{"title":"On a spectral sequence for the action of the Torelli group of genus on the complex of cycles","authors":"A. A. Gaifullin","doi":"10.1070/im9116","DOIUrl":null,"url":null,"abstract":"The Torelli group of a closed oriented surface <inline-formula>\n<tex-math><?CDATA $S_g$?></tex-math>\n<inline-graphic xlink:href=\"IZV_85_6_1060ieqn2.gif\" xlink:type=\"simple\"></inline-graphic>\n</inline-formula> of genus <inline-formula>\n<tex-math><?CDATA $g$?></tex-math>\n<inline-graphic xlink:href=\"IZV_85_6_1060ieqn3.gif\" xlink:type=\"simple\"></inline-graphic>\n</inline-formula> is the subgroup <inline-formula>\n<tex-math><?CDATA $\\mathcal{I}_g$?></tex-math>\n<inline-graphic xlink:href=\"IZV_85_6_1060ieqn4.gif\" xlink:type=\"simple\"></inline-graphic>\n</inline-formula> of the mapping class group <inline-formula>\n<tex-math><?CDATA $\\operatorname{Mod}(S_g)$?></tex-math>\n<inline-graphic xlink:href=\"IZV_85_6_1060ieqn5.gif\" xlink:type=\"simple\"></inline-graphic>\n</inline-formula> consisting of all mapping classes that act trivially on the homology of <inline-formula>\n<tex-math><?CDATA $S_g$?></tex-math>\n<inline-graphic xlink:href=\"IZV_85_6_1060ieqn2.gif\" xlink:type=\"simple\"></inline-graphic>\n</inline-formula>. One of the most intriguing open problems concerning Torelli groups is the question of whether the group <inline-formula>\n<tex-math><?CDATA $\\mathcal{I}_3$?></tex-math>\n<inline-graphic xlink:href=\"IZV_85_6_1060ieqn6.gif\" xlink:type=\"simple\"></inline-graphic>\n</inline-formula> is finitely presented. A possible approach to this problem relies on the study of the second homology group of <inline-formula>\n<tex-math><?CDATA $\\mathcal{I}_3$?></tex-math>\n<inline-graphic xlink:href=\"IZV_85_6_1060ieqn6.gif\" xlink:type=\"simple\"></inline-graphic>\n</inline-formula> using the spectral sequence <inline-formula>\n<tex-math><?CDATA $E^r_{p,q}$?></tex-math>\n<inline-graphic xlink:href=\"IZV_85_6_1060ieqn7.gif\" xlink:type=\"simple\"></inline-graphic>\n</inline-formula> for the action of <inline-formula>\n<tex-math><?CDATA $\\mathcal{I}_3$?></tex-math>\n<inline-graphic xlink:href=\"IZV_85_6_1060ieqn6.gif\" xlink:type=\"simple\"></inline-graphic>\n</inline-formula> on the complex of cycles. In this paper we obtain evidence for the conjecture that <inline-formula>\n<tex-math><?CDATA $H_2(\\mathcal{I}_3;\\mathbb{Z})$?></tex-math>\n<inline-graphic xlink:href=\"IZV_85_6_1060ieqn8.gif\" xlink:type=\"simple\"></inline-graphic>\n</inline-formula> is not finitely generated and hence <inline-formula>\n<tex-math><?CDATA $\\mathcal{I}_3$?></tex-math>\n<inline-graphic xlink:href=\"IZV_85_6_1060ieqn6.gif\" xlink:type=\"simple\"></inline-graphic>\n</inline-formula> is not finitely presented. Namely, we prove that the term <inline-formula>\n<tex-math><?CDATA $E^3_{0,2}$?></tex-math>\n<inline-graphic xlink:href=\"IZV_85_6_1060ieqn9.gif\" xlink:type=\"simple\"></inline-graphic>\n</inline-formula> of the spectral sequence is not finitely generated, that is, the group <inline-formula>\n<tex-math><?CDATA $E^1_{0,2}$?></tex-math>\n<inline-graphic xlink:href=\"IZV_85_6_1060ieqn10.gif\" xlink:type=\"simple\"></inline-graphic>\n</inline-formula> remains infinitely generated after taking quotients by the images of the differentials <inline-formula>\n<tex-math><?CDATA $d^1$?></tex-math>\n<inline-graphic xlink:href=\"IZV_85_6_1060ieqn11.gif\" xlink:type=\"simple\"></inline-graphic>\n</inline-formula> and <inline-formula>\n<tex-math><?CDATA $d^2$?></tex-math>\n<inline-graphic xlink:href=\"IZV_85_6_1060ieqn12.gif\" xlink:type=\"simple\"></inline-graphic>\n</inline-formula>. Proving that it remains infinitely generated after taking the quotient by the image of <inline-formula>\n<tex-math><?CDATA $d^3$?></tex-math>\n<inline-graphic xlink:href=\"IZV_85_6_1060ieqn13.gif\" xlink:type=\"simple\"></inline-graphic>\n</inline-formula> would complete the proof that <inline-formula>\n<tex-math><?CDATA $\\mathcal{I}_3$?></tex-math>\n<inline-graphic xlink:href=\"IZV_85_6_1060ieqn6.gif\" xlink:type=\"simple\"></inline-graphic>\n</inline-formula> is not finitely presented.","PeriodicalId":54910,"journal":{"name":"Izvestiya Mathematics","volume":"196 ","pages":""},"PeriodicalIF":0.8000,"publicationDate":"2021-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"1","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Izvestiya Mathematics","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1070/im9116","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 1
Abstract
The Torelli group of a closed oriented surface of genus is the subgroup of the mapping class group consisting of all mapping classes that act trivially on the homology of . One of the most intriguing open problems concerning Torelli groups is the question of whether the group is finitely presented. A possible approach to this problem relies on the study of the second homology group of using the spectral sequence for the action of on the complex of cycles. In this paper we obtain evidence for the conjecture that is not finitely generated and hence is not finitely presented. Namely, we prove that the term of the spectral sequence is not finitely generated, that is, the group remains infinitely generated after taking quotients by the images of the differentials and . Proving that it remains infinitely generated after taking the quotient by the image of would complete the proof that is not finitely presented.
期刊介绍：
The Russian original is rigorously refereed in Russia and the translations are carefully scrutinised and edited by the London Mathematical Society. This publication covers all fields of mathematics, but special attention is given to:
Algebra;
Mathematical logic;
Number theory;
Mathematical analysis;
Geometry;
Topology;
Differential equations.
求助内容：
应助结果提醒方式：
京公网安备 11010802042870号
