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Study on 3-manifolds and cohomology of subgroups of the mapping class group of a surface

Research Project

Project/Area Number 18K03310
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeMulti-year Fund
Section一般
Review Section Basic Section 11020:Geometry-related
Research InstitutionTokyo Denki University

Principal Investigator

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

Project Period (FY) 2018-04-01 – 2024-03-31
Project Status Completed (Fiscal Year 2023)
Budget Amount *help
¥3,380,000 (Direct Cost: ¥2,600,000、Indirect Cost: ¥780,000)
Fiscal Year 2020: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
Fiscal Year 2019: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
Fiscal Year 2018: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
Keywords写像類群 / LMO関手 / ホモロジーシリンダー / Torelli群 / Reidemeisterトーション / ジョンソン核 / トレリ群 / 有限型不変量 / ハンドル体写像類群 / トポロジー / 3次元多様体
Outline of Final Research Achievements

We study a subgroup of the mapping class group called the Torelli group and the monoid of homology cylinders of a surface. Especially, we showed that there exist torsion elements on the abelianization of a subgroup of the Torelli group called the Johnson kernel. We describe it in terms of the LMO functor, which is a topological invariant of homology cylinders. We also showed that the leading term of the 1-loop part of the LMO functor is described by the Reidemeister-Turaev torsion of homology cylinders.

Academic Significance and Societal Importance of the Research Achievements

曲面の写像類群とは曲面の対称性のことであり、曲面の写像類群の群構造を3次元多様体の観点から研究することは基本的である。Torelli群と呼ばれる写像類群の部分群における降中心列と呼ばれる部分群列と、3次元多様体の中のホモロジーシリンダーのなすモノイドにおけるY降下列と呼ばれる部分モノイド列がある。Torelli群からホモロジーシリンダーへの自然なある単射モノイド準同型があり、これにより降中心列はY降下列に埋め込める。この準同型を通じて、両者の逐次商の構造を調べるとともに、LMO関手と呼ばれるホモロジーシリンダーの位相不変量に1つの幾何的解釈を与えた。

Report

(7 results)
  • 2023 Annual Research Report   Final Research Report ( PDF )
  • 2022 Research-status Report
  • 2021 Research-status Report
  • 2020 Research-status Report
  • 2019 Research-status Report
  • 2018 Research-status Report
  • Research Products

    (15 results)

All 2023 2022 2021 2020 2019 Other

All Int'l Joint Research (2 results) Journal Article (4 results) (of which Int'l Joint Research: 1 results,  Peer Reviewed: 4 results,  Open Access: 1 results) Presentation (7 results) (of which Int'l Joint Research: 1 results,  Invited: 6 results) Remarks (2 results)

  • [Int'l Joint Research] University of Burgundy(フランス)

    • Related Report
      2023 Annual Research Report
  • [Int'l Joint Research] University of Zurich(スイス)

    • Related Report
      2023 Annual Research Report
  • [Journal Article] A non-commutative Reidemeister-Turaev torsion of homology cylinders2023

    • Author(s)
      Yuta Nozaki, Masatoshi Sato and Masaaki Suzuki
    • Journal Title

      Trans. Amer. Math. Soc.

      Volume: 376 Issue: 7 Pages: 5045-5088

    • DOI

      10.1090/tran/8925

    • Related Report
      2022 Research-status Report
    • Peer Reviewed
  • [Journal Article] On the kernel of the surgery map restricted to the 1-loop part2022

    • Author(s)
      Yuta Nozaki, Masatoshi Sato, Masaaki Suzuki
    • Journal Title

      Journal of Topology

      Volume: 15 Issue: 2 Pages: 587-619

    • DOI

      10.1112/topo.12233

    • Related Report
      2021 Research-status Report
    • Peer Reviewed
  • [Journal Article] Abelian quotients of the Y-filtration on the homology cylinders via the LMO functor2021

    • Author(s)
      Yuta Nozaki, Masatoshi Sato, and Masaaki Suzuki
    • Journal Title

      Geometry and Topology

      Volume: -

    • Related Report
      2020 Research-status Report
    • Peer Reviewed
  • [Journal Article] The Meyer function on the handlebody group2020

    • Author(s)
      KUNO Yusuke、SATO Masatosh
    • Journal Title

      TURKISH JOURNAL OF MATHEMATICS

      Volume: 44 Issue: 5 Pages: 1520-1533

    • DOI

      10.3906/mat-1911-67

    • Related Report
      2020 Research-status Report
    • Peer Reviewed / Open Access / Int'l Joint Research
  • [Presentation] LMO関手による写像類群の部分群の可換商2023

    • Author(s)
      佐藤正寿
    • Organizer
      数理科学科談話会・東京理科大学幾何学セミナー
    • Related Report
      2023 Annual Research Report
    • Invited
  • [Presentation] LMO関手によるTorelli群の降中心列の次数商の評価2023

    • Author(s)
      佐藤正寿
    • Organizer
      佐賀大学・学習院大学合同トポロジーセミナー
    • Related Report
      2023 Annual Research Report
    • Invited
  • [Presentation] A non-commutative Reidemeister-Turaev torsion of homology cylinders2023

    • Author(s)
      佐藤正寿
    • Organizer
      微分トポロジーセミナー
    • Related Report
      2022 Research-status Report
    • Invited
  • [Presentation] A non-commutative Reidemeister-Turaev torsion of homology cylinders2022

    • Author(s)
      Masatoshi Sato
    • Organizer
      Topology and Geometry of Low-dimensional Manifolds
    • Related Report
      2021 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] A non-commutative Reidemeister-Turaev torsion of homology cylinders2021

    • Author(s)
      佐藤正寿
    • Organizer
      トポロジー火曜セミナー
    • Related Report
      2021 Research-status Report
    • Invited
  • [Presentation] Abelian quotients of the Y-filtration on the homology cylinders via the LMO functor II2019

    • Author(s)
      佐藤正寿
    • Organizer
      低次元トポロジーin 白神 2019
    • Related Report
      2019 Research-status Report
  • [Presentation] Abelian quotients of the Y-filtration on the homology cylinders via the LMO functor II2019

    • Author(s)
      Masatoshi Sato
    • Organizer
      A one-day workshop on Invariants of homology cylinders
    • Related Report
      2019 Research-status Report
    • Invited
  • [Remarks] 佐藤正寿

    • URL

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

    • Related Report
      2022 Research-status Report
  • [Remarks] 佐藤正寿

    • URL

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

    • Related Report
      2021 Research-status Report 2020 Research-status Report 2019 Research-status Report

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Published: 2018-04-23   Modified: 2025-01-30  

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