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The Homology of the Mapping Class Group J. DIFFERENTIAL GEOMETRY 24 (1986) 1-14 THE HOMOLOGY OF THE MAPPING CLASS GROUP EDWARD Y. MILLER 1. Introduction The mapping class group Γg is the group of components of the groups + Diff (Sg) or orientation preserving diffeomorphisms of a Riemann surface Sg of genus g. Since each component is contractible, there are natural isomor- phisms of integral cohomology groups: (1.1) H*(BΏm+{Sg); z) = H*{BTg; Z). In the context of complex analysis, Γg is called the Teichmuller group. It acts properly and discontinuously on the Teichmuller space T3g~3 with finite isotropy groups. The quotient of this action is the module space Mg of smooth algebraic curves of genus g. Consequently, there is an isomorphism of rational cohomology: (1.2) H*{BTg:A) = H*(Mg:Q). + In this paper we will show that Mg, BTg, and BΌiϊl (Sg) get more and more complicated as the genus g tends to infinity. More precisely, we will prove: Theorem 1.1. Let Q[z2, z4, z6, ] denote the polynomial algebra of genera- tors z2n in dimension In, n = 1,2,3, . There are classes y2, y4, , y2n, '* 2n + with y2n in the 2nth cohomology group H (BΌiίί (Sg)\ Z) such that the homomorphism of algebras sending z2n toy2n is an injection in dimensions less than (g/3). These classes y2n were first introduced by D. Mumford [7]. In the topologi- cal context, they are defined as follows: + Let p: E -> BΌiίΐ (Sg) be the universal Sg bundle with fiber Sg. Let d be the first Chern class of Γ*, the tangent bundle along the fibers of the Received December 25, 1985. 2 EDWARD Y. MILLER fibration p, and p* denote the "integration along the fibers" homomorphism in integral cohomology. The homomorphism p* maps H2n+2{E\Z) to 2n + H (BDiff (Sg); Z) (since the fibers are dimension 2). Define y2n by n+ι (1-3) y2n=P*(d ), where dn+1 is the (n + l)-fold cup product of d. {Note: D. Mumford in [7] defines analogous classes in H *(Mg; Z) by a strictly algebraic process. His classes extend to the closure of the moduli space Mg.) 2 It is useful to utilize Diff(5g, Z) ), the group of orientation preserving 2 diffeomorphism of Sg fixing a chosen disk D in Sg. By taking connected sums of the surfaces Sg and Sh (of genera g and h) along their fixed disks we obtain natural homomorphisms 2 2 2 (1.4) Diff(Sg,/) ) X Diff(SΛ,Z) ) -> Diff(Sg+Λ,Z) ), 2 (1.5) Diff(Sg,Z> ) X(identity) -* Diff(Sg+Λ). In these terms one of the basic results concerning the homology of the mapping class group is the following remarkable theorem of J. Harer [3]. Theorem 1.2 (/. Harer). The induced maps of classifying spaces + 2 + + 2 + (1.6) 5Diff (5g,Z) )^5Diff (5g+Λ), £Diff (Sg,I> )^f?Diff (Sg) give rise to isomorphisms on integral homology in dimensions less than (g/3). Note. Since Theorem 1.2 does not appear in Harer's work in the form stated here we will show in §4 how it follows from his much stronger results [3]. Harer's theorem implies that the rational cohomology of the moduli space Mg stabilizes. Indeed this is true integrally since Mg is a F-manifold whose singularities have codimension that increases with g (see [7]). The algebraic + analog of BΌifί (Sg) is the moduli space of triples (Cp,p,υ) where Cg is a smooth curve of genus g, p is a point on Cg, and υ is a nonzero cotangent vector based at p. By Theorem 1.2 the limit of homology groups is of finite type. The homomorphisms (1.4) induce maps of classifying spaces + 2 + 2 + 2 (1.7) F: BΌiff (Sg9D )xBΌifί (Sh,D )^BΌiίf (Sg+h9D ). These induce a product F+ on the limit and so a Hopf algebra structure on the limit A. + 2 Theorem 1.3. (a) A = Lim H*(B Diff (Sg, D ); Q) under the F^product is a commutative, cocommutative Hopf algebra of finite type. THE HOMOLOGY OF THE MAPPING CLASS GROUP 3 (b) A is the tensor product of a polynomial algebra on even dimensional generators and an exterior algebra on odd dimensional generators. (c) A contains at least one generator x2n in each even dimension 2n9 n = 1,2,3, ••-. As explained in §2, Theorem 1.3 part (a) is implied by general considera- tions. Part (b) then follows from the general structure theory of Hopf algebras over Q of Milnor and Moore (see [6]). Part (c) is proved by explicitly constructing the desired classes x2n and detecting them by means of the universal cohomology classes y2n of Mumford. It is presently an open question whether or not there are nontorsion classes in the odd dimensional homology of the mapping class groups Γg in dimen- sions less than (g/3). Mumford has conjectured that A is the polynomial algebra on precisely the classes x2n, n = 1,2,3, [7] (i.e., one generator in each even-dimension). Quite possibly the number of even dimensional genera- tors might increase exponentially with dimension. From the definition (1.3) the universal classes y2n restricted to 2 + 2 H "(BΌiΐf (Sg9 D )\Z) are compatible under the inclusions (1.6). Conse- quently, they define universal cohomology classes in the inverse limit 2n 2 Lim H (BΌiίf+(Sg, D )\ Z). The main properties of these universal classes y2n are: Lemma 1.4. The classes y2n vanish on the F^-decomposibles of the Hopf algebra A above. By Lemma 1.4 the classes y2n may be used to detect polynomial generators of A. That is, if we construct classes x2n in A with nonzero evaluation by y2n e x φ 9 (i > [yin> 2n\ Q\ then the x2n s are the desired polynomial generators sought in Theorem 1.3, part (c). Dually (again using Harer's Theorem 1.2) Theorem 1.1 is proved. + 2 In view of Harer's Theorem 1.2, H2n(BΌiίf (Sg9 D )\ Z) is isomorphic to + H2n(BΌifi (Sg)\Z) for g. large. Hence to prove Theorem 1.3, part (c) it + suffices to construct explicit classes u2n in H2n(B Diiί (Sg)\ Z) with [y2n, u2n] Φ 0 for g large. The desired examples are provided by Theorem 1.5 below. Theorem 1.5. For each n there is a fϊbration of smooth projective algebraic n+1 n varieties pn: Z -> X with fiber a smooth connected curve, dimc^" = w, [dn+ι, Zn+ι]Φ 0. Here d equals the first Chern class of the tangent bundle along the fibers Γ* topn. The genus of fiber Y" ofpn may be made as large as desired. n+ n+ι n+ι n n The equality [d \ Z ] = [{pn)^d \ X \ = [y2n, X \ follows from the definition of the "integration over the fibers" map (/>„)*• Hence, once Theo- rem 1.5 is proved, Theorem 1.3, part (c) and Theorem 1.1 are proved as explained above. 4 EDWARD Y. MILLER n+ι The construction of pn: Z -> X" of Theorem 1.5 is modeled on the methods of Atiyah [1]. In that paper a more standard detection procedure is suggested. It may be described as follows. The local coefficient system [i/^Fiber; Z)] with its symplectic form via cup product defines a classifying map + 2 L: BΌifϊ (Sg,D ) -+£Sp(2g,Z). + 2 Equivalently, L is the classifying map of the homomorphism Diff (Sg, D ) -> Sp(2g, Z) which records the symplectic homomorphism induced by a diffeo- morphism of the Riemann surface Sg. It is natural to attempt to detect nonzero + 2 classes in B Diff (Sg9 D ) by pulling back classes from i?Sp(2g, Z). This is Atiyah's approach in studying two dimensional classes. The real symplectic group Sp(2g, R) has maximal compact subgroup t/(g), the unitary group. Thus, the inclusion U(g) -> Sp(2g, R) induces a homotopy equivalence /: BU(g) -> £Sp(2g, R) with inverse J~ι. Consequently, the + 2 inclusions and homomorphisms of groups Diff (5g, D ) -> Sp(2g, Z) -» Sp(2g, R) <- U(g) induce a map of classifying spaces + 2 (1.8) G:BΌiff (Sg,D )-+BU{g). Recall that the homology of BU = Lim BU(g) is a polynomial algebra under the Whitney sum on generators zn in dimension 2«; and that the 2n primative characteristic class s(n)(t) = n\ch(n)(t) in H (BU; Z) vanishes on decomposibles with [ch(n)(t\ zn] Φ 0. Here t is the universal bundle over BU. See [1]. + 2 Note that the map G sends the .F-product in Lim B Diff (Sg, D ) to the Whitney sum product of bundles in BU = Lim BU{g). Consequently, G*(C/J(W)(O) vanishes on the ivdecomposibles of A and so may be used to detect possible polynomial generators. The relationship between this detection procedure and the nonmultiplica- tivity of the signature has been elucidated by Atiyah [1]. He shows that the signature of the total space of a (4k — 2) dimensional family X2k~ι of Riemann surfaces can be expressed in terms of the classes G*(ch(n)(t)) evaluated against the characteristic classes of X2k~ι. The relationship between these detection procedures was independently discovered by D. Mumford. It is: + 2 Theorem 1.6. There exist as classes in H*(B Diff (Sg, D ): Q): (1.9) G*(chiH)(t)) = Nn(y2n) + {decomposible) k ι with N2k = 0 and N2k_ι = (-\) - Bk/(2k)\, where Bk is the kth Bernoulli number. THE HOMOLOGY OF THE MAPPING CLASS GROUP 5 Combining the above Theorem 1.1 and 1.4 we have proved the result.
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