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Research on shadow complexities and geometric structures of 3-manifolds

Research Project

Project/Area Number 17K05254
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeMulti-year Fund
Section一般
Research Field Geometry
Research InstitutionHiroshima University

Principal Investigator

Koda Yuya  広島大学, 先進理工系科学研究科(理), 教授 (20525167)

Project Period (FY) 2017-04-01 – 2021-03-31
Project Status Completed (Fiscal Year 2020)
Budget Amount *help
¥4,420,000 (Direct Cost: ¥3,400,000、Indirect Cost: ¥1,020,000)
Fiscal Year 2019: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000)
Fiscal Year 2018: ¥1,560,000 (Direct Cost: ¥1,200,000、Indirect Cost: ¥360,000)
Fiscal Year 2017: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000)
Keywords3 次元多様体 / 4 次元多様体 / シャドウ / 双曲体積 / Goeritz 群 / 安定写像 / 絡み目 / スパイン / 接触構造 / Heegaard 分解 / 写像類群 / 国際情報交換 イタリア / 3次元多様体 / 4次元多様体 / 幾何構造
Outline of Final Research Achievements

A shadow of a 3-manifold is a 2-skeleton of a 4-manifold bounded by the 3-manifold. In this research project, we focused on the complexity defined by counting the number of vertices of shadows, and we have obtained the following results on the topology and geometry of 3 and 4-manifolds: (1) Classification of closed 4-manifolds of shadow complexity 1; (2) Classification of acyclic 4-manifolds with sphere boundary of shadow complexity at most 2; (3) Classification of hyperbolic links in the 3-sphere admitting a stable map with a single codimension-2 singular fiber. Further, as a related topic, we have obtained various interesting properties the Goeritz groups of Heegaard splittings of 3-manifolds.

Academic Significance and Societal Importance of the Research Achievements

空間の数学的モデルである多様体の中でも, 特に低次元 (次元が 4 以下) のものについては, 微分構造と組み合わせ構造が等価であることが知られている. したがって, これらの間のつながりを明示的に記述し, 微分構造に基づいて定義される諸概念から, 計算可能な組み合わせ的な量を引き出すことが原理的に可能である. 本研究課題では, シャドウ複雑度という組み合わせ量を用いて 3 次元多様体の微分構造・幾何構造, 4 次元多様体の微分構造を記述したものであり, 得られた成果は低次元多様体への理解に寄与するものである.

Report

(5 results)
  • 2020 Annual Research Report   Final Research Report ( PDF )
  • 2019 Research-status Report
  • 2018 Research-status Report
  • 2017 Research-status Report
  • Research Products

    (36 results)

All 2021 2020 2019 2018 2017 Other

All Int'l Joint Research (7 results) Journal Article (11 results) (of which Int'l Joint Research: 6 results,  Peer Reviewed: 11 results,  Open Access: 2 results) Presentation (15 results) (of which Int'l Joint Research: 4 results,  Invited: 15 results) Remarks (3 results)

  • [Int'l Joint Research] University of Pisa(イタリア)

    • Related Report
      2020 Annual Research Report
  • [Int'l Joint Research] Hanyang University/Korea University(韓国)

    • Related Report
      2020 Annual Research Report
  • [Int'l Joint Research] Hanyang University/Jeonbuk National University(韓国)

    • Related Report
      2019 Research-status Report
  • [Int'l Joint Research] Hanyang University/Chonbuk National University/Korea University(韓国)

    • Related Report
      2018 Research-status Report
  • [Int'l Joint Research] University of Pisa(イタリア)

    • Related Report
      2018 Research-status Report
  • [Int'l Joint Research] Hanyang University/KIAS(韓国)

    • Related Report
      2017 Research-status Report
  • [Int'l Joint Research] University of Pisa(イタリア)

    • Related Report
      2017 Research-status Report
  • [Journal Article] Four-maifolds with shadow-complexity one2021

    • Author(s)
      Yuya Koda, Bruno Martelli, Hironobu Naoe
    • Journal Title

      Annales de la Faculte des Sciences de Toulouse

      Volume: -

    • Related Report
      2020 Annual Research Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Braid group and leveling of a knot2021

    • Author(s)
      Sangbum Cho, Yuya Koda, Arim Seo
    • Journal Title

      Journal of Topology and Analysis

      Volume: - Issue: 04 Pages: 945-968

    • DOI

      10.1142/s1793525321500114

    • Related Report
      2020 Annual Research Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Goeritz groups of bridge decompositions,2021

    • Author(s)
      Susumu Hirose, Daiki Iguchi, Eiko Kin, Yuya Koda
    • Journal Title

      International Mathematics Research Notices,

      Volume: - Issue: 12 Pages: 9308-9356

    • DOI

      10.1093/imrn/rnab001

    • Related Report
      2020 Annual Research Report
    • Peer Reviewed / Open Access
  • [Journal Article] Shadows of acyclic 4-manifolds with sphere boundary2020

    • Author(s)
      Yuya Koda, Hironobu Naoe
    • Journal Title

      Algebraic & Geometric Topology

      Volume: 20 Issue: 7 Pages: 3707-3731

    • DOI

      10.2140/agt.2020.20.3707

    • Related Report
      2020 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Extending automorphisms of the genus-2 surface over the 3-sphere2020

    • Author(s)
      Kenta Funayoshi, Yuya Koda
    • Journal Title

      The Quarterly Journal of Mathematics

      Volume: 71 Issue: 1 Pages: 175-196

    • DOI

      10.1093/qmathj/haz042

    • Related Report
      2019 Research-status Report
    • Peer Reviewed
  • [Journal Article] Twisted book decompositions and the Goeritz groups2020

    • Author(s)
      Daiki Iguchi, Yuya Koda
    • Journal Title

      Topology and its Applications

      Volume: 272 Pages: 107064-107064

    • DOI

      10.1016/j.topol.2020.107064

    • Related Report
      2019 Research-status Report
    • Peer Reviewed
  • [Journal Article] Disk surgery and the primitive disk complexes of the 3-sphere2020

    • Author(s)
      Sangbum Cho, Yuya Koda, Jung Hoon Lee
    • Journal Title

      Topology and its Applications

      Volume: 272 Pages: 107092-107092

    • DOI

      10.1016/j.topol.2020.107092

    • Related Report
      2019 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Neighborhood equivalence for multibranched surfaces in 3-manifolds2019

    • Author(s)
      Ishihara Kai、Koda Yuya、Ozawa Makoto、Shimokawa Koya
    • Journal Title

      Topology and its Applications

      Volume: 257 Pages: 11-21

    • DOI

      10.1016/j.topol.2019.02.005

    • Related Report
      2019 Research-status Report 2018 Research-status Report
    • Peer Reviewed / Open Access
  • [Journal Article] The mapping class groups of reducible Heegaard splittings of genus two2019

    • Author(s)
      Sangbum Cho, Yuya Koda
    • Journal Title

      Transactions of the American Mathematical Society

      Volume: 371 Issue: 4 Pages: 2473-2502

    • DOI

      10.1090/tran/7375

    • Related Report
      2018 Research-status Report 2017 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] The disk complex and 2-bridge knots2018

    • Author(s)
      Sangbum Cho, Yuya Koda
    • Journal Title

      Journal of Knot Theory and its Ramifications

      Volume: 27 Issue: 04 Pages: 1850027-1850027

    • DOI

      10.1142/s021821651850027x

    • Related Report
      2018 Research-status Report 2017 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Haken spheres for genus two Heegaard splittings2018

    • Author(s)
      Sangbum Cho, Yuya Koda
    • Journal Title

      Mathematical Proceedings of the Cambridge Philosophical Society

      Volume: 165 Issue: 3 Pages: 563-572

    • DOI

      10.1017/s0305004117000718

    • Related Report
      2018 Research-status Report 2017 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Presentation] `Monodromy groups” of Heegaard surfaces of 3-manifolds2020

    • Author(s)
      古宇田 悠哉
    • Organizer
      埼玉大学 談話会 (Zoom)
    • Related Report
      2020 Annual Research Report
    • Invited
  • [Presentation] Turaev のシャドウとその複雑度2020

    • Author(s)
      古宇田 悠哉
    • Organizer
      N-KOOKセミナー (Zoom)
    • Related Report
      2020 Annual Research Report
    • Invited
  • [Presentation] Goeritz groups of bridge decompositions2020

    • Author(s)
      古宇田 悠哉
    • Organizer
      トポロジー火曜セミナー, 東京大学 (Zoom)
    • Related Report
      2020 Annual Research Report
    • Invited
  • [Presentation] Positive flow-spines and contact 3-manifolds2019

    • Author(s)
      古宇田 悠哉
    • Organizer
      広島大学 トポロジー・幾何セミナー
    • Related Report
      2019 Research-status Report
    • Invited
  • [Presentation] Positive flow-spines and contact 3-manifolds2019

    • Author(s)
      Yuya Koda
    • Organizer
      Intelligence of Low-dimensional Topology
    • Related Report
      2019 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] 3・4 次元多様体のシャドウと安定写像2019

    • Author(s)
      古宇田 悠哉
    • Organizer
      研究集会「接触構造, 特異点, 微分方程式及びその周辺」
    • Related Report
      2018 Research-status Report
    • Invited
  • [Presentation] 閉 4 次元多様体のシャドウ複雑度2018

    • Author(s)
      古宇田 悠哉
    • Organizer
      広島大学 トポロジー・幾何セミナー
    • Related Report
      2018 Research-status Report
    • Invited
  • [Presentation] Shadow complexity of 3- and 4-manifolds2018

    • Author(s)
      古宇田 悠哉
    • Organizer
      第65回 トポロジーシンポジウム
    • Related Report
      2018 Research-status Report
    • Invited
  • [Presentation] 閉 4 次元多様体のシャドウ複雑度2018

    • Author(s)
      古宇田 悠哉
    • Organizer
      研究集会「Graphと3次元多様体の研究」
    • Related Report
      2018 Research-status Report
    • Invited
  • [Presentation] 欠陥のトポロジー2018

    • Author(s)
      古宇田 悠哉
    • Organizer
      キラリティー,トポロジー,結び目論第2回研究会
    • Related Report
      2018 Research-status Report
    • Invited
  • [Presentation] Positive flow-spines and contact 3-manifolds2018

    • Author(s)
      Yuya Koda
    • Organizer
      Workshop ``New development of low-dimensional topology"
    • Related Report
      2018 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Positive flow-spines and contact 3-manifolds2018

    • Author(s)
      古宇田 悠哉
    • Organizer
      研究集会「葉層構造の幾何学とその応用」
    • Related Report
      2018 Research-status Report
    • Invited
  • [Presentation] Extending automorphisms of the genus-2 surface over the 3-sphere2018

    • Author(s)
      Yuya Koda
    • Organizer
      Topology Seminar at Chung-Ang University
    • Related Report
      2017 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] The disk complex and 2-bridge knots2017

    • Author(s)
      古宇田悠哉
    • Organizer
      研究集会「Graphと3次元多様体の研究」
    • Related Report
      2017 Research-status Report
    • Invited
  • [Presentation] The mapping class groups of Heegaard splittings for 3-manifolds2017

    • Author(s)
      Yuya Koda
    • Organizer
      Capital Normal University-Hiroshima University Joint conference on Mathematics
    • Related Report
      2017 Research-status Report
    • Int'l Joint Research / Invited
  • [Remarks] 古宇田悠哉のホームページ

    • URL

      https://home.hiroshima-u.ac.jp/ykoda/index_j.html

    • Related Report
      2020 Annual Research Report 2019 Research-status Report
  • [Remarks]

    • URL

      https://home.hiroshima-u.ac.jp/ykoda/

    • Related Report
      2018 Research-status Report
  • [Remarks]

    • URL

      http://home.hiroshima-u.ac.jp/ykoda/

    • Related Report
      2017 Research-status Report

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Published: 2017-04-28   Modified: 2022-01-27  

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