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On the degree-two part of the associated graded of the lower central series of the Torelli group

Quentin Faes, Gwenael Massuyeau, Masatoshi Sato

TL;DR

This work determines the degree-two part of the associated graded of the Torelli group's lower central series, proving Γ_2I/Γ_3I is torsion-free and realized as a lattice inside Λ^2(Λ^3H^Q)/R_2^Q with R_2^Q = ker(B). It then provides an integral description of Γ_2I/Γ_3I and 𝒦/Γ_3I via τ_2 and d″, and proves an embedding of I/Γ_3I into the homology cylinders modulo Y_3-equivalence. The authors combine a diagrammatic, algebraic bracket B with a geometric, IHX-based vanishing analysis to obtain a finite generating system for the relevant kernel K and to prove J vanishes on it, yielding a quadratic presentation at degree two. Furthermore, they relate these Torelli-level structures to the monoid of homology cylinders, extending Johnson-type invariants and Casson-type data to the IC setting and establishing injectivity results that bridge classical mapping class group theory with 3-manifold surgery frameworks.

Abstract

We consider the associated graded $\bigoplus_{k\geq 1} Γ_k \mathcal{I} / Γ_{k+1} \mathcal{I} $ of the lower central series $\mathcal{I} = Γ_1 \mathcal{I} \supset Γ_2 \mathcal{I} \supset Γ_3 \mathcal{I} \supset \cdots$ of the Torelli group $\mathcal{I}$ of a compact oriented surface. Its degree-one part is well-understood by D. Johnson's seminal works on the abelianization of the Torelli group. The knowledge of the degree-two part $(Γ_2 \mathcal{I} / Γ_3 \mathcal{I})\otimes \mathbb{Q}$ with rational coefficients arises from works of S. Morita on the Casson invariant and R. Hain on the Malcev completion of $\mathcal{I}$. Here, we prove that the abelian group $Γ_2 \mathcal{I} / Γ_3 \mathcal{I}$ is torsion-free, and we describe it as a lattice in a rational vector space. As an application, the group $\mathcal{I}/Γ_3 \mathcal{I}$ is computed, and it is shown to embed in the group of homology cylinders modulo the surgery relation of $Y_3$-equivalence.

On the degree-two part of the associated graded of the lower central series of the Torelli group

TL;DR

This work determines the degree-two part of the associated graded of the Torelli group's lower central series, proving Γ_2I/Γ_3I is torsion-free and realized as a lattice inside Λ^2(Λ^3H^Q)/R_2^Q with R_2^Q = ker(B). It then provides an integral description of Γ_2I/Γ_3I and 𝒦/Γ_3I via τ_2 and d″, and proves an embedding of I/Γ_3I into the homology cylinders modulo Y_3-equivalence. The authors combine a diagrammatic, algebraic bracket B with a geometric, IHX-based vanishing analysis to obtain a finite generating system for the relevant kernel K and to prove J vanishes on it, yielding a quadratic presentation at degree two. Furthermore, they relate these Torelli-level structures to the monoid of homology cylinders, extending Johnson-type invariants and Casson-type data to the IC setting and establishing injectivity results that bridge classical mapping class group theory with 3-manifold surgery frameworks.

Abstract

We consider the associated graded of the lower central series of the Torelli group of a compact oriented surface. Its degree-one part is well-understood by D. Johnson's seminal works on the abelianization of the Torelli group. The knowledge of the degree-two part with rational coefficients arises from works of S. Morita on the Casson invariant and R. Hain on the Malcev completion of . Here, we prove that the abelian group is torsion-free, and we describe it as a lattice in a rational vector space. As an application, the group is computed, and it is shown to embed in the group of homology cylinders modulo the surgery relation of -equivalence.
Paper Structure (22 sections, 29 theorems, 138 equations, 3 figures, 1 table)

This paper contains 22 sections, 29 theorems, 138 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Proof of Theorem A
  3. The abelianization of the Torelli group
  4. Nullity of the map $\upsilon$
  5. The "Push" map and its applications
  6. Proof of Theorem B
  7. The map $B$
  8. Strategy for the proof of Theorem B
  9. A finite generating system of $K$
  10. A $G$-generating system of $K \cap U_0$.
  11. A $G$-generating system of $K \cap U_1$.
  12. A $G$-generating system of $K \cap U_2$.
  13. A $G$-generating system of $K \cap U_3$.
  14. Vanishing of $J$ on $K$
  15. BP-maps and PB-maps
  16. ...and 7 more sections

Key Result

Lemma 2.2

As an $\mathcal{S}_p$-module, $B_{\leq 2}(\mathcal{Q})$ is generated by the following elements:

Figures (3)

  • Figure 1: A system of "meridians & parallels" in $\Sigma$
  • Figure 2: The cobounding pair $(\gamma_+,\gamma_-)$, and the bounding simple closed curves $c_1,c_2$
  • Figure 3: The curves $X,Y,Z$ and $U,V,W$ on $\partial(D\times [0,1])$

Theorems & Definitions (60)

  • Remark 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Proposition 2.4
  • proof
  • Example 2.5
  • Example 2.6
  • Remark 3.1
  • ...and 50 more