STABLE COHOMOLOGY OF THE MAPPING CLASS GROUP WITH SYMPLECTIC COEFFICIENTS AND OF THE UNIVERSAL ABEL{JACOBI MAP Eduard Looijenga Abstract. The irreducible representations of the complex symplectic group of genus g are indexed by nonincreasing sequences of integers = (1 2 ) with k = 0 for k > g. A recent result of N.V. Ivanov implies that for a given partition , the cohomology group of a given degree of the mapping class group of genus g with values in the representation associated to is independent of g if g is suciently large. We
prove that this stable cohomology is the tensor product of the stable cohomology of the mapping class group and P a nitely generated graded module over Q c1 : : : c ], where deg(ci ) = 2i and jj = i i . We describe this module explicitly. In the same sense we determine the stable rational cohomology of the moduli space of compact Riemann surfaces with s given ordered distinct (resp. not necessarily distinct) points as well as the stable cohomology of the universal Abel{Jacobi map. These results take into account mixed Hodge structures. j
j
1. Introduction
The mapping class group ; can be dened in terms of a compact connected oriented surface S of genus g on which are given s + r (numbered) distinct points + : it is then the connected component group of the group of orientation (x ) =1 preserving dieomorphisms of S which x each x and are the identity on the tangent space of S at x for i = s + 1 : : : s + r. It is customary to omit the su x r resp. s when it is zero. Harer's stability theorem says essentially that H (; Z) only depends on s if g is large compared to k. For a more precise statement it is convenient to make a denition rst. There is a natural outer homomorphism ; +1 ! ; +1 (that is, an orbit of homomorphisms under the inner automorphism group of the target group, so that there is well-dened map on homology) and there is a forgetful homomorphism ; +1 ! ; . For a coe cient ring R and an integer g0 0, we dene N (g0 R) as the maximal integer N such that both induce isomorphisms on homology with coe cients in R in degree N for all g g0 and s r 0. Harer showed in 3] that N (g Z) 31 g and Ivanov 5], 6] improved this to N (g Z) 1 g ; 1. Recently, Harer proved that N (g Q) 2 g and that N (g Q) 2 g is almost 2 3 3 true: it holds, provided that we restrict to the mapping class groups with r 1 4]. It is likely that in fact N (g Q) 32 g ; 1. We will be mostly concerned with N (g Q) and so we shall write for this number N (g) instead. A consequence of s gr
g
r i i
s
g
g
i
i
k
s gr
s gr
s gr
s g
r
s gr
1991 Mathematics Subject Classication. Primary 32G15, 20F34 Secondary 55N25, 20C99. Key words and phrases. Stable cohomology, mapping class group, Abel{Jacobi map. Typeset by AMS-TEX 1
2
EDUARD LOOIJENGA
the stability property is that for every integer s 0 we have a stable cohomology algebra H (;1 R) (where as before, we omit the superscript s if it is equal to 0). As the notation suggests, this is indeed the cohomology of a group ;1 , namely the group of compactly supported mapping classes of an oriented connected surface of innite genus relative s given numbered points. (The number of ends of this surface may be arbitrary.) Consider the symplectic vector space V := H 1 (S Q). Its symplectic form is preserved by the natural action of the mapping class group ; on V and so any nite dimensional representation U of the algebraic group Sp(V ) can be regarded as a representation of ; in particular we have dened the cohomology groups H (; U ). A basic fact of representation theory is that the isomorphism classes of the irreducible complex representations of Sp(V ) are in a natural bijective correspondence with g-tuples of nonnegative integers (a1 : : : a ). For instance, the kth basis vector (0 : : : 0 1 0 : : : 0) corresponds to the kth exterior power of V . This result goes back to Weyl, who gave in addition a functorial construction of such a representation inside Sym 1 (^1 V ) Sym g (^ V ). He labeled this representation by the sequence (a1 + + a a2 + a3 + + a : : : a ;1 + a a ). We will follow his convention, so for us a numerical partition = (1 2 ) with at most g parts (that is, = 0 for k > g) determines an irreducible representation Sh i(V ). It follows from a recent result of Ivanov 6] that for xed k and , the cohomology groups H (; 1 Sh i(V )) are independent of g if k N (g) ;jj, where P jj := is the size of the partition. We shall give an independent proof of this in the undecorated case r = s = 0 which generalizes in a straightforward manner to the case of arbitrary r and s and we determine these stable cohomology groups as H (;1 Q)-modules at the same time. (1.1) Theorem. For every numerical partition of s, there is a graded nitely generated Qc1 : : : c ]-module B (where c has degree 2i) and a natural homomorphism H (;1 Q) B ! H (; Sh i(V )) which is an isomorphism in degree N (g) ; jj. To describe B , denote the coordinates of s-space Q by u1 : : : u so that Qu1 : : : u ] is its algebra of regular functions. We grade this algebra by giving each coordinate u weight 2. A diagonal of Q is by denition an intersection of the hyperplanes u = u this includes the intersection with empty index set, that is, Q itself. It is clear that a partition P of the set f1 : : : sg determines (and is determined by) a diagonal . Notice that the algebra of regular functions Q ] is a quotient of Qu1 : : : u ] by a graded ideal, so that it inherits a grading. Denote by l (P ) the number of parts of P of cardinality i, and consider s
s
g
g
g
g
g
g
k
g
g
g
g
a
a
g
g
g
g
g
g
g
g
k
g
k
i
g
g
i
s
i
g
g
s
s
s
s
i
i
j
s
P
P
s
i
(1)
M
t; ucodim( s
P
)+2l1(P )+l2 (P ) QP ]
P
where codim(P ) is short for codim( ) and t resp. u has formal degree 1 resp. 2. (The dierence between t2 and u becomes manifest in the Hodge theory: t has Hodge type (0 0), whereas u has Hodge type (1 1).) We regard this as a graded P
STABLE COHOMOLOGY OF MAPPING CLASS GROUPS
3
module over Qu1 : : : u ] that has a natural action of the symmetric group S . We tensorize this module with the signum representation of S and denote the resulting graded Qu1 : : : u ]-module with S -action by B . Now recall that every numerical partition of s determines an equivalence class () of irreducible representations of S and that this gives a bijection between these two sets. For instance, the coarsest partition (s) labels the trivial representation, whereas the nest partition (1 ) corresponds to the signum representation. Passing from a partition to the conjugate partition 0 corresponds to taking the tensor product with the signum representation. In the case at hand we have a decomposition of B into isotypical subspaces: B = B () with B = HomSs (() B ): Clearly, this is also a decomposition into graded Qu1 : : : u ]Ss -submodules. We identify the latter ring with Qc1 : : : c ], where c is the kth elementary symmetric function in the u 's. This completes the description of B . It follows from the work of M. Saito that H (; Sh i(V )) carries a natural mixed Hodge structure. It is known that H (;1 Q) has a natural mixed Hodge structure as well (see (2.5)). We will nd that if we put a Hodge structure on B by giving its degree 2d ; jj-part Hodge type (d d), then the homomorphism of (1.1) is a morphism of mixed Hodge structures. M. Pikaart has recently shown that H (;1 Q) is pure of weight n this implies that H (; Sh i (V )) is pure of weight n + jj in the stable range n N (g) ; jj. Example 1. The s-th symmetric power of V corresponds to Sh i(V ) and so the stable cohomology of ; with values in Sym (V ) is the tensor product of H (;1 ) with B (1 ), that is, the isotypical subspace of expression (1) corresponding to the signum representation of S . Any partition dierent from the partition into singletons is invariant under a transposition and so will not contribute. This leaves us therefore with the (1 )-isotypical subspace of t; u2 QQ u1 : : : u ]. This is the free Qc1 : : : c ]-module generated by the element t; u2 (u ; u ), hence is 1 of the form t; u 2 ( +3)Qc1 : : : c ]. We nd that s
s
s
s
s
s
s
s
s
s
s
s
s
k
i
g
g
n
n
g
g
s
g
g
s
g
g
s
s
s
s
s
s
s
s
s
s
i
i>j
s s
j
s
H (;1 Sym (V1)) = t; u 12 s
s
( +3)H
s s
(;1 Q)c1 : : : c ]: s
Example 2. For g s, the primitive subspace Pr (V ) of the s-th exterior power of V corresponds to Sh1si g and so the stable cohomology of ; with values in Pr (V ) is the tensor product of H (;1 Q) with S -invariant part of (1). This is naturally written as a sum over the numerical partitions of s. Here a numerical partition is best described by means of the exponentialPnotation: (1 1 2 2 3 3 ), where l is the number of parts of cardinality k (so that kl = s). Its contribution is then 1t; k u k max(2 ;1)Qc1 c2 : : : c k ]: If we sum over all sequences (l1 l2 : : : ) of nonnegative integers which become eventually zero, we get s
g
s
g
g
V
g
s
l
k
k
l k
k
1; =0t k
1 l
l
;lk
k
l
u max(2 l
k
k
;1)Q
c1 c2 : : : c ] : l
l
l
4
EDUARD LOOIJENGA
and B(1s) is the t; -part of this expression. For instance, B(12) = t;2(u4 Qc1 c2] u2Qc1]) B(13) = t;3(u6 Qc1 c2 c3] u4Qc1] Qc1] u2Qc1]): Since H 1 (;1 Q) = 0, it follows in particular that H 1 (;1 Sh13i ) has dimension one. (It is easy to see that H 1(;1 Sh i) = 0 for all other numerical partitions .) (1.2) The cohomology groups of ; with values in a symplectic representation have a geometric interpretation as the cohomology of a local system (in fact, of a variation of polarized Hodge structure). The Teichmuller space T of conformal structures on S modulo isotopy is a contractible complex manifold (of complex dimension 3g ; 3, when g 2). The action of ; on it is properly discontinuous and a subgroup of nite index acts freely. The orbit space M := ; nT is naturally interpreted as the coarse moduli space of smooth complex projective curves. Via this interpretation, it gets the structure of a normal quasi-projective variety. It follows from the preceding that M has the rational cohomology of ; . More generally, if U is a rational representation of ; , then we have a natural isomorphism H (; U ) = H (M U) where U is the sheaf on M which is the quotient of the trivial local system T U ! T by the (diagonal) action of ; . This applies in particular to the representations S (V ). This representation appears in the cohomology of degree s := jj of the conguration space of s numbered (not necessarily distinct) points on S . So if C ! M denotes the s-fold ber product of the universal curve, then its sth direct image contains the local system (in the orbifold sense) associated to S (V ) as a direct summand. By a theorem of Deligne, the Leray spectral sequence of the forgetful map C ! M degenerates at the E2-term and thus the stable cohomology of this representation is realized inside H (C Q). This fact will be used in an essential way. Acknowledgements. I thank Dick Hain for discussions and for suggesting to use Deligne's degeneration theorem. I am also grateful to him for pointing out that the way (1.1) was stated in an earlier version could not be correct. His student A. Kabanov obtained related results that should appear soon. I am also indebted to the referee for invaluable comments, in particular for a suggestion for shortening the original proof of (2.3). s
g
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g
g
s g
g
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g
s g
g
s g
g
g
s g
2. Stable cohomology of
M
s g
and
C
s g
We rst state and prove an immediate consequence of the stability theorems. For s 1, the class of a (Dehn) twist about x generates an innite cyclic central subgroup of ; ;1+1 . The quotient group can be identied with ; and so we have a Gysin sequence ! H ;2(; Z);!s H (; Z) ! H (; ;1+1 Z) ! where u 2 H 2 (; Z) is the rst Chern class. Similarly, x determines a rst Chern class u 2 H 2 (; Z) for i = 1 : : : s. These classes are clearly stable and we shall not make any notational distinction between the u 's and their stable representatives. s
s gr
s gr
k
s gr
s gr
s
i
u
k
s gr
k
s gr
i
s gr
i
STABLE COHOMOLOGY OF MAPPING CLASS GROUPS
5
(2.1) Proposition. The stable cohomology algebra over of the mapping class
groups of surfaces with s distinct numbered points is a graded polynomial algebra on the stable cohomology ring of the absolute mapping class groups. More precisely, there is a natural S -equivariant graded ring homomorphism s
H (;1 Z)u1 : : : u ] ! H (; Z) s gr
s
which is an isomorphism in degree N (g Z). In particular, the rational stable cohomology algebra of the mapping class groups of surfaces with s unlabeled points is a graded polynomial H (;1 Q)-algebra on the elementary symmetric functions c1 : : : c of the u 's. Proof. The composite of the forgetful maps ; ;1 ! ; ;1 and ; ;1 ! ; ;1;1 induces an isomorphism on H (; Z) for large g. So in this range the forgetful map induces a surjection H (; ;1 Z) ! H (; ;1 Z). If we feed this in the above Gysin sequence, we get short exact sequence s
i
s gr
s gr
s gr
s gr
k
k
0!H
;2
s gr
k
s gr
(;1 Z);! H (;1 Z) ! H (;1;1 Z) ! 0 u
s
s
s
so that H (;1 Z) = H (;1;1 Z)u] as algebra's. The assertion now follows with induction on s. Let us x a nite set X . We denote by C the moduli space of pairs (C x) where C is a compact Riemann surface of genus g and x : X ! C is a map. Let j : M C be the open subset dened by the condition that x be injective. Just as M is a virtual classifying space for ; , M is one for ;j j in particular, ;j j and M have the same rational cohomology. This enables us to restate (2.1) in more geometric terms. Let C ! M be the universal curve and denote by its relative tangent sheaf. For every i 2 X , the map (C x) 7! x(i) denes a projection of C onto the C denote by the pull-back of under this map. One easily recognizes the rst Chern class of jM as the rst Chern class dened above. So (2.1) implies: (2.2) Proposition. The ring homomorphism s
s
X g
X g
X g
g
g
X g
X g
X g
X g
g
X g
g
g
i
X g
i
H (M Q)u : i 2 X ] ! H (M Q) u g
X g
i
i
7! c1( )jM
X g
i
:
is an isomorphism in degree N (g). We will use this proposition to prove that the rational cohomology of C also stabilizes. We begin with attaching to X a graded commutative Qu : i 2 X ]-algebra. It is convenient to introduce an auxiliary graded commutative Q-algebra A~ rst. The latter is dened by the following presentation: for each nonempty subset I of X , A~ has a generator u of degree two (we also write u for uf g ), and these are subject to the relations u u = u u if i 2 I \ J . So if i 2 I , then u u = u u . It is then easy to see that the Qu : i 2 X ]-submodule generatedQby u is dened by the relations (u ; u )u = 0 whenever i j 2 I . The monomials u I for which I runs over the members of a partition of X form an additive basis of A~ . (To X g
i
X
I
X
I
i
J
i
I
i
J
I
i
i
j
I
I
I
i
I r I
X
I
6
EDUARD LOOIJENGA
make the indexing eective, let us agree that we only allow r to be zero if I is a singleton.) We then let A be the Qu : i 2 X ]-subalgebra of A~ generated by the elements a := uj j;1, where I runs over the subsets of X with at least two elements. These generators obey the relations u a = u a if i j 2 I a a = uj \ j;1a if i 2 I \ J: and it is easy to see we thus obtain a presentation of A as a graded commutative Qu : i 2 X ]-algebra. Notice that as a Qu : i 2 X ]-algebra, A is already generated by the a 's with jI j = 2. For every partition P of X we put a := Q 2 j j2 a (with the convention that a = 1 if P is the partition into singletons). These elements generate A as a Qu : i 2 X ]-module. In fact, M A = Qu : I 2 P ]a : I
I
i
X
I
X
I
i
I
j
I
I
I
J
J
I
i
J
X
i
i
X
I
I
P
P
I
I
P
i
X
I
X
P
j
P X
We give A~ a (rather trivial) Hodge structure: its degree 2p-part has Hodge type (p p). Clearly, A is then a Hodge substructure. Given a partition P of X , then the pairs (C x : X ! C ) for which every member of P is contained in a ber of x dene a closed subvariety i : C (P ) C . Those for which P is the partition dened by x make up a Zariski-open subvariety M (P ) C (P ). Notice that C (P ) resp. M (P ) can be identied with C resp. M (where X=P stands for quotient of X by the equivalence relation dened by P , or rather the set of parts of P ), and that this is a submanifold in the orbifold sense. For every nonempty I X , let P be the partition of X whose parts are I and the singletons in X ; I . So the corresponding orbifold C (P ) has codimension jI j ; 1 in C . (2.3) Theorem. There is an algebra homomorphism : H (M Q) A ! H (C Q) that extends the natural homomorphism H (M Q) ! H (C Q), sends 1 u to c1( ) and sends 1 a to the Poincar e dual of the class of C (P ). This is an S -equivariant algebra homomorphism and is also a morphism of mixed Hodge structures. Moreover, is an isomorphism in degree N (g). Proof. For the rst statement we must show that if I J X have at least two elements, i 2 I \ J , j 2 I and P is a partition of X , then Y i ! (1) = i I !(1) X
X
P
g
g
g
X g
g
X=P g
g
X=P g
I
g
I
X g
X g
g
X g
X
X g
g
i
I
i
g
I
X
X g
P
I0
2P jI 0 j2
P 0
c1( )i I ! (1) = c1( )i I ! (1) i I ! (1)i J ! (1) = c1( )j \ j;1 i i
P
P
P
j
i
P
I
J
PI J
! (1):
The rst identity is geometrically clear and the second follows from the fact that and have isomorphic restrictions to C (P ). To derive the last identity, we use the following lemma (the proof of which is left to the reader): i
j
g
I
STABLE COHOMOLOGY OF MAPPING CLASS GROUPS
7
(2.4) Lemma. Let U and V be closed complex submanifolds of a complex man-
ifold M whose intersection W := U \ V is also a complex manifold. Suppose that any tangent vector of M which is tangent to both U and V is tangent to W . Then i ! (1)i ! (1) = i ! (e), where e is the euler class of the cokernel of the natural monomorphism ! jW jW . Completion of the proof of (2.3). We apply the orbifold version of this lemma to M = C , U = C (P ), V = C (P ) so that W = C (P ). The desired assertion then follows if we use the fact that the bundles appearing in the monomorphism
! jW jW are all direct sums of copies of jW . The second statement of the theorem is clear. To prove the last, let U resp. S denote the union of the strata M (P ) of codim k resp. = k. We prove with induction on k that the homomorphism U
V
W
W
X g
W
g
U
U
V
I
g
J
g
I
V
J
i
k
k
g
M
codim P k
H (M Q) C u : I 2 P ]a g
I
P
! H (U Q) k
is an isomorphism in degree N (g). For k = 0 this is (2.2). If k 1, then consider the Gysin sequence of the pair (U S ): k
k
! H 2 (S Q) ! H (U Q) ! H (U 1 Q) ! : L In degrees N (g) the isomorphism of codim H (M Q) C u : I 2 P ]a onto H (U 1L Q) factorizes over H (U Q). So the Gysin sequence splits in this range. Since codim = H (M Q) C u : I 2 P ] maps isomorphically onto H (S Q) in degree N (g), the theorem follows. ;
n
n
k
k
n
k
n
;
P
n
k
P
;
k
g
I
P
k
g
k
k
I
k
Remark. For a curve C , the image of A in H (C Q) is contained in the Hodge ring of C . If C is general, then it is in fact equal to it. (2.5) A virtual classifying space of ; + is the moduli space of r + s-pointed curves M + . It carries r + s relative tangent bundles 1 : : : + and the total space of the (C ) -bundle M ! M + dened by 1+ : : : + is a virtual classifying space for ; . The comparison maps that enter in the statement of the stability theorem admit simple descriptions in these algebro-geometric terms and so preserve Hodge structures. This is probably well-known, but since we do not know a reference, we explain this. Clearly, the forgetful homomorphism ; +1 ! ; corresponds to the obvious projection M +1 ! M . To exhibit the outer homomorphism ; +1 ! ; +1 , we rst note that M + +1 M11 parametrizes a codimension one stratum of the Knudsen{Deligne{Mumford compactication of M ++1: a pair ((C x1 : : : x + +1 ) (E O)) determines a stable r + s-pointed genus (g + 1)-curve by identifying x + +1 with O. The normal bundle of this stratum is just the exterior tensor product of the relative tangent bundles of the factors. Denote the complement of its zero section by U and let U be a general bre of the projection of U ! M11. We can identify U with M +1 and there is a natural restriction homomorphism r g
r g
s
s
r
r
s gr
s gr
r g
s gr
s
s
s gr
s gr
r g
X
X
X
s g
s
r
r g
r
s
r
s
s gr
s gr s
s
r
s
e
r s g
e
H (M ++1 Q) ! H (U Q) ! H (U Q) = H (M +1 Q): r g
s
e
r s g
8
EDUARD LOOIJENGA
The (C ) -bundles that lie over these spaces determine likewise a homomorphism H (M +1 Q) ! H (M 1+ Q). This is the one we were after. Thus H (;1 Q) acquires a canonical mixed Hodge structure. The composite map r
s g
s g
r
H (;1 Q) A
X
r
! H (; Q) A = H (M Q) A ! H (C g
X
g
X
X g
Q)
is evidently a morphism of mixed Hodge structures. For every i 2 X we have a projection f : C ! C ;f g and an obvious inclusion A ;f g ,! A (with image the linear combinations of monomials in which the u with i 2 I occur with exponent 0). The two are related: (2.6) Lemma. The map f : H (C ;f g Q) ! H (C Q) is covered by the inclusion A ;f g ,! A tensorized with the identity of H (;1 Q). The proof is straightforward. Denote by H (C Q) the quotient of H (C Q) by the subspace spanned by the images of f u f , i 2 X . (2.7) Corollary. Put X
X g
X g
i
i
I
X
i
X g
i
X
i
X g
X
i
X g
i
i
X g
i
A0 :=
M
X
j
P X
Y
(
u2)Qu : I 2 P ]a : I
i
fig2P
P
Then there is a natural homomorphism of mixed Hodge structures
H (;1 Q) A0
X
which is an isomorphism in degree N (g).
! H (C
X g
Q)
3. The Schur{Weyl functor
(3.1) We continue to denote by X a nite nonempty set. The product S comes with an obvious S -action and so there is a resulting action of S on H (S Q). Any dieomorphism f of S induces a dieomorphism of S commuting with this action. Thus is obtained an action of the product ; S on H (S Q). As before, V := H 1 (S Q) and Sp(V ) denotes its symplectic group. It is easily seen that the ; S -action factorizes through one of Sp(V ) S . Let us write the total cohomology H (S Q) as Q V t Qu where u is the canonical class (we assume g 2 here) and t shifts the degree by 1. The Kunneth rule gives an isomorphism X g
X
X g
g
g
g
g
g
X g
X
X g
X
g
g
X
X
g
M
H (S Q) = X g
I J
X I \J =
g
V t u I I
J
g
where t should be thought of Qas a generator of the signum representation of S placed in degree jI j and u = 2 u . The multiplication in the right-hand side I
I
J
j
J
j
STABLE COHOMOLOGY OF MAPPING CLASS GROUPS
9
obeys the Koszul sign rule and is given by contractions stemming from the symplectic form on V . If we dene H (S Q) as in the relative case, that is, as the quotient of H (S Q) by the span of the images of H (S ;f g Q) u H (S ;f g Q), i 2 X , then we see that in terms of the Kunneth decomposition this reduces to the single summand V t . (3.2) Since : C ! M is a projective morphism whose total space is an orbifold, it follows from a theorem of Deligne 1] that its Leray spectral sequence degenerates at the E2-term over Q. In other words, H (C Q) has a canonical decreasing ltration L , the Leray ltration, such that there is a natural isomorphism X g
g
X g
X g
g
i
X g
i
i
X X
X g
g
X g
Gr H (C Q) = H (M R k L
n
X g
k
;k
n
g
Q):
The maps f in (2.6) are strict with respect to the Leray ltrations. So if we combine this with (2.7) we nd: (3.3) Corollary. There is a natural graded S -equivariant map of H (;1 Q)modules H (;1 Q) A0 ! H (; V )t which is an isomorphism in degree N (g). We want to decompose V as a Sp(V ) S -representation. Following Weyl this is done in two steps. The rst step involves Weyl's representation V h i whose denition we presently recall. Let ! 2 V V correspond to the symplectic form on V . For every ordered pair (i j ) 2 X with i 6= j , we have a natural homomorphism V ( ;f g) ! V dened by placing ! in the (i j )-slot. Reversing the order gives minus this map. So the natural assertion is that we have a map i
s
g
X
X
g
X
g
X
g
X
X
g
g
g
g
X
ij
g
g
X
M I
X jI j=2
V ( g
X
;I )
t
I
!V
X
g
:
It is easy to see that this map is injective its cokernel is by denition V h i. Notice that V h i is in a natural way a representation of Sp(V ) S . The second step is the decomposition of V h i: Weyl proved the remarkable fact that X
g
g
X
g
X
X
g
Vh i = X
M
Sh i(V ) () g
g
where runs over the numerical partitions of jX j in at most g parts and () denotes the corresponding equivalence class of irreducible representations of S . In particular, all these irreducible representations appear with multiplicity one. A modern account of the proof and of related results can be found in the book by Fulton and Harris 2]. (3.4) Theorem. If A00 A0 is the Qu : i 2 X ]-submodule dened by X
X
A00 =
M Y
X
j
P X
i
X
(
u2)( i
fig2I
I
Y
2P jI j=2
u )Qu : I 2 P ]a I
I
P
10
EDUARD LOOIJENGA
then there is a natural graded S -equivariant map of H (; Q)u : i 2 X ]modules H (;1 Q) A00 ! H (; V h i)t and this map is an isomorphism in degree N (g). Proof. If I X is a two-element subset, then the Poincar e dual of the hyperdiagonal C (P ) C is u . The restriction of this element to the ber S is P ( 2 u ) + !t . So the map X
g
g
i
X
g
X g
I
I
i
I
X g
I
X
g
X
i
H
;2
(; V ( g
X
g
;I )
)t
X
;I
! H (; V g
X
g
)t
X
dened by multiplication with !t is covered by the map H (;1 Q) A0 ; ! P H (;1 Q) A0 which is multiplication by u ; 2 u . The theorem follows. I
X
I
X
i
I
I
i
Proof of (1.1). We decompose both members of the stable isomorphism (3.4) according to the action of S . Let be a numerical partition of jX j and let 0 be the conjugate partition. According to Weyl's decomposition theorem, the ()-isotypical component of H (; V h i) is H (; Sh i(V )). Note that this is also the isotypical component of type 0 of H (; V h i)t . On the other hand, it is easily seen that the (0 )-isotypical component of A00 can be identied with t B (). Since the Leray spectral sequence (3.2) is a spectral sequence of mixed Hodge structures, the homomorphism of (1.1) is actually a morphism of mixed Hodge structures. The theorem follows. X
X
g
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g
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g
X
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s
X
4. Stable cohomology of the universal Abel{Jacobi map
If we give S a complex structure, then S becomes a compact Riemann surface C of genus g, so that we have dened an Abel{Jacobi map Sym (C ) ! Pic (C ). The induced map on cohomology has been determined by Macdonald: (4.1) Proposition. (Macdonald 7]) Identify H (Pic (C ) Q) with the exterior algebra on V so that the Abel{Jacobi map determines an algebra homomorphism ^ V ! H (Sym (C ) Q). Let ^ V y] ! H (Sym (C ) Q) be the extension that sends the indeterminate y of degree two to the sum of the fundamental classes of the factors. Then this map is surjective and its its kernel is the degree > s-part of the ideal I in ^ V y] generated by fv ^ v0 ; (v:v0 )y : v v0 2 V g. If s > 2g ; 2, then the Abel{Jacobi map is the projectivization of a vector bundle of rank s + 1 ; g over Pic(C ) and we can interpret the image of y as the rst Chern class of the associated line bundle over Sym (C ). Macdonald also expresses the Poincar e duals of the diagonals of C in terms of this presentation. It is our aim to make a corresponding discussion for the universal situation in the stable range. The morphism C ! M denes a relative Picard bundle Pic(C =M ) ! M (in the orbifold sense) whose connected components are still indexed by the degree: Pic (C =M ), k 2 Z. The degree 0-component is called the universal Jacobian and is also denoted J . g
g
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STABLE COHOMOLOGY OF MAPPING CLASS GROUPS
11
(4.2) Lemma. For every k 2 Z, there is a natural isomorphism H (Pic (C =M ) Q) = H (J Q): k
g
g
g
Proof. The relative canonical sheaf denes a section of Pic2 ;2 (C =M ). On a ~ ! M (with Galois group G, say) this section becomes suitable Galois cover M ~ ). This determines divisible by 2g ; 2 and thus produces a section of Pic1 (C~ =M 0 ~ ) an isomorphism Pic (C~ =M = Pic (C~ =M~ ). Although this isomorphism will not in general be G-equivariant, it will dier from any G-translate by a section of Pic0(C~ =M~ ) of nite order, and so the induced map on rational cohomology is G-equivariant. By passing to the G-invariants we obtain an isomorphism H (Pic (C =M ) Q) = H (J Q). One checks that this map does not depend on choices. Let X be a nite nonempty set as before. We wish to determine the subalgebra of S -invariants of A , at least stably. Recall that an additive basis of A consists of Q the set of elements of the form 2 u I , where P runs over the partitions of X and r jI j; 1. Let us dene a partial ordering on the collection of partitions of X by: P Q if P = Q or if for the smallest number k such that the k-element members of P and Q do not coincide every k-element member of P is a k-element member Q Q of Q. This determines a partial ordering on the set of monomials: 2 u I 2 u J if P < Q or if P = Q and r s for all I . The dening relations for A show that a product of two monomials associated to partitions P and Q is a monomial associated to a partition P that dominates both P and Q. operator (acting on A ). Given a Denote by S := 2SX the symmetrizer Q partition P of X , then the smallest terms in 2 S (u I ) are monomials associated to a partition that is a S -translate of P . In this expression the part P associated to P is the subsum corresponding to the partial symmetrizer S := 2SX ( )= . If l is the number of members of P with k elements, then the image of S can be identied with O Qc1 c2 : : : c 1 ] (c k;1Qc1 c2 : : : c k ]): g
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k
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k
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r
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Q
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r
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P
k
P
P
k
l
k
l
l
2 lk >0
Here c in the tensor factor with index k is to be thought of as the lth elementary symmetric function in the u 's with I 2 P and jI j = k (and so has degree 2l) the appearance of c k;1 comes from the condition r jI j; 1. So if we put l
I
k
I
l
C1 := Qc1 c2 ]
O; k
Q
2
M 1
c ;1Qc1 c2 c3 : : : c ] k l
l
l
then we nd:
(4.3) Corollary. There is natural surjective homomorphism of graded algebra's C1 ! (A )SX . Its restriction to the Q-span of all the monomials involving the ( r ) (where c( ) denotes the variable c that occurs in the k th variables c(1 1) : : : cP r tensor power) with k l jX j is a linear isomorphism. If Y is another nite set with jY j jX j, then any injection f : X ,! Y induces an algebra homomorphism A ! A . Since all such injections are in the same X k l
k
k
l
l
l
i
i i
X
Y
12
EDUARD LOOIJENGA
! (A
)SY . In particular, (A~ )SX only depends on jX j. So if we write C~j j for this algebra, then we have a direct system ! C ! C +1 ! . The corollary shows that the limit of this direct system can be identied with C1. It follows from (2.3) that we have an algebra homomorphism
S -orbit they give rise to the same algebra homomorphism (A )S Y
X
X
X
Y
X
s
s
! H ((C ) Q)S = H ((C )S Q) that is an isomorphism in degree N (g). In the limit this yields a homomorphism H (; Q) C ! H ((C )S Q) that is an isomorphism in degree min(2s N (g)). The image of c1 1 1 is H (;1 Q) C
s
s g
s
1
s g
s
s g
1
s
easily seen to be proportional to the element y appearing in Macdonald's theorem. We put
C 01 := Qc2 c3 ]
O; k
Q+
2
1
l
c ;1Qc1 c2 c3 : : : c ] : k
l
l
(4.4) Theorem. The algebra homomorphism above ts in a commutative square of algebra homomorphisms
H (;1 Q) C1
;! ;!
H (;1 Q) C 01
H ((C )Ss Q) s g
"
H (Pic (C =M ) Q) s
g
g
in which the right vertical map is induced by the Abel{Jacobi map. The lower horizontal map is an isomorphism in degree min(s N (g)) so that in the limit we have an isomorphism
H (;1 Q) C 1 0
= M H (; ^ )t : 1
s
=0
1
s
s
Proof. If we combine Macdonald's theorem with the Leray spectral of sequence of the map (C )Ss ! M , then we see that the map H (Pic (C =M ) Q)y] ! H ((C )Ss Q) s g
g
s
g
g
s g
that sends y to c1 1 1 is an isomorphism in degree s. The theorem follows from this. References 1. P. Deligne, Theoreme de Lefschetz et criteres de degenerescence de suites spectrales, Inst. Hautes E tudes Sci. Publ. Math. 35 (1968), 259{278. 2. W. Fulton and J. Harris, Representation theory, Graduate Texts in Math., vol. 129, Springer Verlag, New York, 1991. 3. J. Harer, Stability of the homology of the mapping class groups of orientable surfaces, Ann. of Math. 121 (1985), 215{249.
STABLE COHOMOLOGY OF MAPPING CLASS GROUPS
13
4. J. Harer, Improved stability for the homology of the mapping class groups of surfaces, preprint Duke University (1993). 5. N.V. Ivanov, Complexes of curves and the Teichmuller modular group, Uspekhi Mat. Nauk 42 (1987), 110-126 (Russian) English transl. in Russian Math. Surveys 42 (1987), 55{107. 6. N.V. Ivanov, On the homology stability for Teichmuller modular groups: closed surfaces and twisted coecients, Mapping class groups and moduli spaces of Riemann surfaces (C.F. Bodigheimer and R.M. Hain, eds.), Contemp. Math., vol. 150, AMS, 1993, pp. 149{194. 8. I.G. Macdonald, Symmetric products of an algebraic curve, Topology 1 (1962), 319{343. Faculteit Wiskunde en Informatica, Universiteit Utrecht, PO Box 80.010, 3508 TA Utrecht, The Netherlands
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