gl2g(R) with respect to this
representation, is determined by the branching numbers {ni}. We could also look*
* at
the conjugacy class of f in Sp2g(Z), by considering the action of f on the symp*
*lectic
space H1(Sg; Z), the symplectic structure being given by intersection product. *
*In
case of n = p a prime, this conjugacy question was analyzed by Edmonds and
Ewing [E-E]. The fixed point data of an x 2 g of order p has the form
ffi(x) =< g; p | k1=p; : :;:kq=p >
where 0 < ki < p; the number q here equals the cardinality of the fixed point s*
*et of
any diffeomorphism of order p representing x, and ki is called the type of the *
*i'th
fixed point. Note that q can be computed as the Lefschetz number (x) of x, that
is X
(x) = (-1)itrace(x* : Hi(Sg; Q) ! Hi(Sg; Q)) = q :
The fixed point types come up in the formula for the signature of x, given by
X
sign(x) = Nk(ik + 1)=(ik - 1)
0 0,
and the Euler class e2g(g), are torsion classes. We want to establish the follo*
*wing
result concerning their order.
Tor-Theorem. Denote as before the denominator of B2m =2m by E2m , so
that E2 = 12, E4 = 120, E6 = 252 and so on. Then the following holds.
(1) c2(1) 2 H4(1; Z) and e2(1) 2 H2(1; Z) both have order 12.
(2) c4(2) 2 H8(2; Z) and e4(2) 2 H4(2; Z) both have order 120.
(3) For g > 2, the order of c2g(g) 2 H4g(g; Z) and e2g 2 H2g(g; Z) is either
E2g or 2E2g.
Proof. Because of the results on the order of ci(Z) and en(Z) mentioned
earlier, it suffices to show that c2g(g) has order at least E2g. It follows the*
*n that
e2g(g) has order at least E2g, as
e2g(g)2 = (-1)gc2g(g) :
Also, for g 2 the computations concerning H*(g; Z) rule out the existence of
elements of order 2E2g in H*(g; Z). To get the general lower bound on the order
of c2g(g), we proceed as follows. Suppose that pfi(with fi 1) is the largest
power of a prime p, which divides E2g. Then, by von Staudt's theorem, pfi-1(p -*
* 1)
divides 2g, say
2g = l . pfi-1(p - 1) :
Next, we construct a subgroup
ss g; ss ~=Z=pfiZ
such that the restriction c2g(ss) 2 H2g(ss; Z) of c2g(g) has order pfi. To this*
* end,
we consider the branched covering space
Sg ! S2
5. TORSION IN g AND THE HOMOLOGY REPRESENTATION 21
with two branch points of order pfiand l of order p. We recall the classical c*
*on-
struction of such a covering space. One begins by deleting 2 + l points from t*
*he
2-sphere to get
X = S2 \ {x1; x2; y1; : :y:l}
and chooses a suitable surjective homomorphism
@ : ss1(X) ! Z=pnZ :
To describe @ more explicitly, we choose a presentation of ss1(X) of the form
< u1; u2; v1; : :;:vl| u1u2v1 : :v:l= 1 >
and a generator x 2 Z=pfiZ. Let's restrict to the case p odd and fi > 1; the ot*
*her
cases are similar. Put @(vi) = y for 1 i l, where y denotes a fixed element
of Z=pfiZ of order p, and define @(u1) = x respectively @(u2) = -(x + ly) so
that @ is well defined and surjective. By compactifying the regular covering sp*
*ace
associated to the kernel of @, one obtains a branched covering Sg ! S2, with ge*
*nus
g determined by the Riemann-Hurwitz relation (H-R),
2g - 2 = pfi(-2 + (2 - 2=pfi) + (l - l=p)) ;
yielding 2g = l(p - 1)pfi-1, as desired. The associated covering transformation
group is cyclic of order pfiand acts with 2 + l branch points, two of order pfi*
*and l
of order p. It defines therefore a subgroup ss g, generated by an element z wi*
*th
fixed point data
ffi(z) =< l(p - 1)pfi-1=2 ; pfi| k1=pfi; k2=pfi; k3=p; : :;:k2+l=p > :
Consider now the representation of ss gotten by the composite
ae : ss g ! Sp2g(Z) Gl2g(C) :
We claim that ae is equivalent the sum of l copies of the cyclotomic representa*
*tion
oepfi. This is established by comparing characters. It is easy to see that oe*
*pfiis
induced from the reduced regular representation of a subgroup of order p (just
use the fact that the cyclotomic representation is the unique faithful irreduci*
*ble
representation over Q). Thus, by the well-known formula for the character of an
induced representation, one infers that the character O of oepfiis given by
8
>< 0; if pw 6= 0
O(w) = > -pfi-1; if pw = 0 but w 6= 0
: pfi-1(p - 1); if w = 0:
Here w denotes a general element in Z=pfiZ. On the other hand, if "zdenotes a l*
*ift
of order pfiin Diffeo+ (Sg) of the generator z 2 ss g, then by construction "z*
*has
precisely 2 fixed points (we are still assuming that fi > 1). More generally we*
* can
say that if "zjhas order greater then p, then it has precisely two fixed points*
*, and
when its order is p, it has 2 + lpfi-1fixed points. The Lefschetz number comput*
*es
the number of fixed points FP ("zj) of the map "zj, in case "zjis not the ident*
*ity map,
by
(zj) = 2 - trace(zj*| H1(Sg; Q)) = FP ("zj) :
22 MAPPING CLASS GROUPS, CHARACTERISTIC CLASSES AND BERNOULLI NUMBERS
We infer readily that
8
>< 0; if zpj 6= 1
trace(zj*| H1(Sg; Q) = > -lpfi-1; if zpj = 1 but zj 6= 1
: 2g; if zj = 1:
Comparing with the computation for the trace of the cyclotomic representation, *
*we
see that ae is equivalent to the sum of l copies of oepfi. Since the top Chern *
*class of
oepfihas order pfi(compare Section 4.1), we conclude that the top Chern class
c2g(ae) 2 H4g(ss; Z)
is the l'th power of an element of order pfiand has therefore order pfi. This i*
*mplies
that c2g(g) has order at least pfi, and we are done.
6. The Euler Characteristic
A group is said to be of finite homological type if it has finite virtual *
*coho-
mological dimension and if for every -module M which is finitely generated as an
abelian group, Hi(; M) is finitely generated for every i. For instance, if has*
* finite
vcd and acts properly and simplicially on a finite dimensional simplicial compl*
*ex,
with compact quotient, then is homologically of finite type. Thus, g is of fin*
*ite_
homological type, as we see from its action on the extended Teichm"uller space *
*T g.
If is an arbitrary group of finite homological type, its Euler characteris*
*tic is
defined by
O() = _O()___[;: ]
where denotes a torsion-free subgroup of finite index [ : ] in , and where
X
O() = (-1)idim Q Hi(; Q)
is the topological Euler characteristic of the classifying space of . One chec*
*ks
that O() is well-defined (cf. Brown's book [Brow1]); for background and various
results concerning the Euler characteristic, see also Brown [Brow2-4]). Notice *
*that
for a finite group F one has obviously
O(F ) = _1_|F;|
and for a free group L of rank n, whose classifying space is homotopy equivalen*
*t to
a wedge of n circles, one has
O(L) = 1 - n :
It follows then that for the case of 1 ~=Sl2(Z), whose commutator subgroup is f*
*ree
of rank two and whose abelianized group is of order 12, the Euler characteristi*
*c is
given by
O(Sl2(Z)) = - 1__12:
The denominator of the Euler characteristic is also in the general case closely*
* linked
to the torsion of the group. Indeed, a basic result [Brow2] is the following :
o If a prime power pn divides the denominator of O(), then possesses a
subgroup of order pn
6. THE EULER CHARACTERISTIC 23
One also defines the naive Euler characteristic of a group of finite homologic*
*al
type by X
"O() = (-1)idim Q Hi(; Q) :
Note that for the mapping class group "O(g) is just the topological Euler chara*
*c-
teristic of the moduli space Mg. Brown proved in [Brow4] the following general
relationship between O and "O:
o Let be a group such that for all elements of finite order x 2 the
centralizers C(x) have finite homological type. Then
X
O() = "O() - O(C(x)) ;
x2T
where T denotes a set of representatives for the conjugacy classes of n*
*on-
trivial torsion elements of .
Using this result, and the fact that for an x 2 g of finite order the centraliz*
*er C(x)
can be identified with a mapping class group, Harer-Zagier [Ha-Za] were able to
prove the following amazing result.
(Ha-Za 1) O(g) = ___1___2i-(2g1 - 2g) ; g > 1:
It is convenient, and natural from the point of view of the formula (Ha-Za 1), *
*to
work with the pointed mapping class group "g , which is defined to be the group
of connected components of Diffeo+ (Sg; s0), the group of pointed orientation p*
*re-
serving diffeomorphisms of Sg. The obvious map
"g ! Aut+ (ss1(Sg; s0))
is an isomorphism, compare with (IV) of Section 1. Thus
"1 ~=1 ~=Sl2(Z)
and for g > 1 one has a natural short exact sequence
1 ! ss1(Sg; s0) ! "g ! g ! 1 ;
where we have identified the group of inner automorphisms of ss1(Sg; s0) with
ss1(Sg; s0), a group with trivial center g > 1. It follows then readily, by ta*
*king
in account that O(ss1(Sg)) = 2 - 2g, that
O("g ) = (2 - 2g)O(g); g > 1 :
Thus we can rewrite (Ha-Za 1) to get
(Ha-Za 2) O("g ) = i(1 - 2g) ; g 1 :
Note that the formula is indeed also correct for g = 1, as both sides of the eq*
*uation
equal then - 1_12. It is a classical result that the values of the i-function a*
*t negative
integers can be expressed in terms of Bernoulli numbers as follows:
i(1 - 2g) = - B2g_2g; g 1 :
24 MAPPING CLASS GROUPS, CHARACTERISTIC CLASSES AND BERNOULLI NUMBERS
Our computation of the order of e2g(g) shows now that the denominator
den(O("g )) = den(- B2g_2g) = E2g
corresponds up to possibly a factor two to the order of e2g(g):
o For g 1 the denominator of O("g ) equals the order of e2g(g) or half t*
*hat
order.
In this statement, we could have used equally well the Euler class e2g("g ) in *
*place
of e2g(g), i.e., the Euler class of the flat bundle induced by the composite map
"g ! g ! Sp2g(Z) Sl2g(R) :
Indeed, the cyclic subgroups ss g of order pfi, which were used to detect the *
*order
of e2g(g), are generated by an element which lifts to a periodic diffeomorphism*
* of
Sg with a fixed point, as we see by looking at its fixed point data. It follows*
* that
ss lifts to "g and detects the order of e2g("g ) as well. It would be good to h*
*ave a
more direct way of understanding the relationship between the Euler class and t*
*he
Euler characteristic. Is the order of e2g(g) precisely the denominator of O("g*
* )?
For small values of g, the explicit computations show that this is indeed the c*
*ase:
e2(1) has order 12 and e4(2) has order 120; on the other hand,
O("1 ) = - 1__12; and O("2 ) = _1__120:
7. The Yagita Invariant of the Mapping Class Group
The invariant, which we call the Yagita invariant, was first introduced by *
*Yagita
[Ya] in the case of finite groups, and in Thomas [Th2] for more general groups.*
* It is
related to group actions on products of spheres in a similar way as finite grou*
*ps with
periodic cohomology are related to actions on spheres (a more precise statement*
* is
given below). We recall the definition of the Yagita invariant and some backgro*
*und.
Let be a group of finite virtual cohomological dimension and ss any
subgroup of prime order p. Because ss injects into any finite quotient of the f*
*orm
=, where is a torsion-free normal subgroup of finite index in , the image
Im (Hk (; Z) ! Hk (ss; Z)) of the restriction map in cohomology is non-zero for
some degree k > 0. Reduction mod-p maps H*(ss; Z) onto Fp[u] H*(ss; Fp) with
u a generator in H2(ss; Fp). Thus, there exists a maximum value m = m(ss; ) such
that
Im ((H*(; Z) ! H*(ss; Fp)) Fp[um ] H*(ss; Fp) :
Note that m(ss; ) is bounded by m(ss; =), where denotes as before a torsion-
free normal subgroup of finite index. Since = is finite, we conclude that m(ss;*
* )
is bounded by a bound 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 invariant p() agrees with the p-period *
*of
a p-periodic group and one can show [G-M-X2] that p() has in general the form
l . pk with l dividing 2(p - 1); in particular, for the prime 2 the Yagita inva*
*riant is
a power of 2.
The interest in p() stems from the fact that it provides a lower bound for *
*the
dimension of a complex, which admits a certain type of action of . For instance,
7. THE YAGITA INVARIANT OF THE MAPPING CLASS GROUP 25
using the same reasoning as in [Ya], where only finite groups were considered,
one finds 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 x 2 Rn x (Sm )k i*
*s a
p-torsion-free group, then m + 1 is a multiple of the Yagita invariant p().
The mapping class group g is never 2-periodic for g > 1, since the Krull
dimension (g; 2) is at least two. For an odd prime p and p-periodic g we have
described p(g) earlier. We recall that for an odd prime p and genus g 6 1 mod *
* p,
g is always p-periodic; thus for the discussion of the Yagita invariant of g we*
* will
only need to be concerned with the case g 1 mod p. In [G-M-X2] a complete
result is given in case p is an odd regular prime, and partial results for gene*
*ral
primes. Recall that a prime p is calledpregular_if it does not divide the class*
* number
of the cyclotomic field Q(exp (2ss -1 =p)). A famous criterion of Kummer stat*
*es
that a prime p is regular if and only if p does not divide the numerator of any
Bernoulli number B2i with 2 2i < p - 1. Thus 691, the numerator of B12, is an
example of an irregular prime; the first three irregular primes are 37, 59 and *
*67.
The following terminology was introduced in [G-M-X2].
o Let p be a prime. We say that an integer g satisfies the (p)-condition
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.
We can now state the main results concerning the Yagita invariant of the mapping
class group, as proved in [G-M-X2].
(Y-INVARIANT 1)
Let p be an odd regular prime and assume that g = lpff+ 1 with l prime 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, only a partial result is available, which *
*however
underlines the role of the (p)-condition.
(Y-INVARIANT 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.
For the prime 2, the following result on the Yagita invariant is due to Xia [Xi*
*5].
o For even genus g the Yagita invariant p(g) at the prime 2 equal 4.
We want to sketch the strategy involved in the proofs concerning the Yagita inv*
*ari-
ant of g. As explained, we can assume that p is an odd prime and g = lpff+ 1;
ff > 0 and l prime to p.
First step:
o Show that p(g) is of the form 2(p - 1)pfiwith fi ff - 1.
For this, one observes that g contains a subgroup ss isomorphic to Z=pffZ suita*
*ble
to provide a lower bound for p(g). Indeed, the the covering transformation group
in the cyclic pff-sheeted unramified covering space
Slpff+1! Sl+1 ;
26 MAPPING CLASS GROUPS, CHARACTERISTIC CLASSES AND BERNOULLI NUMBERS
which one can construct by mapping ss1(Sl+1) onto Z=pffZ, is suitable. All gene*
*r-
ators of that group ss g have the same fixed-point data, and they are therefore
conjugate in g. This implies that the restriction map
H*(g; Z) ! H*(ss; Fp)
is zero in dimensions not divisible by 2(p - 1)pff-1, by looking at the action *
*of
Aut(ss) on the cohomology of ss. Therefore, p(g) is a multiple of 2(p - 1)pff-1*
*. On
the other hand, as mentioned earlier, the Yagita invariant of any group of fini*
*te vcd
is always a factor of 2(p - 1)pn for some n 0, and the result follows.
Second step:
o Show that p(g) divides 2(p - 1)pff:
It suffices to show that for every subgroup ss g of order p one can find a num*
*ber
j(ss) prime to p such that the restriction map
ff 2j(ss)pff
H2j(ss)p(g; Z) ! H (ss; Z)
is non-trivial. This can be achieved, for details consult [G-M-X2]. Besides of
studying the restriction of various characteristic classes, one also makes use *
*of the
action of g on the complement of the union of the singular sets in Teichm"uller*
* space
of the actions of all subgroups of g which are conjugate to ss. This complement*
* is
a smooth non-compact manifold, which is used to construct a cohomology element
in H*(g), whose restriction to the cohomology of ss is non-zero.
Third step:
o Settle the case when g satisfies the p-condition.
The p-condition is just the condition needed to be able to construct a subgroup
Z=pff+1Z in g, with a fixed point data for a generator x to be of the form
ffi(x) =< g; pff+1| k1=p; : :;:kl=p > ;
a fixed point data which can be used in a manner similar to the argument in step
one to show that p(g) is a multiple of pff. It follows then that p(g) must be e*
*qual
to 2(p - 1)pff.
Fourth step:
o Show that if p is a regular prime and g does not satisfy the p-conditio*
*n,
then p(g) < 2(p - 1)pff.
At this point we know that p(g) is either equal to 2(p-1)pffor 2(p-1)pff-1. It *
*was
proved in [G-M-X2] that for every subgroup ss of order p in g, p a regular prim*
*e,
one can find a symplectic characteristic class di(g) which restricts non-trivia*
*lly to
H*(ss; Z) and which satisfies 0 < i < pff. This implies that p(g) 6= 2(p - 1)pf*
*fand
thus p(g) = 2(p-1)pff-1. The regularity condition on p is used as follows. The *
*case
of p = 3 can be checked directly, so we can assume p 5. To get the non-vanishi*
*ng
of the symplectic characteristic class one is led to consider the (p - 1)=2 x (*
*p - 1)=2
matrix
0 1 1 1 : : : 1 1
BB 2 4 6 : : : p - 1CC
A = BB 3 6 9 : : : CC:
@ ... ... A
p-1_
2 p - 1 : : :
7. THE YAGITA INVARIANT OF THE MAPPING CLASS GROUP 27
with the (i; j) entry in the rows below the first row being the the smallest po*
*sitive
residue of the product ij mod p. This matrix, which is very closely related to *
*the
matrix considered by Carlitz and Olson in [Ca-Ol] turns out to have
| det(A)| = 2p(p-5)=2. h1 ; p 5 ;
where h1 denotespthe_class number of the maximal real subfield of the cyclotomic
field Q(exp (2ss -1 =p)). It is well-known that p is irregular if and only ifp*
*p_divides_
h1, which is usually called the first factor of the class number of Q(exp (2ss *
* -1 =p)).
But the argument concerning the symplectic characteristic classes requires that
p(p-3)=2- detA ;
which is equivalent with the condition that p be regular.
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