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A study of cohomology of subgroups of the mapping class group and finite type invariants of homology three spheres

Research Project

Project/Area Number 26800034
Research Category

Grant-in-Aid for Young Scientists (B)

Allocation TypeMulti-year Fund
Research Field Geometry
Research InstitutionTokyo Denki University (2015-2018)
Gifu University (2014)

Principal Investigator

SATO Masatoshi  東京電機大学, 未来科学部, 准教授 (10632010)

Research Collaborator ISHIDA Tomohiko  
Project Period (FY) 2014-04-01 – 2019-03-31
Project Status Completed (Fiscal Year 2018)
Budget Amount *help
¥3,250,000 (Direct Cost: ¥2,500,000、Indirect Cost: ¥750,000)
Fiscal Year 2017: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000)
Fiscal Year 2016: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000)
Fiscal Year 2015: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000)
Fiscal Year 2014: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
Keywords写像類群 / ハンドル体 / 円板複体 / 3次元ハンドル体 / 配置空間 / Krizモデル / ハンドル体写像類群 / 有限型不変量 / Torelli群
Outline of Final Research Achievements

We studied mainly on the cohomology group of the handlebody mapping class group. For every genus, we computed its first homology group with coefficients in the first integral homology of the boundary surface of the handlebody. This is a joint work with Tomohiko Ishida, and the results are published in Osaka Journal of Mathematics. In genus two case, we also computed the integral cohomology group of the handlebody group. In genus three case, we gave an upper bound on the rank of the rational cohomology group of the handlebody mapping class group.
We also computed values of LMO functor on some elements in the Torelli group and Johnson kernel.

Academic Significance and Societal Importance of the Research Achievements

低次元トポロジーにおいて曲面の写像類群は2次元の図形の対称性を記述する重要な対象である。また、3次元のすべての図形は3次元の基本的な図形の貼りあわせとして記述されるが、その貼りあわせをつかさどるものが曲面の写像類群である。
本研究の興味は、この曲面の対称性の情報と3次元の図形の不変量の関係であり、貼り合わせを表す2次元図形の対称性という群論的対象を通して、3次元の幾何的対象の情報を記述することが目標である。本研究の1つの成果としてハンドル体写像類群のコホモロジー群という群論的対象の情報が得られた。

Report

(6 results)
  • 2018 Annual Research Report   Final Research Report ( PDF )
  • 2017 Research-status Report
  • 2016 Research-status Report
  • 2015 Research-status Report
  • 2014 Research-status Report
  • Research Products

    (20 results)

All 2019 2017 2016 2015 2014 Other

All Journal Article (1 results) (of which Peer Reviewed: 1 results,  Open Access: 1 results,  Acknowledgement Compliant: 1 results) Presentation (15 results) (of which Int'l Joint Research: 2 results,  Invited: 10 results) Remarks (4 results)

  • [Journal Article] A twisted first homology group of the handlebody mapping class group2017

    • Author(s)
      Tomohiko Ishida, Masatoshi Sato
    • Journal Title

      Osaka Journal of Mathematics

      Volume: 印刷中

    • NAID

      120006336191

    • Related Report
      2016 Research-status Report
    • Peer Reviewed / Open Access / Acknowledgement Compliant
  • [Presentation] トーラス上の点の順序つき配置空間のコホモロジー群について2019

    • Author(s)
      佐藤正寿
    • Organizer
      種々の幾何学的構造と基本群に現れる様々な特性類とその不変量への応用
    • Related Report
      2018 Annual Research Report
  • [Presentation] On the cohomology group of genus 3 handlebody mapping class group2019

    • Author(s)
      Masatoshi Sato
    • Organizer
      Cohomological study of mapping class groups and related topics
    • Related Report
      2018 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] On the homology group of the genus three handlebody mapping class group2017

    • Author(s)
      佐藤正寿
    • Organizer
      Topological invariants in low dimensional topology
    • Related Report
      2017 Research-status Report
    • Invited
  • [Presentation] 種数3のハンドル体写像類群のホモロジー群について2017

    • Author(s)
      佐藤正寿
    • Organizer
      表現論による諸問題への新しいアプローチを目指して
    • Related Report
      2017 Research-status Report
  • [Presentation] On the homology group of the genus three handlebody mapping class group2017

    • Author(s)
      佐藤正寿
    • Organizer
      Wednesday afternoon seminar
    • Related Report
      2017 Research-status Report
    • Invited
  • [Presentation] 種数3のハンドル体写像類群のホモロジー群について2017

    • Author(s)
      佐藤正寿
    • Organizer
      学習院トポロジーセミナー
    • Related Report
      2017 Research-status Report
    • Invited
  • [Presentation] On the LMO functor constructed by Cheptea, Habiro and Massuyeau2016

    • Author(s)
      佐藤正寿
    • Organizer
      Workshop on finite type invariants of 3-manifolds
    • Place of Presentation
      東京大学(東京都, 目黒区)
    • Year and Date
      2016-03-30
    • Related Report
      2015 Research-status Report
    • Invited
  • [Presentation] LMO関手と写像類群の部分群の可換商について2016

    • Author(s)
      佐藤正寿
    • Organizer
      理学セミナー
    • Place of Presentation
      日本女子大学(東京都文京区)
    • Related Report
      2016 Research-status Report
    • Invited
  • [Presentation] 種数2のハンドル体写像類群の整係数コホモロジー環について2015

    • Author(s)
      佐藤正寿
    • Organizer
      東工大トポロジーセミナー
    • Place of Presentation
      東京工業大学(東京都, 目黒区)
    • Year and Date
      2015-11-26
    • Related Report
      2015 Research-status Report
    • Invited
  • [Presentation] On the cohomology ring of the handlebody mapping class group of genus two2015

    • Author(s)
      佐藤正寿
    • Organizer
      トポロジー火曜セミナー
    • Place of Presentation
      東京大学(東京都, 目黒区)
    • Year and Date
      2015-11-24
    • Related Report
      2015 Research-status Report
    • Invited
  • [Presentation] On the cohomology ring of the handlebody mapping class group of genus two2015

    • Author(s)
      Masatoshi Sato
    • Organizer
      Atelier de travail franco-japonais sur la geometrie des groupes modulaires et des espaces de Teichmueller
    • Place of Presentation
      東京大学(東京都, 目黒区)
    • Year and Date
      2015-11-19
    • Related Report
      2015 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] 種数2のハンドル体写像類群のコホモロジー環について2015

    • Author(s)
      佐藤正寿
    • Organizer
      トポロジー金曜セミナー
    • Place of Presentation
      九州大学(福岡県・福岡市)
    • Year and Date
      2015-01-22
    • Related Report
      2014 Research-status Report
    • Invited
  • [Presentation] A twisted first homology of the handlebody mapping class group2015

    • Author(s)
      石田智彦
    • Organizer
      Hurwitz action
    • Place of Presentation
      大阪電気通信大学(大阪府・寝屋川市)
    • Year and Date
      2015-01-10
    • Related Report
      2014 Research-status Report
  • [Presentation] ハンドル体写像類群のねじれ係数1次元ホモロジー群について2014

    • Author(s)
      石田智彦
    • Organizer
      日本数学会秋季総合分科会
    • Place of Presentation
      広島大学(広島県・東広島市)
    • Year and Date
      2014-09-26
    • Related Report
      2014 Research-status Report
  • [Presentation] • A twisted first homology of the handlebody mapping class group2014

    • Author(s)
      石田智彦
    • Organizer
      リーマン面に関連する位相幾何学
    • Place of Presentation
      東京大学(東京都・目黒区)
    • Year and Date
      2014-08-25
    • Related Report
      2014 Research-status Report
  • [Remarks] 佐藤正寿

    • URL

      https://www.cck.dendai.ac.jp/math/~msato/index.html

    • Related Report
      2018 Annual Research Report
  • [Remarks] 佐藤正寿(Masatoshi Sato)

    • URL

      https://www.cck.dendai.ac.jp/math/~msato/index.html

    • Related Report
      2017 Research-status Report 2016 Research-status Report
  • [Remarks] 佐藤正寿(Masatoshi Sato)

    • URL

      https://www.cck.dendai.ac.jp/math/~msato/

    • Related Report
      2015 Research-status Report
  • [Remarks] 佐藤正寿

    • URL

      http://www1.gifu-u.ac.jp/~msato/

    • Related Report
      2014 Research-status Report

URL: 

Published: 2014-04-04   Modified: 2020-03-30  

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