论文标题
关于阿贝里亚化约翰逊内核的扭转亚组的非平凡性
On the non-triviality of the torsion subgroup of the abelianized Johnson kernel
论文作者
论文摘要
Johnson内核是由Dehn Twists生成的封闭式表面的映射类组的子组，沿着简单的封闭曲线。约翰逊内核的理性阿巴利亚化是由Dimca，Hain和Papadima计算得出的，随后由Morita，Sakasai和Suzuki提供了更明确的形式。基于这些结果，Nozaki，Sato和Suzuki使用了3个manifolds的有限型不变性理论来证明Abelianized Johnson kernel的扭力亚组是非平凡的。 在本文中，我们给出了该扭转亚组的非平凡性的纯粹二维证明，并为其基数提供了下限。我们的主要工具是映射类组对表面基本组的Malcev Lie代数的作用。使用相同的无穷小技术，我们还提供了理性的Abelianized Johnson内核的替代图形描述，并且在结果中包括一个带有一个边界成分的方向表面。
The Johnson kernel is the subgroup of the mapping class group of a closed oriented surface that is generated by Dehn twists along separating simple closed curves. The rational abelianization of the Johnson kernel has been computed by Dimca, Hain and Papadima, and a more explicit form was subsequently provided by Morita, Sakasai and Suzuki. Based on these results, Nozaki, Sato and Suzuki used the theory of finite-type invariants of 3-manifolds to prove that the torsion subgroup of the abelianized Johnson kernel is non-trivial. In this paper, we give a purely 2-dimensional proof of the non-triviality of this torsion subgroup and provide a lower bound for its cardinality. Our main tool is the action of the mapping class group on the Malcev Lie algebra of the fundamental group of the surface. Using the same infinitesimal techniques, we also provide an alternative diagrammatic description of the rational abelianized Johnson kernel, and we include in the results the case of an oriented surface with one boundary component.
