[論文レビュー] The abelianization of the Johnson kernel

Alexandru Dimca, Richard Hain+1 more|arXiv (Cornell University)|Jan 7, 2011

Homotopy and Cohomology in Algebraic Topology参考文献 13被引用数 5

ひとこと要約

本稿は、$g \geq 4$ のとき、ジョンソン核 $K_g$ の最初の複素ホモロジー群 $H_1(K_g,\mathbb{C})$ が、非自明でユニポテンツな $H_1(T_g,\mathbb{C})$-加群であることを証明し、$g \geq 6$ のときには、$\operatorname{Sym}_\bullet(H_1(T_g,\mathbb{C}))$-加群としての明示的表示も与える。主な結果は、$H_1(K_g,\mathbb{C})$ が、曲面のホモロジーの対称冪と外冪を含む自然な写像の余核と同一視され、これにより、写像類群の関連する階化リー代数の無限小アレクサンダー不変量と結びつく。

ABSTRACT

We prove that the first complex homology of the Johnson subgroup of the Torelli group $T_g$ is a non-trivial unipotent $T_g$-module for all $g\ge 4$ and give an explicit presentation of it as a $\Sym H_1(T_g,\C)$-module when $g\ge 6$. We do this by proving that, for a finitely generated group $G$ satisfying an assumption close to formality, the triviality of the restricted characteristic variety implies that the first homology of its Johnson kernel $K$ is a nilpotent module over the corresponding Laurent polynomial ring, isomorphic to the infinitesimal Alexander invariant of the associated graded Lie algebra of $G$. In this setup, we also obtain a precise nilpotence test.

研究の動機と目的

• For $g \geq 4$, to determine the structure of $H_1(K_g,\mathbb{C})$ as a module over the Torelli group $T_g$.
• To establish that $H_1(K_g,\mathbb{C})$ is a non-trivial, unipotent $H_1(T_g,\mathbb{C})$-module.
• To give an explicit presentation of $H_1(K_g,\mathbb{C})$ as a $\Gamma_g/K_g$-module when $g \geq 6$, using symmetric algebra actions.
• To relate the homology of the Johnson kernel to the infinitesimal Alexander invariant of the associated graded Lie algebra of the mapping class group.

提案手法

• Use the theory of relative completion and Malcev completion to analyze the Lie algebra of the Torelli group.
• Define a canonical $\mathrm{Sp}(H_\mathbb{C})$-equivariant map $q: \operatorname{Sym}_\bullet(V_\mathbb{C}) \otimes \wedge^3 V_\mathbb{C} \to \operatorname{Sym}_\bullet(V_\mathbb{C}) \otimes Q$, where $Q$ is the irreducible $\mathrm{Sp}(H_\mathbb{C})$-module of highest weight $2\lambda_2$, and study its cokernel.
• Show that the cokernel of $q$ is finite-dimensional when $g \geq 6$, hence a nilpotent module over the symmetric algebra.
• Establish a canonical isomorphism between $H_1(K_g,\mathbb{C})$ and the cokernel of $q$, using the action of $\Gamma_g/K_g$ via the Zariski-dense embedding into $\mathrm{Sp}(H_\mathbb{C}) \ltimes V_\mathbb{C}$.
• Apply the theory of characteristic varieties and the restricted characteristic variety condition to deduce nilpotence and finiteness properties of the homology module.
• Leverage mixed Hodge structures and compatible splittings of the lower central series to lift the action from $\Gamma_g/K_g$ to the Lie algebra level and identify the module structure.

実験結果

リサーチクエスチョン

• RQ1 Is $H_1(K_g,\mathbb{C})$ a non-trivial, unipotent $H_1(T_g,\mathbb{C})$-module for $g \geq 4$?
• RQ2 Can $H_1(K_g,\mathbb{C})$ be explicitly presented as a module over $\operatorname{Sym}_\bullet(H_1(T_g,\mathbb{C}))$ when $g \geq 6$?
• RQ3 What is the relationship between the homology of the Johnson kernel and the infinitesimal Alexander invariant of the associated graded Lie algebra of the Torelli group?
• RQ4 How does the restricted characteristic variety condition imply nilpotence of $H_1(K_g,\mathbb{C})$ as a module over the symmetric algebra?
• RQ5 What role do mixed Hodge structures and compatible splittings play in realizing the $\Gamma_g/K_g$-action on $H_1(K_g,\mathbb{C})$?

主な発見

• For $g \geq 4$, the first complex homology $H_1(K_g,\mathbb{C})$ is a non-trivial, unipotent $H_1(T_g,\mathbb{C})$-module.
• For $g \geq 6$, $H_1(K_g,\mathbb{C})$ is isomorphic to the cokernel of a canonical map $q$, which is a finite-dimensional $\mathrm{Sp}(H_\mathbb{C}) \ltimes V_\mathbb{C}$-module.
• The cokernel of $q$ is a graded $\operatorname{Sym}_\bullet(V_\mathbb{C})$-module and carries a natural action of $\mathrm{Sp}(H_\mathbb{C}) \ltimes V_\mathbb{C}$, making it a nilpotent module.
• The module $H_1(K_g,\mathbb{C})$ is isomorphic to the infinitesimal Alexander invariant of the associated graded Lie algebra of the Torelli group.
• The action of $\Gamma_g/K_g$ on $H_1(K_g,\mathbb{C})$ factors through a Zariski-dense embedding into $\mathrm{Sp}(H_\mathbb{C}) \ltimes V_\mathbb{C}$, and the module structure is compatible with this action.
• The restricted characteristic variety condition implies that $H_1(K_g,\mathbb{C})$ is a nilpotent module over $\operatorname{Sym}_\bullet(H_1(T_g,\mathbb{C}))$, providing a precise nilpotence test.