1 Introduction↩︎

1.1 Finite generation of homology↩︎

In the study of the mapping class group \(\mathop{\mathrm{Mod}}\nolimits_g\) of the genus \(g\) oriented surface \(\Sigma_g\), a fundamental role is played by the Torelli subgroup \(\mathcal{I}_g\le\mathop{\mathrm{Mod}}\nolimits_g\) consisting of those mapping classes that act trivially on \(H_1(\Sigma_g;\mathbb{Z})\). It is an old well-known problem to find out which homology groups \(H_k(\mathcal{I}_g;\mathbb{Z})\) are finitely generated and which are not, see, for instance [1], [2]. A particularly important question is whether these groups are finitely generated in a range of degrees tending to infinity with \(g\). Hain [3] mentioned a folk conjecture that \(H_k(\mathcal{I}_g;\mathbb{Z})\) is finitely generated when \(k\le g-2\). Our first result is exactly the proof of this conjecture.

Along with the closed surface \(\Sigma_g\), we will consider a surface with one puncture \(\Sigma_{g,1}\) and a surface with one boundary component \(\Sigma_g^1\). To treat these three cases together, we write \(\Sigma_{g,p}^b\) for a genus \(g\) oriented surface with \(p\) punctures and \(b\) boundary components, and agree that we omit an index if it is zero. However, throughout the paper we always impose the condition \(b+p\le 1\). We denote by \(\mathop{\mathrm{Mod}}\nolimits_{g,p}^b\) and \(\mathcal{I}_{g,p}^b\) the mapping class group and Torelli group for \(\Sigma_{g,p}^b\), respectively. (Note that \(\mathop{\mathrm{Mod}}\nolimits_g^1\) is the mapping class group for diffeomorphisms of \(\Sigma_g^1\) that fix every point of the boundary.) For definitions and results concerning mapping class groups, see [4].

Previously, the finite generation of homology was known only for \(k=1\) and \(g\ge 3\) (Johnson [5], [6]) and for the second rational homology groups \(H_2(\mathcal{I}_{g,p}^b;\mathbb{Q})\) when \(g\ge 5\) (Minahan [7] for \(g\ge51\), and Minahan and Putman [8] for all \(g\ge 5\)). Note that even for \(k=2\) our result is stronger than that of Minahan and Putman. Firstly, we prove finite generation of the second homology group starting from \(g=4\) instead of \(g=5\). Secondly, we do this with integer coefficients rather than rational ones.

Note that the homology groups of the Torelli groups are not finitely generated in general. The first infiniteness result for Torelli groups was obtained by McCullough and Miller [9] who showed that \(\mathcal{I}_2\) is not finitely generated. Mess [10] proved that \(\mathcal{I}_2\) is an infinitely generated free group, and hence has not finitely generated \(H_1\). The first general result of non‑finite generation of higher homology groups was obtained by Akita [11] who showed that \(\dim H_*(\mathcal{I}_g;\mathbb{Q})=\infty\), provided that \(g\ge 7\). The groups \(H_3(\mathcal{I}_3;\mathbb{Z})\) and \(H_4(\mathcal{I}_3;\mathbb{Z})\) were shown to be not finitely generated by Johnson and Millson (unpublished, see [10]) and Hain [12], respectively. Later, Bestvina, Bux, and Margalit [13] proved that the cohomological dimension of \(\mathcal{I}_g\) is equal to \(3g-5\), and the top homology group \(H_{3g-5}(\mathcal{I}_g;\mathbb{Z})\) is not finitely generated. Finally, the author [14] extended the non‑finite generation statement for \(H_k(\mathcal{I}_g;\mathbb{Z})\) to all \(k\) in range \(2g-3\le k\le 3g-6\).

1.2 Algebraicity↩︎

We put \(H=H_1(\Sigma_{g,p}^b;\mathbb{Z})\) and choose an isomorphism \(H\cong\mathbb{Z}^{2g}\) that takes the intersection form to the standard unimodular skew-symmetric form \((\cdot,\cdot)\) on \(\mathbb{Z}^{2g}\). (Such an isomorphism exists, since \(b+p\le 1\).) The action of the mapping class group \(\mathop{\mathrm{Mod}}\nolimits_{g,p}^b\) on \(H\) preserves the intersection form and hence defines a short exact sequence \[1\longrightarrow \mathcal{I}_{g,p}^b\longrightarrow\mathop{\mathrm{Mod}}\nolimits_{g,p}^b\longrightarrow\mathrm{Sp}_{2g}(\mathbb{Z})\longrightarrow 1.\] Hence all homology groups \(H_k(\mathcal{I}_{g,p}^b;\mathbb{Z})\) are \(\mathrm{Sp}_{2g}(\mathbb{Z})\)-modules.

We recall some facts on arithmetic groups that we will need, see [17], [18]. We always consider the following setup:

1. \(\mathbf{G}\) is a connected semisimple algebraic subgroup of \(\mathbf{GL}_n\) defined over \(\mathbb{Q}\),

2. \(\Gamma\) is a Zariski dense arithmetic subgroup of its rational points \(\mathbf{G}(\mathbb{Q})\), where arithmeticity means that \(\Gamma\) is commensurable with \(\mathbf{G}(\mathbb{Z})=\mathbf{G}(\mathbb{Q})\cap\mathrm{GL}_n(\mathbb{Z})\).

Recall that a rational representation of \(\mathbf{G}\) on a finite-dimensional \(\mathbb{Q}\)-vector space \(V\) is a morphism of algebraic groups \(\mathbf{G}\to\mathbf{GL}(V)\). A representation \(\rho\colon\Gamma\to\mathrm{GL}(V)\) is called algebraic if it can be obtained from a rational representation of \(\mathbf{G}\) (defined over \(\mathbb{Q}\)) by taking the rational points and then restricting to \(\Gamma\). (Note that an extension of \(\rho\) to a rational representation of \(\mathbf{G}\) is unique whenever it exists, since \(\Gamma\) is Zariski dense in \(\mathbf{G}(\mathbb{Q})\).) More generally, a representation \(\rho\) is called almost algebraic if there exists a finite-index subgroup \(\Lambda\le\Gamma\) such that the restriction of \(\rho\) to \(\Lambda\) is algebraic.

In this paper, we will only be interested in the case \(\mathbf{G}=\mathbf{Sp}_{2g}\) and \(\Gamma=\mathrm{Sp}_{2g}(\mathbb{Z})\) with \(g\ge 2\). In this case, Bass, Milnor, and Serre [19] proved that any representation of \(\Gamma\) on a finite-dimensional vector space over \(\mathbb{Q}\) is almost algebraic.

Recall that over a field of characteristic \(0\) all rational representations of \(\mathbf{Sp}_{2g}\) are semisimple and the irreducible rational representations of \(\mathbf{Sp}_{2g}\) are indexed by partitions \(\lambda=[\lambda_1\ldots\lambda_m]\) of length \(m\le g\), where \(\lambda_1\ge \cdots\ge\lambda_m\ge 1\); see [20], [21], and also a nice summary in [22]. We denote by \(\mathbf{V}_{\lambda}\) or \(\mathbf{V}_{\lambda_1,\ldots,\lambda_m}\) the irreducible representation that corresponds to a partition \(\lambda\). When we need to indicate the dependence of \(g\) in notation, we write \(\mathbf{V}_{\lambda}(g)\) instead of \(\mathbf{V}_{\lambda}\).

It is an important question whether the representations of \(\mathrm{Sp}_{2g}(\mathbb{Z})\) on the rational homology groups \(H_k(\mathcal{I}_{g,p}^b;\mathbb{Q})\) are algebraic without having to pass to a finite-index subgroup. The algebraicity was known for \(H_1(\mathcal{I}_{g,p}^b;\mathbb{Q})\) with \(g\ge 3\) by a result of Johnson [6]. Minahan and Putman [8] have recently proved that \(H_2(\mathcal{I}_{g,p}^b;\mathbb{Q})\) is algebraic for \(g\ge 6\). Our next result is as follows.

1.3 Explicit description of the cohomology ring↩︎

Kupers and Randal-Williams [23] obtained the following conditional result: they computed the algebraic part \(H^*\bigl(\mathcal{I}_g^1;\mathbb{Q}\bigr)^{\mathrm{alg}}\) of the rational cohomology ring \(H^*\bigl(\mathcal{I}_g^1;\mathbb{Q}\bigr)\) in stable range \(*\ll g\) under the assumption that the spaces \(H^*\bigl(\mathcal{I}_g^1;\mathbb{Q}\bigr)\) are finite-dimensional in stable range. Theorem 1 ensures that this assumption holds, and Theorem 2 shows that the result of Kupers and Randal-Williams gives a description of the whole rational cohomology ring \(H^*\bigl(\mathcal{I}_g^1;\mathbb{Q}\bigr)\) in stable range \(*\ll g\). The stable range in which the explicit description holds was incorrectly determined in [23]. A correction is given in [24]: it is shown there that the aforementioned description holds for \(*\le \frac{g-4}{4}\).

To state the result thus obtained we need the construction of twisted Miller–Morita–Mumford classes. Let \(\mathop{\mathrm{Diff}}\nolimits(\Sigma_g,D)\) denote the topological group of diffeomorphisms of \(\Sigma_g\) that are equal to the identity near a specified two-dimensional disk \(D\subseteq \Sigma_g\); we endow this group with the \(C^{\infty}\)-topology. Let \(\mathop{\mathrm{Tor}}\nolimits(\Sigma_g,D)\le \mathop{\mathrm{Diff}}\nolimits(\Sigma_g,D)\) be the corresponding (topological) Torelli group consisting of all diffeomorphisms that act trivially on \(H_1(\Sigma_g;\mathbb{Z})\). Let \[\Sigma_g\stackrel{f}{\longrightarrow} E\stackrel{\pi}{\longrightarrow} B\mathop{\mathrm{Tor}}\nolimits(\Sigma_g,D)\] be the universal \(\Sigma_g\)-bundle over \(B\mathop{\mathrm{Tor}}\nolimits(\Sigma_g,D)\), let \(T_{\pi}E\to E\) be its vertical tangent bundle, and let \(s\colon B\mathop{\mathrm{Tor}}\nolimits(\Sigma_g,D)\to E\) be the section of \(\pi\) determined by the centre of the disk \(D\). By studying the Serre spectral sequence for \(\pi\), one can prove (see [23] for details) that the map \(f^*\colon H^1(E;\mathbb{Q})\to H^1(\Sigma_g;\mathbb{Q})\) is a surjection and has a unique right inverse \(\iota\colon H^1(E;\mathbb{Q})\to H^1(\Sigma_g;\mathbb{Q})\) satisfying \(s^*\circ \iota=0\). Then for \(m\in\mathbb{Z}_{\ge0}\) and \(v_1,\ldots,v_r\in H^1(\Sigma_g;\mathbb{Q})\), the twisted Miller–Morita–Mumford classes are defined by \[\label{eq95MMM} \kappa_{e^m}(v_1\wedge \cdots\wedge v_r)=\pi_!\bigl(e^m(T_{\pi}E)\iota(v_1)\cdots\iota(v_r)\bigr)\in H^{2m+r-2}\bigl(B\mathop{\mathrm{Tor}}\nolimits(\Sigma_g,D);\mathbb{Q}\bigr),\tag{1}\] where \(e\) denotes the Euler class and \(\pi_!\) is the Gysin homomorphism for \(\pi\). Finally, it is known (see [25]) that the connected component of the identity in the group \(\mathop{\mathrm{Diff}}\nolimits(\Sigma_g,D)\) is contractible. Hence \(B\mathop{\mathrm{Tor}}\nolimits(\Sigma_g,D)\simeq B\mathcal{I}_g^1\) and we obtain the classes \[\kappa_{e^m}(v_1\wedge \cdots\wedge v_r)\in H^{2m+r-2}\bigl(\mathcal{I}_g^1;\mathbb{Q}\bigr).\]

Combining Theorem A from [23] with our Theorems 1 and 2 and taking into account the corrections made in [24], we immediately obtain the following result.

As a consequence of Theorem 3, an explicit calculation of the \(\mathrm{Sp}_{2g}(\mathbb{Z})\)-representation \(H^3(\mathcal{I}_g^1;\mathbb{Q})^{\mathrm{alg}}\) from [23] and [24] also becomes a result about the whole group \(H^3(\mathcal{I}_g^1;\mathbb{Q})\). (A similar result for the second rational homology of \(\mathcal{I}_g^1\) was already known by the algebraicity result of Minahan and Putman in this case, see [8].)

After we prove finite-dimensionality and algebraicity of \(H^*(\mathcal{I}_g^1;\mathbb{Q})\) in stable range, and thus establish the presentation from Theorem 3, we can obtain further consequences from the results of [23], which were originally conditional results about the maximal algebraic subrepresentation \(H^*(\mathcal{I}_g^1;\mathbb{Q})^{\mathrm{alg}}\) (under the condition of finite dimensionality), but now become unconditional results about the whole ring \(H^*(\mathcal{I}_g^1;\mathbb{Q})\). In the next two subsections we obtain two such consequences: we prove Morita’s conjecture on the invariant part of the cohomology of Torelli groups and the uniform representation stability for \(\left\{H_k(\mathcal{I}_g^1;\mathbb{Q})\right\}_{g=1}^{\infty}\).

1.4 Morita’s conjecture↩︎

A remarkable theorem by Madsen and Weiss [29] says that \[\lim_{g\to\infty}H^*\bigl(\mathop{\mathrm{Mod}}\nolimits_g^1;\mathbb{Q}\bigr)=\mathbb{Q}[e_1,e_2,e_3,\ldots],\] where \(e_m\)’s are the Miller–Morita–Mumford classes. As we have already mentioned (see Remark 3), the odd Miller–Morita–Mumford classes \(e_{2n+1}\) vanish when restricted to the Torelli group. An old conjecture due to Morita, see [27] (cf. [30]) asserts that any class \(e_{2n}\) is nontrivial in the rational cohomology of \(\mathcal{I}_g\) (or \(\mathcal{I}_g^1\)) for sufficiently large \(g\), and moreover, the \(\mathrm{Sp}_{2g}(\mathbb{Z})\)-invariant part of the cohomology of \(\mathcal{I}_g^1\) stabilizes and we have an isomorphism \[\lim_{g\to\infty}H^*\bigl(\mathcal{I}_g^1;\mathbb{Q}\bigr)^{\mathrm{Sp}_{2g}(\mathbb{Z})}=\mathbb{Q}[e_2,e_4,e_6,\ldots].\] We prove this conjecture.

1.5 Representation stability.↩︎

In [31] Church and Farb introduced a concept of uniform representation stability. In the case of \(\mathrm{Sp}_{2g}(\mathbb{Q})\)-representations, the definition is as follows.

Let \(\{V_g\}\) be a sequence of rational \(\mathrm{Sp}_{2g}(\mathbb{Q})\)-representations equipped with linear maps \(\varphi_g\colon V_g\to V_{g+1}\) that is compatible in the sense that \(\varphi_g(\gamma v)=f_g(\gamma)\varphi_g(v)\) for all \(\gamma\in \mathrm{Sp}_{2g}(\mathbb{Q})\) and \(v\in V_g\), where \(f_g\) is the standard inclusion \(\mathrm{Sp}_{2g}(\mathbb{Q})\hookrightarrow \mathrm{Sp}_{2g+2}(\mathbb{Q})\). The sequence \(\{V_g\}\) is said to be uniformly representation stable if there exists some \(N\) such that each of the following conditions holds:

1. Injectivity: The map \(\varphi_g\colon V_g\to V_{g+1}\) is injective for all \(g\ge N\),

2. Surjectivity: The induced map \(\mathop{\mathrm{Ind}}\nolimits_{\mathrm{Sp}_{2g}(\mathbb{Q})}^{\mathrm{Sp}_{2g+2}(\mathbb{Q})}V_g\to V_{g+1}\) is surjective for all \(g\ge N\),

3. Multiplicity stability: Write \[V_g\cong\bigoplus_{\lambda}c_{\lambda,g}\mathbf{V}_{\lambda}(g).\] Then the multiplicities \(c_{\lambda,g}\) are independent of \(g\) for all \(g\ge N\). In particular, for any \(\lambda\) for which \(\mathbf{V}_{\lambda}(N)\) is not defined \(c_{\lambda,g}=0\) for all \(g\ge N\).

The uniform representation stability for \(\left\{ H_1(\mathcal{I}_g^1;\mathbb{Q})\right\}_{g=1}^{\infty}\) follows immediately from a classical result of Johnson [6]. The uniform representation stability for \(\left\{ H_2(\mathcal{I}_g^1;\mathbb{Q})\right\}_{g=1}^{\infty}\) was proved by Minahan and Putman, see [8], [22]. We extend this result to all homology groups. By Theorem 2 each space \(H_k(\mathcal{I}_g^1;\mathbb{Q})\) with \(k\le g-2\) becomes a rational \(\mathrm{Sp}_{2g}(\mathbb{Q})\)-representation.

The sum \(|\lambda|=\lambda_1+\cdots+\lambda_m\) is called the degree of an irreducible representation \(\mathbf{V}_{\lambda}(g)\); it is equal to the smallest number \(d\) such that \(\mathbf{V}_{\lambda}(g)\) appears in \(H(g)^{\otimes d}\), where \(H(g)\) is the tautological representation of \(\mathrm{Sp}_{2g}(\mathbb{Q})\).

The injectivity and multiplicity stability properties follow immediately from Theorems 1 and 2 and the results from [23] (with the corrections from [24]). Surjectivity, however, requires additional work, see Proposition 34. Note that for \(k=2\) the surjectivity property was already known from a result of Boldsen and Dollerup [32], but for \(k>2\) no such result is available. The author does not know whether it is possible to extract the surjectivity property starting from \(g=4k+4\) from the results of [23] and [24].

1.6 Outline of the proofs of Theorems 1 and 2↩︎

For \(x\in\mathbb{Z}^{2g}\), the transformation \(t_x\in\mathrm{Sp}_{2g}(\mathbb{Z})\) defined by \[t_xy=y+(x,y)x,\] as well as any of its powers \(t_x^{\xi}\), is called a symplectic transvection.

Our approach is partially inspired by the author’s proof (see [33]) of the finite generation of the abelianization \((\mathcal{K}_g^b)^{\mathrm{ab}}\) for \(g\ge 3\) and \(b\in\{0,1\}\), where \(\mathcal{K}_g^b\le\mathcal{I}_g^b\) is the Johnson kernel, i. e., the subgroup generated by all Dehn twists about separating simple closed curves.

A key role in our approach is played by the following property of \(\mathrm{Sp}_{2g}(\mathbb{Z})\)-modules, which we shall call transvection unipotence.

The main result on which Theorems 1 and 2 rely is as follows.

The proof of this theorem is based on the study of a spectral sequence \[\label{eq95ss95intro} E^1_{p,q}=\bigoplus_{\sigma\in\mathfrak{X}_p}H_q(\mathcal{I}_{\sigma};\mathbb{Z}) \;\Longrightarrow \;H_{p+q}^{\mathcal{I}}\bigl(\mathcal{C}_x(\Sigma_g);\mathbb{Z}\bigr)\tag{2}\] associated with the action of \(\mathcal{I}=\mathcal{I}_{g,p}^b\) on the complex of homologous curves \(\mathcal{C}_x(\Sigma_g)\) defined by Putman [34]. Here \(\mathfrak{X}_p\) is a set of representatives for \(\mathcal{I}\)-orbits of \(p\)-cells of \(\mathcal{C}_x(\Sigma_g)\). Minahan [35] proved that the complex \(\mathcal{C}_x(\Sigma_g)\) is \((g-3)\)-acyclic, which implies that the spectral sequence 2 converges to the homology of \(\mathcal{I}\) in dimensions \(k<g-2\), and in dimension \(g-2\), the resulting limit group surjects onto \(H_{g-2}(\mathcal{I};\mathbb{Z})\). We will prove that the transvection \(t_x\) acts trivially on the spectral sequence 2 . The required unipotence condition ?? will follow, since the \(E^{\infty}\) page is the associated graded of some filtration in the homology of \(\mathcal{I}\).

It turns out that Theorem 6 is already sufficient to deduce Theorem 2 from Theorem 1. Indeed, we will prove in Section 10 that any finite-dimesional transvection unipotent representation of \(\mathrm{Sp}_{2g}(\mathbb{Z})\) with \(g\ge 2\) is algebraic.

The second ingredient in the proof of Theorem 1 will be results on bounded elementary generation and finite width for the groups \(\mathrm{Sp}_{2g}(\mathbb{Z})\), where \(g\ge 2\).

We need the following result that is due to Tavgen [36]. (This result is stated in [36] in a slightly different form; we will explain how to extract it from the results of that paper in Section 5.)

A key observation is that this proposition easily implies that any finitely generated transvection unipotent \(\mathrm{Sp}_{2g}(\mathbb{Z})\)-module is finitely generated as an abelian group. Kassabov and Putman [37] proved that the groups \(H_2(\mathcal{I}_{g,p}^b;\mathbb{Z})\) are finitely generated as \(\mathrm{Sp}_{2g}(\mathbb{Z})\)-modules. Combining this result with Theorem 6 and Proposition 9, we obtain the assertion of Theorem 1 for \(k=2\).

However, in the case \(k>2\) this argument does not work, because we initially have no result on the finite generation of the \(\mathrm{Sp}_{2g}(\mathbb{Z})\)-module \(H_k(\mathcal{I}_{g,p}^b;\mathbb{Z})\). Therefore, we need to learn how to do without this result. We will do this as follows.

We proceed by induction on \(k\), so we assume that all homology groups \(H_i(\mathcal{I}_{g,p}^b;\mathbb{Z})\) with \(i<k\) are finitely generated. Raghunathan [38] and Ivanov [39] proved the finite generation of all homology groups (even with twisted coefficients) of the groups \(\mathrm{Sp}_{2g}(\mathbb{Z})\) and \(\mathop{\mathrm{Mod}}\nolimits_{g,p}^b\), respectively. Then studying the Lyndon–Hochschild–Serre spectral sequence for the short exact sequence \[1\longrightarrow \mathcal{I}_{g,p}^b\longrightarrow\mathop{\mathrm{Mod}}\nolimits_{g,p}^b\longrightarrow\mathrm{Sp}_{2g}(\mathbb{Z})\longrightarrow 1,\] we will prove that the \(k\)th homology group \(H_k(\mathcal{I}_{g,p}^b;\mathbb{Z})\) satisfies the following condition:

For any finite-index subgroup \(\Lambda\le\mathrm{Sp}_{2g}(\mathbb{Z})\) and any \(\Lambda\)-module \(A\) that is finitely generated as an abelian group, the coinvariant group \(\left(H_k\bigl(\mathcal{I}_{g,p}^b;\mathbb{Z}\bigr)\otimes A\right)_{\Lambda}\) is finitely generated.

In general, for an \(\mathrm{Sp}_{2g}(\mathbb{Z})\)-module \(M\), the condition of having finitely generated coinvariants \((M\otimes A)_{\Lambda}\) for all \(\Lambda\) and \(A\) as described above is much weaker than the condition of being finitely generated as an \(\mathrm{Sp}_{2g}(\mathbb{Z})\)-module. Nevertheless, we will prove that for transvection unipotent modules these two conditions are equivalent, so the finite generation of coinvariants suffices to show that \(H_k\bigl(\mathcal{I}_{g,p}^b;\mathbb{Z}\bigr)\) is a finitely generated abelian group. This will complete the proof of Theorem 1.

1.7 Structure of the paper↩︎

Section 2 gathers the necessary preliminaries on a spectral sequence associated with the action of a group on a CW complex. Section 3 is the core of the paper: we study the spectral sequence for the action of the Torelli group on a complex of homologous curves and prove Theorem 6. In Section 4 we construct universal \(\mathcal{T}\)-unipotent modules and prove their finite generation in the case of groups of finite width. Section 5 contains the proof of Theorem 1 for \(k=2\). The proof of Theorem 1 for \(k>2\) occupies Sections 6–9. More precisely, in Section 6 we prove the finite generation of the coinvariants of the homology of Torelli groups. Section 7 provides an auxiliary representation‑theoretic construction. In Section 8 we recall the necessary facts about arithmetic groups and their representations. Finally, in Section 9 we obtain a general result of finite generation for \(\mathcal{T}\)-unipotent \(\Gamma\)-modules with finitely generated coinvariants, where \(\Gamma\) is an arithmetic group of finite width (Theorem 26). As a result, we prove Theorem 1 in the general case. Section 10 contains the proof of Theorem 2 on algebraicity of the rational homology of Torelli groups in the stable range. Finally, in Sections 11 and 12 we prove Theorems 5 and 4, respectively.

1.8 Conventions and notation↩︎

In this paper, we always work with left modules. Modules over the group ring \(\mathbb{Z}[G]\) will be called \(G\)-modules. The symbol \(\otimes\) will denote the tensor product over \(\mathbb{Z}\) (or, equivalently, over \(\mathbb{Q}\) for \(\mathbb{Q}\)-vector spaces) unless explicitly stated otherwise.

An element \(a\) of an abelian group \(A\) is called primitive if it cannot be written in the form \(nb\) where \(b\in A\) and \(n>1\).

Throughout the paper we consider only surfaces \(\Sigma_{g,p}^b\) with \(b+p\le 1\).

1.9 Acknowledgements↩︎

The author is grateful to Denis Osipov, Vasilii Rozhdestvenskii, and Andrei Vladimirov for fruitful discussions.

2 Spectral sequence↩︎

All statements in this section hold for homology with coefficients in any commutative ring \(R\); therefore, we omit the coefficients from the notation.

Assume that a group \(G\) acts cellulary and without rotations on a regular CW complex \(X\). Then we have the following spectral sequence (see [40]): \[\label{eq95spec95seq} E^1_{p,q}=\bigoplus_{\sigma\in\mathfrak{X}_p}H_q(G_{\sigma}) \;\Longrightarrow \;H_{p+q}^G(X).\tag{3}\] Here \(\mathfrak{X}_p\) is a set of representatives for \(G\)-orbits of \(p\)-cells of \(X\), \(G_{\sigma}\) denotes the stabilizer of \(\sigma\), and \(H^G_*(X)\) is the equivariant homology of \(X\). ‘Without rotations’ means that if \(g\in G\) fixes a cell \(\sigma\) setwise, then \(g\) fixes every point of \(\sigma\). The convergency of 3 means that, for each \(k\), there is a filtration \[\label{eq95filtration} 0=\mathcal{F}_{-1,k+1}\subseteq\mathcal{F}_{0,k}\subseteq\mathcal{F}_{1,k-1}\subseteq\cdots\subseteq\mathcal{F}_{k,0}=H_k^G(X)\tag{4}\] such that \(E^{\infty}_{p,q}=\mathcal{F}_{p,q}/\mathcal{F}_{p-1,q+1}\).

The spectral sequence 3 is functorial in the following sense (see [40] and also [41]):

1. The inclusion \(\iota_{\sigma}\colon H_q(G_{\sigma})\hookrightarrow E^1_{p,q}\) is independent of the choice of representatives for all other \(G\)-orbits of cells of \(X\). Moreover, if we replace \(\sigma\) with another representative \(g\sigma\) of the same \(G\)-orbit, then we get a commutative diagram \[\label{eq95cd95iota} \begin{tikzcd} H_q(G_{\sigma}) \arrow[rd,hookrightarrow,"{\iota_{\sigma}}"] \arrow[dd,"{g_*}","\cong"'] & \\ & E^1_{p,q} \\ H_q(G_{g\sigma}) \arrow[ru,hookrightarrow,"{\iota_{g\sigma}}"'] \end{tikzcd}\tag{5}\] where the isomorphism \(g_*\) is induced by the conjugation by \(g\). In particular, the summands in the direct sum decompositions of \(E^1_{p,q}\) are independent of the choice of a set \(\mathfrak{X}_p\).

2. Suppose that \(G\) acts cellulary and without rotations on \(X\), \(H\) acts cellulary and without rotations on \(Y\), \(\varphi\colon G\to H\) is a homomorphism, and \(f\colon X\to Y\) is a \(\varphi\)-equivariant map of CW complexes that maps every cell of \(X\) homeomorphically onto a cell of \(Y\). Then the pair \(\theta=(\varphi,f)\) induces a morphism \[E^r_{p,q}(\theta)\colon E^r_{p,q}(G,X)\to E^r_{p,q}(H,Y)\] of the corresponding spectral sequences 3 such that

   • \(E^1_{p,q}(\theta)\) coincides with the sum of the homomorphisms \[H_q(\varphi|_{G_{\sigma}})\colon H_q(G_{\sigma})\to H_q(H_{f(\sigma)}).\]

   • the map \(H_*(\theta)\colon H_*^G(X)\to H_*^H(Y)\) sends \(\mathcal{F}_{p,q}(G,X)\) to \(\mathcal{F}_{p,q}(H,Y)\) for all \(p\) and \(q\), and \(E^{\infty}_{p,q}(\theta)\) is the associate graded homomorphism.

The functoriality, in particular, implies the following proposition.

3 Proof of Theorem 6↩︎

The assertion of Theorem 6 for \(g\le 2\) is trivial, so we always assume that \(g\ge 3\).

Consider a closed surface \(\Sigma_g\) of genus \(g\). The complex of curves \(\mathcal{C}(\Sigma_g)\) is a simplicial complex whose vertices are isotopy classes of essential simple closed curves on \(\Sigma_g\), and a tuple \(\langle c_0,\ldots,c_k\rangle\) forms a simplex whenever the isotopy classes \(c_0,\ldots,c_k\) admit pairwise disjoint representatives. Recall that a simple closed curve is called essential if it is homotopically nontrivial. In what follows, we do not distinguish between a simple closed curve and its isotopy class.

Choose a primitive homology class \(x\in H_1(\Sigma_g;\mathbb{Z})\). The complex of homologous curves, defined by Putman [34] and denoted \(\mathcal{C}_x(\Sigma_g)\), is the full subcomplex of \(\mathcal{C}(\Sigma_g)\) spanned by all simple closed curves \(c\) that represent the homology class \(x\). Minahan [35] proved that the complex \(\mathcal{C}_x(\Sigma_g)\) is \((g-3)\)-acyclic, that is, \(\widetilde{H}_k\bigl(\mathcal{C}_x(\Sigma_g);\mathbb{Z}\bigr)=0\) for all \(k\le g-3\).

The Torelli group \(\mathcal{I}_g\) acts on \(\mathcal{C}_x(\Sigma_g)\) cellularly and without rotations. In the cases of a surface with one puncture \(\Sigma_{g,1}\) or a surface with one boundary component \(\Sigma_g^1\), we define the actions of \(\mathcal{I}_{g,1}\) and \(\mathcal{I}_g^1\) on \(\mathcal{C}_x(\Sigma_g)\) via the homomorphisms \(\mathcal{I}_{g,1}\twoheadrightarrow \mathcal{I}_g\) and \(\mathcal{I}_g^1\twoheadrightarrow \mathcal{I}_g\) obtained by filling in the puncture and attaching a disk to the boundary of \(\Sigma_g^1\), respectively.

Let us now prove Theorem 6. Let \(\mathcal{I}\) be any of the three Torelli groups \(\mathcal{I}_g\), \(\mathcal{I}_{g,1}\), or \(\mathcal{I}_g^1\), and let \(\mathop{\mathrm{Mod}}\nolimits\) be the corresponding mapping class group (namely \(\mathop{\mathrm{Mod}}\nolimits_g\), \(\mathop{\mathrm{Mod}}\nolimits_{g,1}\), or \(\mathop{\mathrm{Mod}}\nolimits_g^1\), respectively). Set \(H=H_1(\Sigma_g;\mathbb{Z})\). We want to prove that \[\label{eq95annih2} (t_x-1)^{k+1}H_k(\mathcal{I};\mathbb{Z})=0,\qquad k\le g-2,\tag{6}\] for all \(x\in H\). Since \(t_{nx}-1=t_x^{n^2}-1\) is divisible by \(t_x-1\), it suffices to prove 6 for only primitive classes \(x\).

Fix a primitive class \(x\in H\) and consider the action of \(\mathcal{I}\) on the corresponding complex of homologous curves \(\mathcal{C}_x(\Sigma_g)\). Let \[E^1_{p,q}=\bigoplus_{\sigma\in\mathfrak{X}_p}H_q\left(\mathcal{I}_{\sigma};\mathbb{Z}\right) \;\Longrightarrow \;H_{p+q}^{\mathcal{I}}\bigl(\mathcal{C}_x(\Sigma_g);\mathbb{Z}\bigr)\] be the corresponding spectral sequence 3 and \[\label{eq95filtration2} 0=\mathcal{F}_{-1,k+1}\subseteq\mathcal{F}_{0,k}\subseteq\mathcal{F}_{1,k-1}\subseteq\cdots\subseteq\mathcal{F}_{k,0}=H_k^{\mathcal{I}}\bigl(\mathcal{C}_x(\Sigma_g);\mathbb{Z}\bigr)\tag{7}\] the corresponding filtration such that \(E^{\infty}_{p,q}=\mathcal{F}_{p,q}/\mathcal{F}_{p-1,q+1}\).

Let \(\mathcal{G}\) denote the preimage of the infinite cyclic group \(\langle t_x\rangle\) under the surjective homomorphism \(\varphi\colon\mathop{\mathrm{Mod}}\nolimits\twoheadrightarrow \mathrm{Sp}(H)=\mathrm{Sp}_{2g}(\mathbb{Z})\). Then we have a short exact sequence \[1\longrightarrow\mathcal{I}\longrightarrow\mathcal{G}\stackrel{\varphi}{\longrightarrow}\langle t_x\rangle\longrightarrow 1.\] Since \(t_x(x)=x\), the action of \(\mathcal{I}\) on \(\mathcal{C}_x(\Sigma_g)\) extends to a cellular action of \(\mathcal{G}\). Moreover, this action is without rotations, since for any simplex \(\sigma=\langle c_0,\ldots,c_p\rangle\) and any connected component \(S\) of \(\Sigma_g\setminus(c_0\cup\cdots\cup c_p)\), the transvection \(t_x\) preserves the subgroup \(H_1(S;\mathbb{Z})\subseteq H\). By Proposition 10 the group \(\langle t_x\rangle =\mathcal{G}/\mathcal{I}\) acts by automorphisms of the spectral sequence \(E^r_{p,q}\).

We now proceed with the proof of Theorem 6. Since the action of \(t_x\) on the \(E^{\infty}\) page is the associated graded of the action of \(t_x\) on filtration 7 , we obtain that \[(t_x-1)\mathcal{F}_{p,q}\subseteq\mathcal{F}_{p-1,q+1}\] for all \(p\) and \(q\). Therefore, \[(t_x-1)^{k+1}H_k^{\mathcal{I}}(\mathcal{C}_x(\Sigma_g);\mathbb{Z})=0\qquad \text{for all }k.\] Finally, since by the above mentioned result of Minahan the complex \(\mathcal{C}_x(\Sigma_g)\) is \((g-3)\)-acyclic, we see that the natural homomorphism \[H_k^{\mathcal{I}}\bigl(\mathcal{C}_x(\Sigma_g);\mathbb{Z}\bigr)\to H_k\bigl(\mathcal{I};\mathbb{Z}\bigr)\] is an isomorphism for \(k\le g-3\) and a surjection for \(k=g-2\). Consequently, \[(t_x-1)^{k+1}H_k\bigl(\mathcal{I};\mathbb{Z}\bigr)=0\qquad \text{for }k\le g-2.\] This completes the proof of Theorem 6.

4 Universal \(\mathcal{T}\)-unipotent modules and groups of finite width↩︎

Let \(\Gamma\) be a group and \(\mathcal{T}\subseteq\Gamma\) a conjugation-invariant subset. The following is a generalization of Definition 7.

We need the following construction of a universal \(\mathcal{T}\)-unipotent module of unipotency index \(n\). Fix a number \(n\) and consider the following two-sided ideal of the group ring \(\mathbb{Z}[\Gamma]\): \[J_n=\sum_{\tau\in\mathcal{T}}(\tau-1)^{n}\mathbb{Z}[\Gamma]=\sum_{\tau\in\mathcal{T}}\mathbb{Z}[\Gamma](\tau-1)^{n}.\] We set \[\label{eq95Un} \mathcal{U}_n=\mathbb{Z}[\Gamma]/J_n.\tag{8}\] Multiplication from the left by elements of \(\Gamma\) endows \(\mathcal{U}_n\) with the structure of a left \(\mathcal{T}\)-unipotent \(\Gamma\)-module. (Note that \(\mathcal{U}_n\) has also a structure of a right \(\Gamma\)-module but this structure will not be interesting to us.)

The module \(\mathcal{U}_n\) will be referred to as the universal \(\mathcal{T}\)-unipotent module of unipotency index \(n\). The image of the element \(1\in\mathbb{Z}[\Gamma]\) in \(\mathcal{U}_n\) will be denoted by \(u\) and called the fundamental generator of \(\mathcal{U}_n\). Of course, the ideal \(J_n\) and the module \(\mathcal{U}_n\) depend on the set \(\mathcal{T}\), but we prefer to suppress this dependence from the notation. The obvious universality property for \(\mathcal{U}_n\) is as follows.

Now, assume that \(\Gamma\) is a group of finite width, see Definition 8. We need the following easy proposition.

5 Bounded generation of \(\mathrm{Sp}_{2g}(\mathbb{Z})\) and proof of Theorem 1 for \(k=2\)↩︎

In this section we first explain how the following proposition can be extracted from the paper [36] by Tavgen, and then prove Theorem 1 for \(k=2\). For all definitions and facts concerning Chevalley groups, see [42].

A group \(\Gamma\) is said to be boundedly generated by a subset \(A\subseteq \Gamma\) if there exists a constant \(K\) such that every element \(\gamma\in \Gamma\) can be written as \(\gamma=a_1\cdots a_k\) with \(a_i\in A\) and \(k\le K\). Carter and Keller [43] proved that the group \(\mathrm{SL}_n(\mathbb{Z})\) is boundedly generated by elementary matrices, provided that \(n\ge 3\). Tavgen [36] then generalized this result to all Chevalley groups \(G(\Phi,\mathbb{Z})\), where \(\Phi\) is a reduced irreducible root system of rank \(\ge 2\). Namely, he proved that \(G(\Phi,\mathbb{Z})\) is boundedly generated by the set of all root unipotents (also called root elements) \(t_{\alpha}^{\xi}=\exp(\xi X_{\alpha})\), where \(\alpha\in\Phi\), \(\xi\in\mathbb{Z}\), and \(X_{\alpha}\) is the element corresponding to \(\alpha\) of the Chevalley basis for the Lie algebra \(\mathfrak{g}(\Phi)\). In particular, this result holds for all groups \(\mathrm{Sp}_{2g}(\mathbb{Z})=G(C_g,\mathbb{Z})\) with \(g\ge 2\). (A somewhat weaker result for \(\mathrm{Sp}_{2g}(\mathbb{Z})\) with \(g\ge 3\) and a larger set of generators was earlier obtained by Zakiryanov [44].)

Since \(\Phi\) is a finite set, Tavgen’s result immediately implies that \(\mathrm{Sp}_{2g}(\mathbb{Z})=\langle t_{\alpha_1}\rangle\cdots \langle t_{\alpha_m}\rangle\) for some finite sequence of (not necessarily distinct) root unipotents \(t_{\alpha_i}=\exp(X_{\alpha_i})\). Indeed, one can take for \(\alpha_1,\ldots,\alpha_m\) any finite sequence of roots that contain as subsequences all possible sequences of roots of length \(K\). For the following standard facts, see, for instance, [45]:

• If \(\alpha\) is a long root, then \(t_{\alpha}\) is a symplectic transvection \(t_x\) for some primitive \(x\in\mathbb{Z}^{2g}\).

• If \(\alpha\) is a short root, then \(t_{\alpha}\) is a so-called Eichler–Siegel–Dickson transvection, i. e., a transformation of the form \[t_{x,y}z=z+(x,z)y+(y,z)x,\] where \(x\) and \(y\) are primitive vectors in \(\mathbb{Z}^{2g}\) with \((x,y)=0\). But \[t_{x,y}=t_{x+y}t_x^{-1}t_y^{-1}\] and the symplectic transvections \(t_{x+y}\), \(t_x\), and \(t_y\) pairwise commute. Hence, \[\langle t_{\alpha}\rangle \subseteq \langle t_{x+y}\rangle\langle t_{x}\rangle\langle t_{y}\rangle.\]

Proposition 17 follows.

6 Finite generation of the coinvariants↩︎

The aim of the present section is to prove the following proposition. We fix \(g\), \(b\), and \(p\) with \(b+p\le 1\).

Recall that a group \(G\) is of type FL if \(\mathbb{Z}\) has a finite free resolution over \(\mathbb{Z}[G]\), and \(G\) is of type VFL if \(G\) has a finite-index subgroup of type FL. If \(G\) is of type VFL and \(A\) is a \(G\)-module that is finitely generated as an abelian group, then \(H_k(G;A)\) is finitely generated for every \(k\). Moreover, if \(G\) is a group of type VFL, then any finite-index subgroup of \(G\) is also of type VFL. These and other basic facts on groups of types FL and VFL can be found in [46].

We will need the following two classical results.

7 Quasiprojection \(q\)↩︎

This section contains a representation-theoretic construction, which is then used in Section 9 to prove Theorem 26. This construction will be needed for us in the case where \(\Gamma\) is an arithmetic group. However, the construction itself does not rely on arithmeticity, so in this section \(\Gamma\) is an arbitrary group.

For a group \(\Gamma\) and a commutative ring \(R\), we denote by \(\varepsilon\colon R[\Gamma]\to R\) the augmentation homomorphism. If \(V\) is an \(R[\Gamma]\)-module and \(q\in R[\Gamma]\), then, with some abuse of notation, we denote by \(q\) also the map \(V\to V\) given by \(v\mapsto qv\).

To prove Lemma 21 we need the following auxiliary lemma.

8 Preliminaries on arithmetic groups↩︎

As in the Introduction, let \(\mathbf{G}\) be a connected semisimple algebraic subgroup of \(\mathbf{GL}_n\) defined over \(\mathbb{Q}\) and let \(\Gamma\) be a Zariski dense arithmetic subgroup of \(\mathbf{G}(\mathbb{Q})\). We shall restrict ourselves to groups that have the following property.

Almost Algebraic Representations (AAR) Property: Every representation of \(\Gamma\) on a finite-dimensional \(\mathbb{Q}\)-vector space is almost algebraic.

The only example of interest to us will be the groups \(\mathrm{Sp}_{2g}(\mathbb{Z})\) with \(g\ge 2\). In this particular case, the AAR property was established by Bass, Milnor, and Serre [19], see also [47]. Note, however, that in fact this property is known for a wide class of arithmetic groups, including all arithmetic subgroups of simple algebraic groups of \(\mathbb{Q}\)-rank \(\ge2\) defined over \(\mathbb{Q}\) (see [17]; in this case the property follows from Margulis Superrigidity Theorem), as well as all groups possessing a certain form of the congruence subgroup property (see [48]).

We need the following well known consequences of the AAR property. In the next two corollaries, \(\Gamma\) is a Zariski dense arithmetic subgroup of a connected semisimple algebraic group \(\mathbf{G}(\mathbb{Q})\) defined over \(\mathbb{Q}\) such that that \(\Gamma\) possesses the AAR property.

Corollary 24 is deduced from the AAR property in [19]. Corollary 25 follows from a computation made by Serre [49] in the case of \(\mathbf{SL}_2\); for an explanation, why this works in the general case, see Proposition 5.1 and its proof in [48].

9 Proof of Theorem 1↩︎

In this section we formulate an algebraic theorem (Theorem 26) that will enable us to prove finite generation over \(\mathbb{Z}\) of a transvection unipotent \(\mathrm{Sp}_{2g}(\mathbb{Z})\)-module \(M\), relying not on the finite generation of \(M\) over \(\mathbb{Z}\bigl[\mathrm{Sp}_{2g}(\mathbb{Z})\bigr]\) but on a much weaker condition—the finite generation of suitable coinvariant groups \((M\otimes A)_{\Lambda}\), where \(\Lambda\le\mathrm{Sp}_{2g}(\mathbb{Z})\) is a subgroup of finite index. This theorem may be of further interest not only for the groups \(\mathrm{Sp}_{2g}(\mathbb{Z})\) but also for the groups \(\mathrm{SL}_n(\mathbb{Z})\), in view of possible applications to automorphisms of free groups. We therefore state it in some reasonable generality.

Before proving this theorem, we deduce Theorem 1 from it.

In the remainder of this section, we prove Theorem 26. To do this we first need to study some properties of the universal \(\mathcal{T}\)-unipotent modules from Section 4. We assume that \(\Gamma\) and \(\mathcal{T}\) are as in Theorem 26.

Fix a positive integer \(n\). Let \(\mathcal{U}_n\) be the universal \(\mathcal{T}\)-unipotent \(\Gamma\)-module of unipotency index \(n\) given by 8 . By Proposition 16 the module \(\mathcal{U}_n\) is a finitely generated abelian group. We denote by \(\mathop{\mathrm{tors}}\nolimits(\mathcal{U}_n)\) the torsion subgroup of \(\mathcal{U}_n\), and set \[\begin{gather} \mathcal{U}_n'=\mathcal{U}_n/\mathop{\mathrm{tors}}\nolimits(\mathcal{U}_n),\\ \mathcal{V}_n=\mathop{\mathrm{Hom}}\nolimits_{\mathbb{Z}}(\mathcal{U}'_n,\mathcal{U}_n). \end{gather}\] (As usually, if \(M\) and \(N\) are \(\Gamma\)-modules, then the structure of a \(\Gamma\)-module on \(\mathop{\mathrm{Hom}}\nolimits_{\mathbb{Z}}(M,N)\) is defined by \((\gamma\xi)(z)=\gamma\cdot\xi(\gamma^{-1}z)\), where \(\gamma\in\Gamma\), \(\xi\in\mathop{\mathrm{Hom}}\nolimits_{\mathbb{Z}}(M,N)\), and \(z\in M\).)

Since \(\mathcal{U}_n\) is a finitely generated abelian group, we have that \(\mathcal{V}_n\) is also a finitely generated abelian group. Moreover, by Corollary 24 the \(\mathbb{Q}[\Gamma]\)-module \(\mathcal{V}_n\otimes\mathbb{Q}\) is semisimple. Hence by Lemma 21 there exists a quasiprojection \(q_n\in \mathbb{Z}[\Gamma]\) such that \(\varepsilon(q_n)\ne 0\), \(q_n\mathcal{V}_n\subseteq\mathcal{V}_n^{\Gamma}\) and the map \(q_n\colon \mathcal{V}_n\to\mathcal{V}_n^{\Gamma}\) is \(\Gamma\)-invariant. We fix such a quasiprojection.

Let \(a_n\) be the largest order of an element of \(\mathop{\mathrm{tors}}\nolimits(\mathcal{U}_n)\). In other words, \(a_n\) is the smallest positive integer such that \(a_n\mathop{\mathrm{tors}}\nolimits(\mathcal{U}_n)=0\). Set \[b_n=|\varepsilon(q_n)|,\qquad c_n=a_nb_n.\]

10 Algebraicity of rational homology groups↩︎

11 Representation stability↩︎

In this section, we present some constructions and results from the paper by Kupers and Randal-Williams [23] (with corrections from [24]) and then prove Theorem 5. In [23], higher‑dimensional Torelli groups were studied, namely the Torelli groups of diffeomorphisms of the manifolds \(\#^gS^n\times S^n\); we present all constructions and results only in the case that interests us, that is, the ordinary Torelli groups, i. e., \(n=1\).

Set \(H(g)=H^1(\Sigma_g^1;\mathbb{Q})\), and let \((a,b)=\left\langle ab,\bigl[\Sigma_g^1,\partial \Sigma_g^1\bigr]\right\rangle\) be the standard skew-symmetric form on \(H(g)\). Throughout this section we use the fact that all rational representations of \(\mathrm{Sp}_{2g}(\mathbb{Q})\) are self-dual; in particular, the \(\mathrm{Sp}_{2g}(\mathbb{Q})\)-representations \(H_k(\mathcal{I}_g^1;\mathbb{Q})\) and \(H^k(\mathcal{I}_g^1;\mathbb{Q})\) are isomorphic to each other in the stable range \(k\le g-2\). Also we may not distinguish between irreducible algebraic representations of \(\mathrm{Sp}_{2g}(\mathbb{Z})\) and irreducible rational representations of \(\mathrm{Sp}_{2g}(\mathbb{Q})\), since \(\mathrm{Sp}_{2g}(\mathbb{Z})\) is Zariski dense in \(\mathrm{Sp}_{2g}(\mathbb{Q})\).

We present the construction of graded spaces \(\mathcal{P}(S)_{\ge0}\) and \(\mathcal{P}(S)_{\ge0}'\) of certain labelled partitions of a finite set \(S\) and describe their relation to the twisted Miller–Morita–Mumford classes following [23]. A large part of this construction goes back to the work of Kawazumi [50].

A labelled partition of a finite set \(S\) is a finite collection \(P=\{(P_{\alpha},m_{\alpha})\}_{\alpha\in I}\) such that

• \(P_{\alpha}\) are (possibly empty) subsets of \(S\) that are pairwise disjoint and whose union is \(S\),

• the labels \(m_{\alpha}\) are nonnegative integers.

We do not impose any ordering on the set of parts. Therefore, partitions that differ from each other by a reindexing of the parts are considered the same. However, partitions that differ by the addition of empty parts are considered different.

We assign to a finite set \(S\) the \(\mathbb{Q}\)-vector space \(\mathcal{P}(S)_{\ge0}\) with basis the set of labelled partitions \(P=\{(P_{\alpha},m_{\alpha})\}_{\alpha\in I}\) such that

1. each part of size \(0\) has label \(\ge 3\),

2. each part of size \(1\) has label \(\ge 1\).

Also, let \(\mathcal{P}(S)'_{\ge0}\) be the \(\mathbb{Q}\)-vector space defined in the same way but with an additional restriction:

1. each part of size \(2\) has label \(\ge 1\).

Introduce gradings on the spaces \(\mathcal{P}(S)_{\ge0}\) and \(\mathcal{P}(S)'_{\ge0}\) by setting the degree of each labelled partition to \(\sum_{\alpha\in I}(2m_{\alpha}+|P_{\alpha}|-2)\). In the case of \(\mathcal{P}(S)'_{\ge0}\), the summand \(2m_{\alpha}+|P_{\alpha}|-2\) corresponding to each part \(P_{\alpha}\) is positive. In the case of \(\mathcal{P}(S)_{\ge0}\), a partition may contain parts of size \(2\) and label \(0\) which provide zero impact to the degree.

There is a natural map (see [23]) \[\label{eq95PhiS} \Phi_{g,S}\colon \mathcal{P}(S)_{\ge0}\otimes\det\mathbb{Q}^S \to \left(H^*\bigl(\mathcal{I}_g^1;\mathbb{Q})\otimes H(g)^{\otimes S}\right)^{\mathrm{Sp}_{2g}(\mathbb{Z})}.\tag{10}\] This map can be written explicitly as follows. Let \(P=\{(P_{\alpha},m_{\alpha})\}_{\alpha\in I}\) be a labelled partition of \(S\) that satisfies (i) and (ii). To each part \(P_{\alpha}\) assign the element \[\begin{gather} \label{eq95phi95alpha} \varphi_{g,\alpha}=\sum_{j_1,\ldots,j_{|P_{\alpha}|}=1}^{2g}\kappa_{e^{m_{\alpha}}}\left(a_{j_1}\wedge\cdots\wedge a_{j_{|P_{\alpha}|}}\right)\otimes a_{j_1}^{\#}\otimes\cdots\otimes a_{j_{|P_{\alpha}|}}^{\#}\\{}\in H^{2m_{\alpha}+|P_{\alpha}|-2}\bigl(\mathcal{I}_g^1;\mathbb{Q}\bigr)\otimes H(g)^{\otimes|P_{\alpha}|}. \end{gather}\tag{11}\] Here \(a_1,\ldots,a_{2g}\) is a basis of \(H(g)\) and \(a_1^{\#},\ldots,a_{2g}^{\#}\) is the dual basis of \(H(g)\) characterized by \((a_i^{\#},a_j)=\delta_{ij}\). It is clear that the right-hand side of 11 is independent of the choice of the basis \(a_1,\ldots,a_{2g}\). We then multiply all of the elements \(\varphi_{g,\alpha}\): \[\label{eq95PhiS95form} \Phi_{g,S}(P)=\prod_{\alpha\in I}\varphi_{g,\alpha}\in \left(H^*\bigl(\mathcal{I}_g^1;\mathbb{Q})\otimes H(g)^{\otimes |S|}\right)^{\mathrm{Sp}_{2g}(\mathbb{Z})}.\tag{12}\] Note that \(\varphi_{g,\alpha}\) have even degrees; hence, the result is independent of the order in which we take the product. Finally, we would like to replace \(H(g)^{\otimes |S|}\) with \(H(g)^{\otimes S}\). This produces a sign depending on the ordering of the elements of \(S\) up to an even permutation. Thus, the map \(\Phi_{g,S}\) becomes well‑defined after adding the factor \(\det\mathbb{Q}^S\) as in 10 .

Now, set \[\Omega=\bigl[\kappa_{e^{2m}}\mid m>1\bigr]=\mathbb{Q}[e_3,e_5,e_7,\ldots].\] The space \(\mathcal{P}(S)_{\ge0}\) is a graded \(\Omega\)-module with each \(\kappa_{e^{2m}}\) acting by adding an empty part with label \(m\). (The absence of \(e_1=\kappa_{e^2}\) in \(\Omega\) corresponds to the fact that empty parts with label \(2\) are not allowed.) Since the odd Miller–Morita–Mumford classes \(e_{2m-1}=\kappa_{e^{2m}}\) vanish in the rational cohomology of Torelli groups, the homomorphism \(\Phi_{g,S}\) induces a homomorphism

\[\widehat{\Phi}_{g,S}\colon \mathbb{Q}\otimes_{\Omega}\mathcal{P}(S)_{\ge0}\otimes\det\mathbb{Q}^S \longrightarrow \left(H^*\bigl(\mathcal{I}_g^1;\mathbb{Q})\otimes H(g)^{\otimes S}\right)^{\mathrm{Sp}_{2g}(\mathbb{Z})}.\]

We need the following three propositions.

Let \(\mathfrak{S}_d\) denote the symmetric group, and let \(S^{\lambda}\) denote the irreducible \(\mathfrak{S}_d\)-representation corresponding to a partition \(\lambda\) of \(d\).

Propositions 31 and 32 were obtained by Kupers and Randal-Williams as conditional results under the assumption that the cohomology groups \(H^k(\mathcal{I}_g^1;\mathbb{Q})\) are finite-dimensional in the specified stable range. Moreover, Proposition 32 was only a result about the algebraic part of \(H^k(\mathcal{I}_g^1;\mathbb{Q})\), not about the whole \(H^k(\mathcal{I}_g^1;\mathbb{Q})\). In view of our Theorems 1 and 2, these propositions now become unconditional results about the whole representations \(H^k(\mathcal{I}_g^1;\mathbb{Q})\). As conditional results:

• Proposition 31 is contained in Section 4 of [23] together with the correction of the stable range in [24],

• Proposition 32 follows from Theorem 4.1 and Proposition 2.16 from [23] (see also the argument at the very beginning of Section 6), together with the corrections made in Theorem 4.1\('\) and Propositions 2.16\('\) from [24].

Let \(i_g\colon \Sigma_g^1\hookrightarrow\Sigma_{g+1}^1\) be the standard embedding. Then we have the pullback homomorphisms \(i^*_g\colon H(g+1)\to H(g)\) and \(i^*_g\colon H^1(\mathcal{I}_{g+1}^1;\mathbb{Q})\to H^1(\mathcal{I}_g^1;\mathbb{Q})\).

12 Proof of Morita’s conjecture↩︎

In this section we prove Theorem 4. Below the words ‘in the stable range’ always mean ‘in the range of degrees \(*\le \frac{2g-2}{3}\)’.

We start with the case of \(\mathcal{I}_g^1\). Taking \(S=\varnothing\) in Proposition 31, we obtain an isomorphism in the stable range: \[\widehat{\Phi}_{g,\varnothing}\colon \mathbb{Q}\otimes_{\Omega}\mathcal{P}(\varnothing)_{\ge0} \stackrel{\cong}{\longrightarrow} H^*\bigl(\mathcal{I}_g^1;\mathbb{Q})^{\mathrm{Sp}_{2g}(\mathbb{Z})}.\] The basis of \(\mathcal{P}(\varnothing)_{\ge0}\) consists of all unordered finite collections of labels \(\{m_{\alpha}\}\) with each \(m_{\alpha}\ge 3\). Tensoring by \(\mathbb{Q}\) over \(\Omega\) kills all (empty) parts with even labels. Hence the basis of \(\mathbb{Q}\otimes_{\Omega}\mathcal{P}(\varnothing)_{\ge0}\) consists of all collections \(\{m_{\alpha}\}\) with each \(m_{\alpha}=2n_{\alpha}+1\) odd and \(\ge 3\). Formula 12 then reads as \[\widehat{\Phi}_{g,\varnothing}(\{m_{\alpha}\})=\prod_{\alpha}e_{2n_{\alpha}}.\] In the stable range, since \(\widehat{\Phi}_{g,\varnothing}\) is an isomorphism, it follows that the ring \(H^*(\mathcal{I}_g^1;\mathbb{Q})^{\mathrm{Sp}_{2g}(\mathbb{Z})}\) is generated by the even Miller–Morita–Mumford classes, and moreover, all monomials in them are linearly independent, hence the required equality \[H^*\bigl(\mathcal{I}_g^1;\mathbb{Q}\bigr)^{\mathrm{Sp}_{2g}(\mathbb{Z})}=\mathbb{Q}[e_2,e_4,e_6,\ldots].\]

We now consider the case of \(\mathcal{I}_{g,1}\). With some abuse of notation, we denote by \(e_k\) the Miller–Morita–Mumford classes in the cohomology of each of the three groups \(\mathcal{I}_g^1\), \(\mathcal{I}_{g,1}\), and \(\mathcal{I}_g\).

We have a fibration sequence \[B\mathcal{I}_g^1\stackrel{f}{\longrightarrow} B\mathcal{I}_{g,1}\stackrel{h}{\longrightarrow}B\mathrm{SO}(2).\] The Leray–Hirsch property for this fibration sequence in the stable range on maximal algebraic subrepresetations was established in [23] under the condition that the rational cohomology groups are finite-dimensional in this stable rank. Since we now know from Theorems 1 and 2 that the rational cohomology are finite-dimensional and algebraic, we obtain the Leray–Hirsch property in the stable range in full generality. Consequently, we have an isomorphism of \(H^*(B\mathrm{SO}(2);\mathbb{Q})\)-modules \[H^*(\mathcal{I}_{g,1};\mathbb{Q})\cong H^*\bigl(\mathcal{I}_g^1;\mathbb{Q}\bigr)\otimes H^*(B\mathrm{SO}(2);\mathbb{Q}).\] Moreover, this isomorphism sends the class \(\epsilon\in H^2(\mathcal{I}_{g,1};\mathbb{Q})\) (defined in Remark 6) to \(1\otimes u\), where \(u\in H^2(B\mathrm{SO}(2);\mathbb{Q})\) is the standard generator. This holds because \(\epsilon\) restricts trivially to \(\mathcal{I}_g^1\) and \(h\) is the classifying map for the bundle \(\eta\) from Remark 6. It follows that \(\epsilon\) is a regular element of \(H^*(\mathcal{I}_{g,1};\mathbb{Q})\), i. e. not a zero-divisor in the stable range, and \[f^*\colon H^*(\mathcal{I}_{g,1};\mathbb{Q})\twoheadrightarrow H^*\bigl(\mathcal{I}_g^1;\mathbb{Q}\bigr)\] is the quotient by the ideal \((\epsilon)\). Since the classes \(\epsilon, e_2,e_4,\ldots\) in \(H^*(\mathcal{I}_{g,1};\mathbb{Q})\) are \(\mathrm{Sp}_{2g}(\mathbb{Z})\)-invariant, we obtain the required equality \[H^*(\mathcal{I}_{g,1};\mathbb{Q})=\mathbb{Q}[\epsilon,e_2,e_4,e_6,\ldots].\]

Finally, consider the case of \(\mathcal{I}_g\). We have a fibration sequence \[\Sigma_g\longrightarrow B\mathcal{I}_{g,1}\stackrel{\pi}{\longrightarrow} B\mathcal{I}_g.\] The pullback \[\pi^*\colon H^*(\mathcal{I}_g;\mathbb{Q})\to H^*(\mathcal{I}_{g,1};\mathbb{Q})\] and the Gysin homomorphism \[\pi_!\colon H^*(\mathcal{I}_{g,1};\mathbb{Q})\to H^{*-2}(\mathcal{I}_g;\mathbb{Q})\] are \(\mathrm{Sp}_{2g}(\mathbb{Z})\)-equivariant. Moreover, \(\pi_!(\epsilon)=\chi(\Sigma_g)=2-2g\) and hence \[\pi_!\bigl(\epsilon(\pi^*y)\bigr)=(2-2g)y\] for all \(y\in H^*(\mathcal{I}_g;\mathbb{Q})\). Since we are working in the stable range \(*\le \frac{2g-2}{3}\), we may assume that \(g>1\). Then \(\pi^*\) is injective and \(\pi_!\) is surjective. Since we know that the cohomology groups considered are finite-dimensional algebraic representations of \(\mathrm{Sp}_{2g}(\mathbb{Z})\) and hence are semisimple, we see that \[\begin{align} H^*(\mathcal{I}_g;\mathbb{Q})^{\mathrm{Sp}_{2g}(\mathbb{Z})}&=\pi_!\left(H^*(\mathcal{I}_{g,1};\mathbb{Q})^{\mathrm{Sp}_{2g}(\mathbb{Z})}\right)\\ {}&=\pi_!\left(\mathbb{Q}[\epsilon,e_2,e_4,e_6,\ldots]\right)\\ {}&=\pi_!\left(\mathbb{Q}[e_2,e_4,e_6,\ldots]\right)\qquad\quad (\text{since }\pi_!(\epsilon)=2-2g)\\&=\mathbb{Q}[e_2,e_4,e_6,\ldots]. \end{align}\] Moreover, the classes \(e_{2k}\) are algebraically independent in the stable range, since \(\pi^*\) is a monomorphism.

References↩︎