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Study of geometric structures of singularities and manifolds based on polyhedra

Research Project

Project/Area Number 19K03499
Research Category

Grant-in-Aid for Scientific Research (C)

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

Principal Investigator

ISHIKAWA Masaharu  慶應義塾大学, 経済学部(日吉), 教授 (10361784)

Project Period (FY) 2019-04-01 – 2024-03-31
Project Status Completed (Fiscal Year 2023)
Budget Amount *help
¥4,290,000 (Direct Cost: ¥3,300,000、Indirect Cost: ¥990,000)
Fiscal Year 2021: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000)
Fiscal Year 2020: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000)
Fiscal Year 2019: ¥1,690,000 (Direct Cost: ¥1,300,000、Indirect Cost: ¥390,000)
Keywords特異点 / 多様体 / 3次元多様体 / 4次元多様体 / 無限遠の特異点 / 多面体 / 2次元結び目 / 接触構造
Outline of Research at the Start

本研究では,Turaevのshadowや3次元多様体のflow-spineに着目し,多様体上のファイバー束構造とその境界に現れるオープンブック分解および接触構造の研究についての新しい枠組みを,多面体を軸に構築する.Shadowやspineは,多面体の組み合わせ的構造を利用して,3次元空間や4次元空間の表示を与える手法である.各面に向きの構造を入れることで,安定写像のStein分解や3次元多様体のflow-spineといった,より豊富な構造との対応を与えることができる.さらに詳細な情報を与えることで,レフシェッツ束や接触構造などのより深い幾何構造との対応を与えることが研究の目的である.

Outline of Final Research Achievements

We studied 3- and 4-dimensional manifolds and singular plane curves by using 2-dimensional polyhedra such as flow spines and shadows. Flow spines are related to contact structures on 3-dimensional manifolds, and shadows are related to plane curves and line arrangements. We showed that a presentation of the fundamental group of a plane curve singularity can be obtained from a real morsification of the singularity by doubling the real curve on the 2-dimensional plane and making it to be a shadow in a suitable way. This is a generalization of the Wirtinger presentation of the fundamental group of a knot complement. In the study of flow spines and contact structures, we focused on the abalone and studied its transversality with Seifert fibrations. We also studied atypical fibers of polynomial maps, called singularities at infinity, and circle actions on the 4-dimensional sphere.

Academic Significance and Societal Importance of the Research Achievements

多様体および多様体間の写像は数学のみならず自然科学全般において重要な道具であり、そこに現れる特異点も重要な研究対象である。これらの情報を表すためにグラフや実平面曲線が用いられることが多いが、より複雑な研究を行う場合には、2次元多面体を用いる必要がある。それがスパインであり、シャドウである。本研究では、接触構造や代数曲線といった幾何構造を多面体を用いて表示し、そこから得られる情報により多様体や多様体間の写像の情報を得るという枠組みを構築している。これらの結果は、3次元多様体と4次元多様体、さらに高次元の多様体を結び付ける研究の基礎として、今後、重要な役割を果たすものである。

Report

(6 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
  • Research Products

    (27 results)

All 2023 2022 2021 2020 2019 Other

All Int'l Joint Research (3 results) Journal Article (10 results) (of which Int'l Joint Research: 2 results,  Peer Reviewed: 7 results) Presentation (13 results) (of which Int'l Joint Research: 6 results,  Invited: 8 results) Book (1 results)

  • [Int'l Joint Research] VAST(ベトナム)

    • Related Report
      2022 Research-status Report
  • [Int'l Joint Research] Institute of Mathematics, VAST(ベトナム)

    • Related Report
      2020 Research-status Report
  • [Int'l Joint Research] Institute of Mathematics, VAST/Dalat University(ベトナム)

    • Related Report
      2019 Research-status Report
  • [Journal Article] Presentation of the fundamental groups of complements of shadows2023

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

      Essays in Geometry Dedicated to Norbert A’Campo

      Volume: - Pages: 557-588

    • Related Report
      2023 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Positive flow-spines and contact 3-manifolds, II2023

    • Author(s)
      Ishii Ippei、Ishikawa Masaharu、Koda Yuya、Naoe Hironobu
    • Journal Title

      Annali di Matematica Pura ed Applicata (1923 -)

      Volume: - Issue: 3 Pages: 2091-2126

    • DOI

      10.1007/s10231-023-01400-4

    • Related Report
      2023 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Positive flow-spines and contact 3-manifolds2023

    • Author(s)
      Ippei Ishii, Masaharu Ishikawa, Yuya Koda, Hironobu Naoe
    • Journal Title

      Annali di Matematica Pura ed Applicata (1923 -)

      Volume: - Issue: 5 Pages: 2091-2126

    • DOI

      10.1007/s10231-023-01314-1

    • Related Report
      2022 Research-status Report
    • Peer Reviewed
  • [Journal Article] シャドウ補空間の基本群の表示2022

    • Author(s)
      石川昌治、古宇田悠哉、直江央寛
    • Journal Title

      研究集会「結び目の数理 IV 」報告集

      Volume: - Pages: 299-309

    • Related Report
      2021 Research-status Report
  • [Journal Article] S-stable foliations on flow-spines with transverse Reeb flow2021

    • Author(s)
      Shin Handa, Masaharu Ishikawa
    • Journal Title

      Hiroshima Mathematical Journal

      Volume: 51 Pages: 77-99

    • Related Report
      2020 Research-status Report
    • Peer Reviewed
  • [Journal Article] Milnor fibration, A'Campo's divide and Turaev's shadow2020

    • Author(s)
      Masaharu Ishikawa, Hironobu Naoe
    • Journal Title

      Singularities ― Kagoshima 2017, Proceedings of the 5th Franco- Japanese-Vietnamese Symposium on Singularities, World Scientific Publishing

      Volume: - Pages: 95-116

    • Related Report
      2020 Research-status Report
    • Peer Reviewed
  • [Journal Article] アワビとザイフェルト束の横断性について2020

    • Author(s)
      石川昌治
    • Journal Title

      Hakone Seminar

      Volume: 35 Pages: 9-30

    • Related Report
      2020 Research-status Report
  • [Journal Article] Alternating knots with large boundary slope diameter2020

    • Author(s)
      Masaharu Ishikawa, Thomas W. Mattman, Kazuya Namiki, Koya Shimokawa
    • Journal Title

      Contemporary Mathematics

      Volume: 760 Pages: 207-216

    • DOI

      10.1090/conm/760/15292

    • Related Report
      2020 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Bifurcation sets of real polynomial functions of two variables and Newton polygons2019

    • Author(s)
      石川昌治, Tat-Thang Nguyen, Tien-Son Pham
    • Journal Title

      Journal of the Mathematical Society of Japan

      Volume: 71 Issue: 4 Pages: 1201-1222

    • DOI

      10.2969/jmsj/80518051

    • NAID

      130007733324

    • ISSN
      0025-5645, 1881-1167, 1881-2333
    • Related Report
      2019 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] 3次元多様体の接触構造とフロースパイン2019

    • Author(s)
      石川昌治
    • Journal Title

      Hakone Seminar

      Volume: 34 Pages: 7-31

    • Related Report
      2019 Research-status Report
  • [Presentation] Positive flow-spines and contact 3-manifolds2023

    • Author(s)
      石川昌治
    • Organizer
      Singularities, arrangements, and low-dimensional topology, III
    • Related Report
      2023 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] On the fundamental groups of branched twist spins2023

    • Author(s)
      石川昌治
    • Organizer
      Breadth in low-dimensional topology
    • Related Report
      2022 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Positive flow-spines and contact 3-manifolds2022

    • Author(s)
      石川昌治
    • Organizer
      JSPS-VAST Bilateral Joint Research Project Workshop "Singularities, arrangements, and low-dimensional topology", III
    • Related Report
      2022 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] シャドウ補空間の基本群の表示2021

    • Author(s)
      石川昌治
    • Organizer
      研究集会「結び目の数理 IV 」
    • Related Report
      2021 Research-status Report
  • [Presentation] Stable map complexity and hyperbolic volumes of 3-manifolds2021

    • Author(s)
      石川昌治
    • Organizer
      Boston University / Keio University / Tsinghua University Workshop 2021
    • Related Report
      2021 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] アワビとザイフェルト束の横断性について2020

    • Author(s)
      石川昌治
    • Organizer
      東北結び目セミナー2020
    • Related Report
      2020 Research-status Report
  • [Presentation] フロースパインと接触構造2020

    • Author(s)
      石川昌治
    • Organizer
      接触構造、特異点、微分方程式及びその周辺
    • Related Report
      2019 Research-status Report
    • Invited
  • [Presentation] 3次元多様体の正フロースパインと接触構造2019

    • Author(s)
      石川昌治
    • Organizer
      大阪大学トポロジーセミナー
    • Related Report
      2019 Research-status Report
    • Invited
  • [Presentation] 3次元多様体の正フロースパインと接触構造2019

    • Author(s)
      石川昌治
    • Organizer
      談話会(埼玉大学)
    • Related Report
      2019 Research-status Report
    • Invited
  • [Presentation] Polyhedral presentation of Milnor fibers2019

    • Author(s)
      石川昌治
    • Organizer
      Seminar in Institute of Mathematics, VAST
    • Related Report
      2019 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] レンズ空間の正フロースパインと接触構造について2019

    • Author(s)
      石川昌治
    • Organizer
      北陸結び目セミナー 2019
    • Related Report
      2019 Research-status Report
  • [Presentation] アワビとホップ束について2019

    • Author(s)
      石川昌治
    • Organizer
      第42回箱根セミナ(2019)
    • Related Report
      2019 Research-status Report
  • [Presentation] Complexity of contact 3-manifolds2019

    • Author(s)
      石川昌治
    • Organizer
      Hyperplane arrangements and Japanese-Australian Workshop on Real and Complex Singularities
    • Related Report
      2019 Research-status Report
    • Int'l Joint Research
  • [Book] Singularities ― Kagoshima 2017, Proceedings of the 5th Franco- Japanese-Vietnamese Symposium on Singularities2020

    • Author(s)
      Masaharu Ishikawa, Shoji Yokura (Editors-in-Chief)
    • Total Pages
      397
    • Publisher
      World Scientific Publishing
    • Related Report
      2020 Research-status Report

URL: 

Published: 2019-04-18   Modified: 2025-01-30  

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