On the degree-two part of the associated graded of the lower central series of the Torelli group
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866917271113302016 |
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| author | Faes, Quentin Massuyeau, Gwenael Sato, Masatoshi |
| author_facet | Faes, Quentin Massuyeau, Gwenael Sato, Masatoshi |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_07981 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the degree-two part of the associated graded of the lower central series of the Torelli group Faes, Quentin Massuyeau, Gwenael Sato, Masatoshi Geometric Topology Group Theory 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. |
| title | On the degree-two part of the associated graded of the lower central series of the Torelli group |
| topic | Geometric Topology Group Theory |
| url | https://arxiv.org/abs/2407.07981 |