Torsion elements in the associated graded of the Y $Y$ -filtration of the monoid of homology cylinders

IF 0.8 2区数学 Q2 MATHEMATICS

Journal of Topology Pub Date : 2025-06-30 DOI:10.1112/topo.70028

Yuta Nozaki, Masatoshi Sato, Masaaki Suzuki

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引用次数: 0

Abstract

Clasper surgery induces the $Y$ -filtration ${Y_{n} IC}_{n}$ over the monoid of homology cylinders, which serves as a 3-dimensional analogue of the lower central series of the Torelli group of a surface. In this paper, we investigate the torsion submodules of the associated graded modules of these filtrations. To detect torsion elements, we introduce a homomorphism on $Y_{n} IC / Y_{n + 1}$ induced by the degree $n + 2$ part of the LMO functor. Additionally, we provide a formula that computes this homomorphism under clasper surgery, and use it to demonstrate that every nontrivial torsion element in $Y_{6} IC / Y_{7}$ has order 3.

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同调圆柱体单线的Y$ Y$滤过的相关梯度中的扭转元

Clasper手术诱导Y$ Y$滤除{Y n IC} n$ \rbrace Y_n\mathcal {IC}\rbrace _n$，它是曲面托雷利群的下中心级数的三维模拟。在本文中，我们研究了这些滤波的相关梯度模的扭转子模。为了检测扭转元素，我们引入了Y n IC /Y n+1 $Y_n\数学{IC}/Y_{n+1}$上的一个由n阶诱导的同态LMO函子的+2$ n+2$部分。此外，我们还提供了一个在clasper手术下计算这个同态的公式，并用它证明了y6 IC / y7 $Y_6\mathcal {IC}/Y_7$中的每一个非平凡扭转元素都是3阶的。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Journal of Topology 数学-数学

CiteScore

2.00

自引率

9.10%

发文量

审稿时长

>12 weeks

期刊介绍： The Journal of Topology publishes papers of high quality and significance in topology, geometry and adjacent areas of mathematics. Interesting, important and often unexpected links connect topology and geometry with many other parts of mathematics, and the editors welcome submissions on exciting new advances concerning such links, as well as those in the core subject areas of the journal. The Journal of Topology was founded in 2008. It is published quarterly with articles published individually online prior to appearing in a printed issue.