The second rational homology of the Torelli group is finitely generated

TL;DR

The paper proves that the second rational homology $H_2(\mathcal{I}_g;\mathbb{Q})$ of the Torelli group is finite dimensional for genus $g\ge 51$, providing a partial obstruction to finite presentability. It develops a robust framework combining the Johnson homomorphism, the complex of homologous curves $\mathcal{C}_{[a]}(S_g)$, and Bestvina--Margalit tori to control $H_2$-classes via stable and unstable generators. A central technical tool, Proposition \text{gengrpprop}, translates fixed-point and coinvariant data for multicurve stabilizers into a finiteness conclusion for the ambient representation; the authors then verify these hypotheses through a detailed analysis of BM tori, abelian cycles, and the equivariant spectral sequence. The results disassemble the potential infinite contributions to $H_2(\mathcal{I}_g;\mathbb{Q})$ into finitely generated pieces supported on subsurfaces and controlled by symplectic group actions, ultimately yielding the finiteness result. This advances our understanding of finite presentability questions for the Torelli group in high genus and contributes to the broader program of mapping class group homology.

Abstract

We prove that second rational homology of the Torelli group of an orientable closed surface of genus g is finite dimensional for g at least 51. This rules out the simplest obstruction to the Torelli group being finitely presented and provides a partial answer to a question of Bestvina.

Figures (11)

• Figure 1: Page 1 of $\mathbb E_{*,*}^*$ for the action of $\mathcal{I}_g$ on $\mathcal{C}_{[a]}(S_g)$. All coefficients are in $\mathbb Q$.
• Figure 2: The construction of dual cells.
• Figure 3: The bounding pair map $T_{c}T_{c'}^{-1}$ and edge $e = e_{v,w}$.
• Figure 4: The curves $a_0,a_1,a_2$, $b$, and $c_i$
• Figure 5: A 3--cell containing lifts of the edges $x,y,z,yz$ as in Lemma \ref{['lemmaaltcohomolauxadd']}. $x$ lifts to $(a,a_1)$, $y$ lifts to $(a_1,a_2)$, $z$ lifts to $(a_2,a_3)$ and $yz$ lifts to $(a_1,a_3)$.

Theorems & Definitions (109)

• Theorem A
• Theorem B
• Proposition 1.1
• Proposition 1.2
• Proposition 2.1
• Lemma 2.2
• proof
• proof : Proof of Proposition \ref{['gengrpprop']}
• Lemma 3.1
• proof