Abstract

This series of papers is aimed towards the calculation of the cohomology of the mapping class group of a closed oriented surface of genus two. This is all 2, 3 and 5-torsion (Theorem 1.1, p. 6). The mod 5 cohomology is given in the first paper (Theorem 1.2, p. 7), the mod 3 cohomology in the second (Theorem 1.1, p. 29), and the mod 2 cohomology in the third (Theorem 1.1, p. 86). Along the way, we investigate many interesting properties of mapping class groups. This involves us in the study of the braid groups, modular representations of symmetric groups, and configuration spaces.

Subject classification Primary: 57M20, 20J05 Secondary: 20C20, 20C30, 20F36, 20F38, 55R99, 55T99, 57T99

Key words Mapping class group, braid group, cohomology, symmetric group, configuration space.

Introduction

D. J. Benson  F. R. Cohen

Let M_{g,k}^{n} denote an orientable surface (i.e., differential manifold of dimension two) of genus g with k boundary components and n punctures. The mapping class group Γ_{g,k}^{n} is defined to be the discrete group π_{0}Diff^{+}(M_{g,k}^{n}) where Diff^{+} denotes the group of orientation preserving self-diffeomorphisms of the surface. In other words, an element of Γ_{g,k}^{n} can be thought of as an isotopy class of self-diffeomorphisms. Note that elements of Γ_{g,k}^{n} are allowed to permute the punctures, so that there is a surjective map Γ_{g,k}^{n} → → Σ_{n} where Σ_{n} denotes the symmetric group on n letters. In fact, we are only interested in the case k = 0, but it should be noted that for k > 0 the self-diffeomorphisms are required to fix the boundary pointwise.

In this series of papers we study homological properties of the groups Γ_{0,0}^{n}, Γ_{2,0}^{0} and a certain subgroup Δ_{g} of Γ_{g,0}^{0} which was defined by Birman and Hilden [2] as follows. We express M_{g,0}^{0} as a double cover of the sphere S^{2}, branched at 2g + 2 points. Each self-diffeomorphism of S^{2} preserving the set of branch points lifts in two ways to a selfdiffeomorphism of M_{g,0}^{0}, and Δ_{g} is defined to be the subgroup of Γ_{g,0}^{0} represented by these lifts. Thus there is a (non-split) central extension

1 → Z/2 → Δ_{g} → Γ_{0,0}^{2g+2} → 1
∩
Γ_{g,0}^{0}

The group Δ_{g} is generated by Dehn twists α_{1}, …, α_{g}, β_{1}, …, β_{g} and α_{g+1} determined by circles as shown in the picture below.

By results of Waynryb [16], α_{1}, …, α_{g}, β_{1}, …, β_{g} and δ give generators for Γ_{g,0}^{0}. Thus Δ_{2} = Γ_{2,0}^{0} and Δ_{g} is a proper subgroup of Γ_{g,0}^{0} for g ≥ 3.

Dehn twists satisfy braid relations as follows. If the circles corresponding to σ and τ do not intersect then στ = τσ, while if σ and τ intersect once transversally then στσ = τστ. Thus Δ_{g} can be viewed as a quotient of the braid group B_{2g+2} on 2g + 2 strings.

The group Γ_{g,0}^{0} acts on H^{1}(M_{g,0}^{0}, Z) ≅ Z^{2g}, preserving the intersection pairing H^{1} × H^{1} → H^{2}, and so we have a natural map Γ_{g,0}^{0} → Sp_{2g}(Z). The following diagram commutes

Γ_{g,0}^{0} → → Sp_{2g}(Z) → → Sp_{2g}(Z/2)
∪  ∪
B_{2g+2} → → Δ_{g} → → Γ_{2g+2,0}^{0} → → Σ_{2g+2}

and for g = 2 this displays the isomorphism Σ_{6} ≅ Sp_{4}(Z/2)^{1}.

Earle and Eells [6] have shown that for g ≥ 2, the connected component of the identity, Diff_{0}^{+}(M_{g,0}^{0}), is contractible. Thus BΓ_{g,0}^{0} = BDiff^{+}(M_{g,0}^{0}) classifies oriented bundles with fibre M_{g,0}^{0}, and so the cohomology of Γ_{g,0}^{0} gives characteristic classes for such bundles. In general, Γ_{g,0}^{0} is closely related to the moduli space of algebraic curves of genus g, but we shall not go into that subject here.

Some of the first homological calculations for Γ_{g,0}^{0} were given by John Harer, who determined H_{2}(Γ_{g,0}^{0}; Z) for g ≥ 5 [8]. Harer also proved a beautiful stabilization theorem [9]. Using the natural pairing Γ_{g,1}^{0} × Γ_{k,1}^{0} → Γ_{g+k,1}^{0}, Ed Miller constructed an infinite family of elements in the homology of the colimit Γ = lim_{g^{→}} Γ_{g,1}^{0} [14]. Morita gave similar rational

classes [15], while Glover and Mislin [7] and Charney and Lee [4] gave some torsion classes related to the homotopy theorists’ space Im J. Indeed, Cohen and Charney observed that BΓ has Im J_{(1/2)} as a stable retract [3]. However the results in this paper are primarily geared to the study of Γ_{2,0}^{0}.

Our calculations are broken up into three papers. The first one, denoted I, is titled “Artin’s braid group and the homology of certain subgroups of the mapping class group.” The second one, denoted II, is titled “Specht modules and the cohomology of mapping class groups.” The third one, denoted III, is titled “The mod 2 cohomology of the mapping class group for a surface of genus two.” In these three papers we study the cohomology of Γ_{0,0}^{n} and Δ_{g} via the two group extensions

(1) 1 → Z/2 → Δ_{g} → Γ_{0,0}^{2g+2} → 1 and

(2) 1 → K_{n} → Γ_{0,0}^{n} → Σ_{n} → 1.

The cohomology of K_{n} follows fairly directly from some work of W. Magnus [13] dating from 1934. The other ingredient is the cohomology of the pure braid group as an algebra over the symmetric group (i.e., as an algebra on which the symmetric group acts as algebra automorphisms). The cohomology algebra H^{∗}(K_{n}; Z) as an algebra over the symmetric group is given in Theorem 1.5 of I. It is finitely generated and torsion free with Poincaré series (1 + 2t)(1 + 3t) ⋯ (1 + (n − 2)t). It is generated as an algebra by elements of degree one B_{ij}, n ≥ i > j ≥ 2, i ≥ 4, and all relations follow from the one relation B_{42}B_{43} = 0 together with the action of Σ_{n}, associativity and commutativity. From this description, H^{1}(K_{n}; Z) is identified in Theorem 1.3 of II as the Specht module S^{(n−2,2)} as defined in James [10]. Similar information is given for H^{2}(K_{n}; Z) and H^{3}(K_{n}; Z).

1It is probably worth remarking at this stage, although we make no use of it in this paper, that in the case of g = 3 the Dehn twists satisfy braid relations for a braid group of type E_{7}. The natural map from this braid group to the Weyl group W (E_{7}) ≅ Z/2 × Sp_{6}(Z/2) is the product of the sign representation and the composite B(E_{7}) → → Γ_{3,0}^{0} → → Sp_{6}(Z) → → Sp_{6}(Z/2).

Using the identification of the local coefficient system for the extension (2), we analyse the Lyndon-Hochschild-Serre spectral sequence for this extension. Some direct consequences are listed in Theorems 1.1–1.3 of I:

1. H̄_{∗}(Γ_{0,0}^{n}; Z) is all p-torsion for primes p with p ≤ n.

2. The maps Γ_{2,0}^{0} → → Γ_{0,0}^{6} → → Z/5 induce isomorphisms H_{∗}(Γ_{2,0}^{0}; F_{5}) ≅ H_{∗}(Γ_{0,0}^{6}; F_{5}) ≅ H_{∗}(Z/5; F_{5}), and

3. There are embeddings

H_{∗}(B_{n}/center; F_{2}) → H_{∗}(Γ_{0,0}^{n}; F_{2}) and
H_{∗}(B_{2g+2}/center; F_{2}) → H_{∗}(Δ_{g}; F_{2})

where B_{n} denotes Artin’s braid group on n strings, n ≡ 2 (4), and g ≡ 0 (2).

Using apparently different methods, Lee and Weintraub first showed in [11] that H̄_{∗}(Γ_{2,0}^{0}; Z) is all 2, 3 and 5 torsion.

A curious example of the results above is given by the classifying space BΓ_{0,0}^{4}. This space admits a stable decomposition as a one-point union X_{2} ∨ X_{3} where X_{2} is 2-local and X_{3} is 3-local. The 2-local stable type of X_{2} is that of BΣ_{4} ∨ ΣM(2) where ΣM(2) is the homotopy direct limit of the self-map of Σ(RP^{∞} ∧ RP^{∞}) given by the track sum e_{2} = ( \begin{matrix}1 & 0 \\ 0 & 1 \end{matrix} ) + ( \begin{matrix}1 & 1 \\ 1 & 0 \end{matrix} ) + ( \begin{matrix}1 & 1 \\ 0 & 1 \end{matrix} ) + ( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} ); notice that e_{2} is the second Steinberg idempotent associated to GL_{2}(F_{2}). The space X_{3} is K(Z/3, 1). Finally, there is a natural map Γ_{0,0}^{n} → GL_{n−1}(Z) given by the action of Γ_{0,0}^{n} on H_{1}(S^{2} − {n points}; Z). The composite stable map ΣM(2) → BΓ_{0,0}^{4} → BGL_{3}(Z) is inessential and thus the cohomology of ΣM(2) gives a family of exotic characteristic classes for Γ_{0,0}^{4}-bundles which arise from 2-fold iterates of Steenrod operations Sq^{a}Sq^{b}, a ≥ 2b. These remarks are contained in Theorem 1.6 of I.

The diagrammatic methods of Benson and Carlson [1] are crucial in paper II where the calculations of the cohomology of Γ_{0,0}^{6} are carried out completely. In characteristic 3, this gives the complete answer for the cohomology of Γ_{2,0}^{0}, while in characteristic 2 an ambiguity in the action of the Steenrod algebra on the cohomology of Γ_{0,0}^{6} prevents us from computing the spectral sequence converging to the cohomology of Γ_{2,0}^{0}. The main tool in the calculation is a resolution of H^{i}(K_{n}; Z), i = 1, 2, 3, in terms of the exterior powers of H^{1}(K_{n}; Z) and certain other Specht modules. After a rather intricate calculation of H^{∗}(Σ_{6}; H^{∗}(K_{6}; F_{p})), some results of Quillen are used to calculate the differentials in the Lyndon-Hochschild-Serre spectral sequence for the extension (1).

Finally in III the ambiguity in the action of the Steenrod algebra on the mod 2 cohomology of Γ_{0,0}^{6} is resolved using configuration space techniques, and the cohomology of Γ_{2,0}^{0} is calculated. The results of these calculations may be stated as follows:

There are algebra extensions

4. 1 → H^{∗}(Σ_{3}; F_{3}) → H^{∗}(Γ_{2,0}^{0}; F_{3}) → V → 1 where the Poincaré series for V is 1 + t^{3} + t^{4} + t^{5},

5. 1 → F_{2}[a_{2}, a_{3}] → H^{∗}(Γ_{0,0}^{6}; F_{2}) → W → 1 where the degree of a_{i} is i and the Poincaré series for W is 1 + t + 2t^{2} + 3t^{3} + 2t^{4} + t^{5} and 6. The Poincaré series for H^{∗}(Γ_{2,0}^{0}; F_{2}) is (1 + t^{2} + 2t^{3} + t^{4} + t^{5})/(1 − t)(1 − t^{4}).

In a subsequent paper with C.-F. Bödigheimer and M. Peim, we shall give a complete calculation of the additive structure for H^{∗}(Γ_{0,0}^{n}; F_{p}) for all primes p. A large part of the mod 2 cohomology of Γ_{0,0}^{2n} is given by that of B_{2n}/(center) while the cohomology of Γ_{0,0}^{2n+1} is additively isomorphic to that of B_{2n+1}/(center).

The author of I would like to take this opportunity to correct an error which inadvertently appeared in F. R. Cohen, Homology of mapping class groups for surfaces of low genus, Contemp. Math., The Lefschetz Centennial Conference, Proc. on Algebraic Topology, 58.11 pages 21–30, where some of the above calculations were described. Namely, it was claimed that the natural maps B_{g} → Γ_{g,1}^{0} induce a double loop map Ω_{0}^{2}S^{2} → BΓ^{+}. This statement is incorrect as the results in the thesis of John Maginnis [12] show.

We include a (perhaps relevant) digression. Consider Artin’s braid group B_{n}. The group B_{n} admits an epimorphism B_{n} → Σ_{n} with kernel P_{n}, the pure braid group. Thus the homology of P_{n} is a module over the group ring of Σ_{n}. There is an epimorphism of Σ_{n}-modules H_{∗}(P_{n}; Z) → H_{∗}(K_{n}; Z) which is split as a map of abelian groups [I Thm 1.5, proof]. The homology of P_{n} is concentrated in degrees q with 0 ≤ q ≤ n − 1. The representations of Σ_{n} given by the homology of P_{n} have some very curious properties which we now elaborate. Let F be a field and M a graded module over F which is concentrated in degrees at least one. Given a graded vector space V, define σ^{k}V to be V with all degrees raised by k. Define

Λ(M) = ⨁_{n≥0} H_{n−1}(P_{n}; F) ⊗_{Σ_{n}} M^{⊗n}
P (M) = ⨁_{n≥0} H_{∗}(P_{n}; F) ⊗_{Σ_{n}} M^{⊗n}, and
K(M) = ⨁_{n≥0} H_{∗}(K_{n}; F) ⊗_{Σ_{n}} M^{⊗n}

where M^{⊗n} = \underbrace{M ⊗_{F} ⋯ ⊗_{F} M}_{n} and, by convention, H_{0}(G; F) ⊗_{Σ_{1}} M^{⊗0} = F. It is pointed

out in [5] that σ^{+1}Λ(M) is a graded Lie algebra which is isomorphic to L[σ^{+1}M] as a Lie algebra, where L[ · ] denotes the free graded Lie algebra. Furthermore, P (M) comes equipped with the structure of a symmetric algebra which is isomorphic to the symmetric algebra generated by σ^{−1}L[σ^{+1}M]. As there is a natural epimorphism P (M) → K(M), it would be interesting to see whether K(M) admits a similar interpretation in terms of Lie algebras.

References

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[10] G. D. James. The representation theory of the symmetric groups. Springer lecture notes in Mathematics 682, Springer-Verlag, Berlin/New York 1978.

[11] R. Lee and S. Weintraub. Cohomology of Sp_{4}(Z) and related groups and spaces. Topology 24 (1985), 391–410.

[12] J. Maginnis. Thesis, Stanford University, 1987.

[13] W. Magnus. Über Automorphismen von Fundamentalgruppen berandeter Flächen. Math. Ann., 109 (1934), 617–646.

[14] E. Miller. The homology of the mapping class group. J. Diff. Geometry 24 (1986), 1–14.

[15] S. Morita. Characteristic classes of surface bundles. Inv. Math., 90 (1987), 551–577.

[16] B. Wajnryb. A simple presentation for the mapping class group of an orientable surface. Israel J. Math., 45 (1983), 157–174.