{"title":"Torsion elements in the associated graded of the \u0000 \u0000 Y\u0000 $Y$\u0000 -filtration of the monoid of homology cylinders","authors":"Yuta Nozaki, Masatoshi Sato, Masaaki Suzuki","doi":"10.1112/topo.70028","DOIUrl":"https://doi.org/10.1112/topo.70028","url":null,"abstract":"<p>Clasper surgery induces the <span></span><math>\u0000 <semantics>\u0000 <mi>Y</mi>\u0000 <annotation>$Y$</annotation>\u0000 </semantics></math>-filtration <span></span><math>\u0000 <semantics>\u0000 <msub>\u0000 <mrow>\u0000 <mo>{</mo>\u0000 <msub>\u0000 <mi>Y</mi>\u0000 <mi>n</mi>\u0000 </msub>\u0000 <mi>IC</mi>\u0000 <mo>}</mo>\u0000 </mrow>\u0000 <mi>n</mi>\u0000 </msub>\u0000 <annotation>$lbrace Y_nmathcal {IC}rbrace _n$</annotation>\u0000 </semantics></math> 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 <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <msub>\u0000 <mi>Y</mi>\u0000 <mi>n</mi>\u0000 </msub>\u0000 <mi>IC</mi>\u0000 <mo>/</mo>\u0000 <msub>\u0000 <mi>Y</mi>\u0000 <mrow>\u0000 <mi>n</mi>\u0000 <mo>+</mo>\u0000 <mn>1</mn>\u0000 </mrow>\u0000 </msub>\u0000 </mrow>\u0000 <annotation>$Y_nmathcal {IC}/Y_{n+1}$</annotation>\u0000 </semantics></math> induced by the degree <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>n</mi>\u0000 <mo>+</mo>\u0000 <mn>2</mn>\u0000 </mrow>\u0000 <annotation>$n+2$</annotation>\u0000 </semantics></math> 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 <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <msub>\u0000 <mi>Y</mi>\u0000 <mn>6</mn>\u0000 </msub>\u0000 <mi>IC</mi>\u0000 <mo>/</mo>\u0000 <msub>\u0000 <mi>Y</mi>\u0000 <mn>7</mn>\u0000 </msub>\u0000 </mrow>\u0000 <annotation>$Y_6mathcal {IC}/Y_7$</annotation>\u0000 </semantics></math> has order 3.</p>","PeriodicalId":56114,"journal":{"name":"Journal of Topology","volume":"18 3","pages":""},"PeriodicalIF":0.8,"publicationDate":"2025-06-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144520190","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}