The top homology group of the genus 3 Torelli group

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":0,"journal":{"name":"","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-08-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"","FirstCategoryId":"100","ListUrlMain":"https://onlinelibrary.wiley.com/doi/10.1112/topo.12308","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}

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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。

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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