The top homology group of the genus 3 Torelli group

IF 0.8 2区数学 Q2 MATHEMATICS Journal of Topology Pub Date : 2023-08-26 DOI:10.1112/topo.12308

Igor A. Spiridonov

{"title":"The top homology group of the genus 3 Torelli group","authors":"Igor A. Spiridonov","doi":"10.1112/topo.12308","DOIUrl":null,"url":null,"abstract":"<p>The Torelli group of a genus <math>\n <semantics>\n <mi>g</mi>\n <annotation>$g$</annotation>\n </semantics></math> oriented surface <math>\n <semantics>\n <msub>\n <mi>Σ</mi>\n <mi>g</mi>\n </msub>\n <annotation>$\\Sigma _g$</annotation>\n </semantics></math> is the subgroup <math>\n <semantics>\n <msub>\n <mi>I</mi>\n <mi>g</mi>\n </msub>\n <annotation>$\\mathcal {I}_g$</annotation>\n </semantics></math> of the mapping class group <math>\n <semantics>\n <mrow>\n <mi>Mod</mi>\n <mo>(</mo>\n <msub>\n <mi>Σ</mi>\n <mi>g</mi>\n </msub>\n <mo>)</mo>\n </mrow>\n <annotation>${\\rm Mod}(\\Sigma _g)$</annotation>\n </semantics></math> consisting of all mapping classes that act trivially on <math>\n <semantics>\n <mrow>\n <msub>\n <mi>H</mi>\n <mn>1</mn>\n </msub>\n <mrow>\n <mo>(</mo>\n <msub>\n <mi>Σ</mi>\n <mi>g</mi>\n </msub>\n <mo>,</mo>\n <mi>Z</mi>\n <mo>)</mo>\n </mrow>\n </mrow>\n <annotation>${\\rm H}_1(\\Sigma _g, \\mathbb {Z})$</annotation>\n </semantics></math>. The quotient group <math>\n <semantics>\n <mrow>\n <mi>Mod</mi>\n <mrow>\n <mo>(</mo>\n <msub>\n <mi>Σ</mi>\n <mi>g</mi>\n </msub>\n <mo>)</mo>\n </mrow>\n <mo>/</mo>\n <msub>\n <mi>I</mi>\n <mi>g</mi>\n </msub>\n </mrow>\n <annotation>${\\rm Mod}(\\Sigma _g) / \\mathcal {I}_g$</annotation>\n </semantics></math> is isomorphic to the symplectic group <math>\n <semantics>\n <mrow>\n <mi>Sp</mi>\n <mo>(</mo>\n <mn>2</mn>\n <mi>g</mi>\n <mo>,</mo>\n <mi>Z</mi>\n <mo>)</mo>\n </mrow>\n <annotation>${\\rm Sp}(2g, \\mathbb {Z})$</annotation>\n </semantics></math>. The cohomological dimension of the group <math>\n <semantics>\n <msub>\n <mi>I</mi>\n <mi>g</mi>\n </msub>\n <annotation>$\\mathcal {I}_g$</annotation>\n </semantics></math> equals to <math>\n <semantics>\n <mrow>\n <mn>3</mn>\n <mi>g</mi>\n <mo>−</mo>\n <mn>5</mn>\n </mrow>\n <annotation>$3g-5$</annotation>\n </semantics></math>. The main goal of the present paper is to compute the top homology group of the Torelli group in the case <math>\n <semantics>\n <mrow>\n <mi>g</mi>\n <mo>=</mo>\n <mn>3</mn>\n </mrow>\n <annotation>$g = 3$</annotation>\n </semantics></math> as <math>\n <semantics>\n <mrow>\n <mi>Sp</mi>\n <mo>(</mo>\n <mn>6</mn>\n <mo>,</mo>\n <mi>Z</mi>\n <mo>)</mo>\n </mrow>\n <annotation>${\\rm Sp}(6, \\mathbb {Z})$</annotation>\n </semantics></math>-module. We prove an isomorphism\n\n </p>","PeriodicalId":56114,"journal":{"name":"Journal of Topology","volume":null,"pages":null},"PeriodicalIF":0.8000,"publicationDate":"2023-08-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Topology","FirstCategoryId":"100","ListUrlMain":"https://onlinelibrary.wiley.com/doi/10.1112/topo.12308","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}

引用次数: 0

Abstract

The Torelli group of a genus $g$ oriented surface $Σ_{g}$ is the subgroup $I_{g}$ of the mapping class group $Mod (Σ_{g})$ consisting of all mapping classes that act trivially on $H_{1} (Σ_{g}, Z)$ . The quotient group $Mod (Σ_{g}) / I_{g}$ is isomorphic to the symplectic group $Sp (2 g, Z)$ . The cohomological dimension of the group $I_{g}$ equals to $3 g - 5$ . The main goal of the present paper is to compute the top homology group of the Torelli group in the case $g = 3$ as $Sp (6, Z)$ -module. We prove an isomorphism

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属3 Torelli群的顶部同源群

g属$g$面向曲面Σg $\Sigma _g$的Torelli群是映射类组Mod(Σg) ${\rm Mod}(\Sigma _g)$的子群Ig $\mathcal {I}_g$，该映射类组由H1(Σg,Z) ${\rm H}_1(\Sigma _g, \mathbb {Z})$上的所有映射类组成。商群Mod(Σg)/Ig ${\rm Mod}(\Sigma _g) / \mathcal {I}_g$与辛群Sp(2g,Z) ${\rm Sp}(2g, \mathbb {Z})$同构。基团Ig $\mathcal {I}_g$的上同维数为3g−5 $3g-5$。本文的主要目的是计算g=3 $g = 3$情况下Torelli群的顶同调群为Sp(6,Z) ${\rm Sp}(6, \mathbb {Z})$‐模。证明了一个同构H4(I3,Z) = IndS3 × SL(2,Z)×3Sp(6,Z)Z, $$\begin{equation*} \hspace*{4pc}{\rm H}_4(\mathcal {I}_3, \mathbb {Z}) \cong {\rm Ind}^{{\rm Sp}(6, \mathbb {Z})}_{S_3 \ltimes {\rm SL}(2, \mathbb {Z})^{\times 3}} \mathcal {Z}, \end{equation*}$$，其中Z $\mathcal {Z}$是Z3 $\mathbb {Z}^3$与其对角子群Z $\mathbb {Z}$的商，具有置换群S3 $S_3$的自然作用(SL(2,Z)×3 ${\rm SL}(2, \mathbb {Z})^{\times 3}$的作用是平凡的)。我们还构造了组H4(I3,Z) ${\rm H}_4(\mathcal {I}_3, \mathbb {Z})$的显式生成器和关系集。

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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.

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Issue Information Stated SL( n $n$ )-skein modules and algebras A combinatorial take on hierarchical hyperbolicity and applications to quotients of mapping class groups Degenerations of k $k$ -positive surface group representations Regularity of limit sets of Anosov representations