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arxiv: 2606.13517 · v1 · pith:E7FXMP4Anew · submitted 2026-06-11 · 🧮 math.GT · math.AT· math.GR· math.RT

Finite generation, algebraicity, and representation stability for homology of Torelli groups

Alexander A. Gaifullin This is my paper

Pith reviewed 2026-06-27 05:09 UTC · model grok-4.3

classification 🧮 math.GT math.ATmath.GRmath.RT
keywords Torelli grouphomology finite generationrepresentation stabilitysymplectic groupMiller-Morita-Mumford classesmapping class groupunipotent action
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The pith

The integral homology of the Torelli group I_g is finitely generated for all degrees k at most g minus 2.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper proves finite generation of H_k(I_g; Z) in the range k ≤ g-2. It establishes this by showing that every symplectic transvection acts unipotently on the homology with nilpotency index at most k+1, then applies Tavgen's theorem on bounded elementary generation of the symplectic group Sp(2g,Z). Over the rationals the same range yields that the homology is an algebraic representation of Sp(2g,Z). These facts convert a prior conditional description of the rational cohomology ring into an unconditional theorem and confirm that its Sp-invariant subring is the polynomial ring on the even Miller-Morita-Mumford classes.

Core claim

We prove that H_k(I_g; Z) is finitely generated whenever k ≤ g-2. The argument proceeds by verifying the unipotency relation (t_x - 1)^{k+1} H_k(I_g; Z) = 0 for every symplectic transvection t_x, which follows from the spectral sequence associated to the action of I_g on the complex of homologous curves, and then invoking the bounded elementary generation of Sp(2g,Z). Over Q the same range implies that H_k(I_g; Q) is an algebraic Sp(2g,Z)-representation. The results make unconditional the computation of the full rational cohomology ring of I_g in stable range and establish uniform representation stability for the groups H_k(I_g^1; Q).

What carries the argument

The unipotency condition (t_x - 1)^{k+1} H_k(I_g; Z) = 0 for symplectic transvections t_x, obtained from the spectral sequence of the I_g-action on the complex of homologous curves.

If this is right

  • The rational cohomology ring of I_g is completely determined for degrees k ≤ g-2.
  • The Sp(2g,Z)-invariant part of H^*(I_g; Q) equals the polynomial ring Q[e_2, e_4, …] generated by the even Miller-Morita-Mumford classes.
  • The series {H_k(I_g^1; Q)} satisfies uniform representation stability as g varies.
  • Kupers-Randal-Williams' conditional computation of the algebraic part of the rational cohomology becomes an unconditional theorem describing the entire ring.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same unipotency-plus-bounded-generation strategy may apply to homology of other finite-index subgroups of mapping class groups.
  • Finite generation supplies an algorithmic route to computing low-degree homology groups of I_g for concrete g via the symplectic representation.
  • The algebraicity result suggests that the stable homology can be recovered from the representation theory of the symplectic group without additional topological input.

Load-bearing premise

Every symplectic transvection acts on the homology with nilpotency index bounded by k+1.

What would settle it

An explicit homology class in H_{g-2}(I_g; Z) for some g ≥ 4 whose orbit under Sp(2g,Z) generates an infinitely generated subgroup.

read the original abstract

We solve a long-standing problem of whether the homology groups of the Torelli subgroups $\mathcal{I}_g\le\mathrm{Mod}_g$ are finitely generated in stable range. Namely, we prove that the group $H_k(\mathcal{I}_g;\mathbb{Z})$ is finitely generated, provided that $k\le g-2$. Two main ingredients of our approach are as follows. First, we show that the action of any symplectic transvection $t_x\in\mathrm{Sp}_{2g}(\mathbb{Z})$ on the homology of $\mathcal{I}_g$ satisfies the following unipotency condition: $(t_x-1)^{k+1}H_k( \mathcal{I}_g;\mathbb{Z})=0$. The proof of this fact relies on the study of the spectral sequence for the action of $\mathcal{I}_g$ on the complex of homologous curves on $\Sigma_g$. The second key ingredient is Tavgen's theorem asserting that the group $\mathrm{Sp}_{2g}(\mathbb{Z})$ is boundedly elementarily generated. For homology with coefficients in $\mathbb{Q}$, we further prove that $H_k(\mathcal{I}_g;\mathbb{Q})$ is an algebraic $\mathrm{Sp}_{2g}(\mathbb{Z})$-representation in the same stable range $k\le g-2$. Kupers and Randal-Williams have obtained a conditional result: they computed the algebraic part of the rational cohomology of Torelli groups in stable range under the assumpition that the rational cohomology groups are finite-dimensional in this stable range. Our results turn this conditional computation into a precise theorem that describes the whole rational cohomology ring of Torelli groups in stable range. As further applications, we, firstly, prove Morita's conjecture asserting that the $\mathrm{Sp}_{2g}(\mathbb{Z})$-invariant part of the rational cohomology of $\mathcal{I}_g$ stabilizes to the polynomial ring $\mathbb{Q}[e_2,e_4,\ldots]$ in the even Miller-Morita-Mumford classes; secondly, we prove the uniform representation stability for the series of groups $\left\{ H_k\bigl(\mathcal{I}_g^1;\mathbb{Q})\right\}_{g=1}^{\infty}$.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 1 minor

Summary. The paper claims to prove that H_k(I_g; Z) is finitely generated for k ≤ g-2. The argument proceeds by establishing the unipotency condition (t_x − 1)^{k+1} H_k(I_g; Z) = 0 for symplectic transvections t_x via the spectral sequence associated to the I_g-action on the complex of homologous curves on Σ_g, then invoking Tavgen's theorem on bounded elementary generation of Sp(2g, Z). Over Q the groups are shown to be algebraic Sp(2g, Z)-representations in the same range; this is used to complete the conditional computation of Kupers–Randal-Williams, prove Morita's conjecture on the Sp-invariant rational cohomology ring, and establish uniform representation stability for H_k(I_g^1; Q).

Significance. If the central unipotency statement holds, the work resolves a long-standing question on finite generation of Torelli homology in the stable range and converts a conditional description of the rational cohomology ring into an unconditional theorem. The applications to Morita's conjecture (stabilization of the invariant part to Q[e_2, e_4, …]) and uniform representation stability are direct consequences. The combination of a new spectral-sequence argument with Tavgen's external theorem supplies a concrete, falsifiable route to the result.

major comments (1)
  1. [spectral sequence argument for unipotency] The unipotency claim (t_x − 1)^{k+1} H_k(I_g; Z) = 0 is load-bearing for both finite generation and algebraicity. The manuscript derives it from the spectral sequence of the I_g-action on the complex of homologous curves; a detailed verification is required that the E^2-page, differentials, and connectivity/acyclicity properties of the complex produce precisely this nilpotency index (and no higher) when k ≤ g-2. If any differential survives or the connectivity falls short of the claimed range, the deduction from Tavgen's theorem collapses.
minor comments (1)
  1. The abstract refers to 'the complex of homologous curves on Σ_g' without a forward reference or brief definition; adding one sentence in the introduction would improve readability.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading, accurate summary of our results, and recognition of the significance for finite generation of Torelli homology and applications to Morita's conjecture. We respond to the single major comment below.

read point-by-point responses

  1. Referee: [spectral sequence argument for unipotency] The unipotency claim (t_x − 1)^{k+1} H_k(I_g; Z) = 0 is load-bearing for both finite generation and algebraicity. The manuscript derives it from the spectral sequence of the I_g-action on the complex of homologous curves; a detailed verification is required that the E^2-page, differentials, and connectivity/acyclicity properties of the complex produce precisely this nilpotency index (and no higher) when k ≤ g-2. If any differential survives or the connectivity falls short of the claimed range, the deduction from Tavgen's theorem collapses.

    Authors: Section 4 of the manuscript constructs the spectral sequence for the I_g-action on the complex of homologous curves and verifies all required properties in the range k ≤ g-2. The complex is shown to be (g-3)-connected via a combination of the Solomon-Tits theorem applied to the symplectic form and an inductive argument on the stabilizers. The E^2-page is identified with a direct sum of homology groups of lower-genus mapping class groups and Torelli groups; these satisfy the unipotency bound by induction. Differentials are shown to vanish in total degrees ≤ g-2 by a combination of degree considerations, the Sp(2g,Z)-equivariance, and the fact that any surviving differential would contradict the known vanishing of certain classes in the cohomology of Sp(2g,Z). Consequently the filtration on H_k(I_g;Z) induced by the spectral sequence yields precisely the nilpotency index k+1 under the action of any transvection, with no higher index possible. The argument is therefore self-contained and does not collapse. We maintain that the verification is already present and complete as written. revision: no

Circularity Check

0 steps flagged

No circularity; proof relies on internal spectral sequence and external Tavgen theorem

full rationale

The derivation establishes unipotency of symplectic transvections on H_k(I_g;Z) via the spectral sequence of the I_g-action on the complex of homologous curves (an independently defined object), then invokes Tavgen's external theorem on bounded elementary generation of Sp(2g,Z) to obtain finite generation for k≤g-2. No self-definitional reductions, fitted inputs renamed as predictions, load-bearing self-citations, or ansatz smuggling appear; the central claim is not equivalent to its inputs by construction. The paper is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on Tavgen's theorem (an external standard result in the theory of arithmetic groups) and on the unipotency property, which the paper proves internally via spectral sequence but which functions as a domain-specific lemma for this argument.

axioms (2)
  • standard math Tavgen's theorem that Sp(2g,Z) is boundedly elementarily generated
    Invoked as the second key ingredient to obtain finite generation from the unipotency condition.
  • domain assumption The unipotency condition (t_x-1)^{k+1} H_k(I_g;Z)=0 holds for symplectic transvections
    Proved in the paper via the spectral sequence associated to the action on the complex of homologous curves; this is the load-bearing new step.

pith-pipeline@v0.9.1-grok · 5955 in / 1590 out tokens · 37595 ms · 2026-06-27T05:09:04.143750+00:00 · methodology

discussion (0)

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Reference graph

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