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族のゲージ理論の境界付き4次元多様体への拡張

Research Project

Project/Area Number 21K13785
Research Category

Grant-in-Aid for Early-Career Scientists

Allocation TypeMulti-year Fund
Review Section Basic Section 11020:Geometry-related
Research InstitutionThe University of Tokyo

Principal Investigator

今野 北斗  東京大学, 大学院数理科学研究科, 助教 (20845614)

Project Period (FY) 2021-04-01 – 2025-03-31
Project Status Granted (Fiscal Year 2023)
Budget Amount *help
¥4,680,000 (Direct Cost: ¥3,600,000、Indirect Cost: ¥1,080,000)
Fiscal Year 2024: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
Fiscal Year 2023: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000)
Fiscal Year 2022: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
Fiscal Year 2021: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
Keywords写像類群 / 多様体のモジュライ空間 / 特性類 / 族のゲージ理論 / エキゾチック微分同相写像 / Seiberg-Witten方程式 / 4次元多様体 / Dehnツイスト / 微分同相群 / 群作用 / 同相群 / コンタクト構造 / ゲージ理論 / Yang-Mills反自己双対方程式
Outline of Research at the Start

ゲージ理論を境界付き4次元多様体の族に対して展開する.これは,私がこれまで国内外の共同研究者の協力を得つつ取り組んできた閉4次元多様体の族のゲージ理論の拡張となると予想される.より具体的には,境界付き4次元多様体の滑らかな族に対するゲージ理論的な制約を得るのが理論的な目標となる.そのために,Floer理論的な構成を3次元多様体の族に対して行い,またそこに値を持つ境界付き4次元多様体の族の相対不変量をも定義する必要がある.
期待される応用は,3次元多様体の族や3次元多様体上の群作用が,それを境界として持つ4次元多様体の族やその上の群作用に滑らかに拡張するための障害の構成である.

Outline of Annual Research Achievements

今年度の第一の成果は,4次元多様体の写像類群と4次元多様体の族の特性類のある種の無限性に関するものである.4以外の全ての次元で,単連結閉多様体の写像類群は有限生成であることが知られている.これとは対照的に,単連結閉4次元多様体であって,その写像類群が無限生成であるようなものが存在することを示した.さらに, 高次元単連結多様体Xに対して,分類空間BDiff(X)のホモロジーは各次数で有限生成であることが予想されており,偶数次元では実際に正しいことが知られている.この4次元類似が,ホモロジーのどの次数でも成立しないことを示した.上述の写像類群の無限生成性はこの結果の帰結である.証明は,Seiberg-Witten方程式の族を用いた新しい特性類の無限族を構成・計算することでなされる.

第二の成果は,Seifert3次元多様体に沿ったエキゾチックDehnツイストに関するものである(Abhishek Mallick氏・谷口正樹氏との共同研究).従来,4次元多様体のエキゾチック微分同相写像は,(Euler数が)比較的大きな4次元多様体上で見つかって来た.我々は,可縮なコンパクト4次元多様体が(相対的な意味で)エキゾチックな微分同相を許容することを初めて示した.そのような例は,Seifert 3次元多様体に沿ったエキゾチックDehnツイストとして得られる.Seifert 3次元多様体は自然なS^1作用を許容するが,これを用いてDehnツイストの類似を考えることができる.既にKronheimerとMrowkaが,S^3に沿ったDehnツイストがある4次元多様体に対してエキゾチックとなることを示していた.我々は,S^3以外のSeifert 3次元多様体に沿ったDehn twistがエキゾチックな微分同相の例を与えることを初めて示した.証明には族のSeiberg-Witten理論を用いる.

Current Status of Research Progress
Current Status of Research Progress

2: Research has progressed on the whole more than it was originally planned.

Reason

第一の成果は4次元の特殊性を新たな角度から抉り出すもので,4次元多様体の微分同相群に対する基本的な事実となると考えている.第二の成果の内,特に可縮な4次元多様体上のエキゾチックDehnツイストは,境界付き4次元多様体のエキゾチック微分同相写像として興味深いものである.
一方,境界付き4次元多様体の族であって境界への制限が非自明な族となる場合の研究について,以前から執筆を進めている論文は年度内に完成しなかった.

Strategy for Future Research Activity

境界付き4次元多様体の族であって境界への制限が非自明な族となる場合の研究について,論文執筆を進めていく予定である.

Report

(3 results)
  • 2023 Research-status Report
  • 2022 Research-status Report
  • 2021 Research-status Report
  • Research Products

    (60 results)

All 2023 2022 2021 Other

All Int'l Joint Research (8 results) Journal Article (8 results) (of which Int'l Joint Research: 3 results,  Peer Reviewed: 8 results) Presentation (42 results) (of which Int'l Joint Research: 15 results,  Invited: 41 results) Remarks (2 results)

  • [Int'l Joint Research] Rutgers University(米国)

    • Related Report
      2023 Research-status Report
  • [Int'l Joint Research] Princeton university(米国)

    • Related Report
      2023 Research-status Report
  • [Int'l Joint Research] 清華大学(中国)

    • Related Report
      2023 Research-status Report
  • [Int'l Joint Research] Stony Brook University(米国)

    • Related Report
      2023 Research-status Report
  • [Int'l Joint Research] Rutgers University(米国)

    • Related Report
      2022 Research-status Report
  • [Int'l Joint Research] Georgia Institute of Technology(米国)

    • Related Report
      2022 Research-status Report
  • [Int'l Joint Research] 清華大学(中国)

    • Related Report
      2022 Research-status Report
  • [Int'l Joint Research] Georgia Institute of Technology(米国)

    • Related Report
      2021 Research-status Report
  • [Journal Article] Positive scalar curvature and homology cobordism invariants2023

    • Author(s)
      Hokuto Konno and Masaki Taniguchi
    • Journal Title

      Journal of Topology

      Volume: 16 Issue: 2 Pages: 679-719

    • DOI

      10.1112/topo.12299

    • Related Report
      2023 Research-status Report
    • Peer Reviewed
  • [Journal Article] A note on exotic families of 4-manifolds2023

    • Author(s)
      Tsuyoshi Kato, Hokuto Konno, and Nobuhiro Nakamura
    • Journal Title

      Proceedings of the American Mathematical Society

      Volume: 151 Pages: 2695-2705

    • DOI

      10.1090/proc/16356

    • Related Report
      2022 Research-status Report
    • Peer Reviewed
  • [Journal Article] The groups of diffeomorphisms and homeomorphisms of 4-manifolds with boundary2022

    • Author(s)
      Hokuto Konno, Masaki Taniguchi
    • Journal Title

      Advances in Mathematics

      Volume: 409 Pages: 108627-108627

    • DOI

      10.1016/j.aim.2022.108627

    • Related Report
      2022 Research-status Report
    • Peer Reviewed
  • [Journal Article] A note on generalized Thurston-Bennequin inequalities2022

    • Author(s)
      Nobuo Iida, Hokuto Konno, and Masaki Taniguchi
    • Journal Title

      International Journal of Mathematics

      Volume: 33 Issue: 14

    • DOI

      10.1142/s0129167x22500896

    • Related Report
      2022 Research-status Report
    • Peer Reviewed
  • [Journal Article] A cohomological Seiberg?Witten invariant emerging from the adjunction inequality2022

    • Author(s)
      Konno Hokuto
    • Journal Title

      Journal of Topology

      Volume: 15 Issue: 1 Pages: 108-167

    • DOI

      10.1112/topo.12215

    • Related Report
      2022 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] On the Bauer-Furuta and Seiberg-Witten invariants of families of 4‐manifolds2022

    • Author(s)
      Baraglia David、Konno Hokuto
    • Journal Title

      Journal of Topology

      Volume: 15 Issue: 2 Pages: 505-586

    • DOI

      10.1112/topo.12229

    • Related Report
      2022 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Constraints on families of smooth 4-manifolds from Pin-(2)-monopole2022

    • Author(s)
      Hokuto Konno, Nobuhiro Nakamura
    • Journal Title

      Algebraic & Geometric Topology

      Volume: -

    • Related Report
      2021 Research-status Report
    • Peer Reviewed
  • [Journal Article] A note on the Nielsen realization problem for K3 surfaces2021

    • Author(s)
      David Baraglia, Hokuto Konno
    • Journal Title

      Proceedings of the American Mathematical Society

      Volume: - Pages: 4079-4087

    • DOI

      10.1090/proc/15544

    • Related Report
      2023 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Presentation] From families Seiberg-Witten theory to corks2023

    • Author(s)
      Hokuto Konno
    • Organizer
      微分トポロジー '24
    • Related Report
      2023 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Infiniteness of 4-dimensional mapping class groups and characteristic classes2023

    • Author(s)
      Hokuto Konno
    • Organizer
      Topology seminar at the University of Texas at Austin
    • Related Report
      2023 Research-status Report
    • Invited
  • [Presentation] Spaces of positive scalar curvature metrics on 4-manifolds2023

    • Author(s)
      Hokuto Konno
    • Organizer
      Analysis, Geometry and Topology of Positive Scalar Curvature Metrics
    • Related Report
      2023 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Families Froyshov invariant2023

    • Author(s)
      Hokuto Konno
    • Organizer
      Brandeis Topology Seminar
    • Related Report
      2023 Research-status Report
    • Invited
  • [Presentation] Infiniteness of 4-dimensional mapping class groups and characteristic classes2023

    • Author(s)
      Hokuto Konno
    • Organizer
      Rutgers Geometry/Topology seminar
    • Related Report
      2023 Research-status Report
    • Invited
  • [Presentation] Families Froyshov invariant2023

    • Author(s)
      Hokuto Konno
    • Organizer
      Topology Seminar, Michigan State University
    • Related Report
      2023 Research-status Report
    • Invited
  • [Presentation] Diffeomorphism groups and gauge theory2023

    • Author(s)
      Hokuto Konno
    • Organizer
      RTG Seminar, Michigan State University
    • Related Report
      2023 Research-status Report
    • Invited
  • [Presentation] Families Froyshov invariant2023

    • Author(s)
      Hokuto Konno
    • Organizer
      Columbia Symplectic Geometry and Gauge Theory Seminar
    • Related Report
      2023 Research-status Report
    • Invited
  • [Presentation] Exotic Dehn twists on 4-manifolds2023

    • Author(s)
      Hokuto Konno
    • Organizer
      UMass Geometry and Topology Seminar
    • Related Report
      2023 Research-status Report
    • Invited
  • [Presentation] Exotic Dehn twists on 4-manifolds2023

    • Author(s)
      Hokuto Konno
    • Organizer
      AMS Sectional Meeting
    • Related Report
      2023 Research-status Report
    • Invited
  • [Presentation] Homological instability for moduli spaces of 4-manifolds2023

    • Author(s)
      Hokuto Konno
    • Organizer
      Mapping class groups: pronilpotent and cohomological approaches
    • Related Report
      2023 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Exotic Dehn twists on 4-manifolds2023

    • Author(s)
      Hokuto Konno
    • Organizer
      Gauge Theory and Topology: in Celebration of Peter Kronheimer's 60th Birthday
    • Related Report
      2023 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Exotic Dehn twists on 4-manifolds2023

    • Author(s)
      Hokuto Konno
    • Organizer
      Gauge theory and its application to geometry and low-dimensional topology
    • Related Report
      2023 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Exotic Dehn twists on 4-manifolds2023

    • Author(s)
      Hokuto Konno
    • Organizer
      Topology of 3- and 4-dimensional manifolds
    • Related Report
      2023 Research-status Report
    • Invited
  • [Presentation] Seiberg-Witten theory and moduli spaces of 4-manifolds and metrics2023

    • Author(s)
      Hokuto Konno
    • Organizer
      Differentialgeometrie im Grossen
    • Related Report
      2023 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Homological instability for moduli spaces of 4-manifolds2023

    • Author(s)
      Hokuto Konno
    • Organizer
      2023 Georgia Topology Conference
    • Related Report
      2023 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] An exotic diffeomorphism of a contractible 4-manifold2023

    • Author(s)
      Hokuto Konno
    • Organizer
      Boston Graduate Topology Seminar
    • Related Report
      2023 Research-status Report
    • Invited
  • [Presentation] Homological instability for moduli spaces of 4-manifolds2023

    • Author(s)
      Hokuto Konno
    • Organizer
      Symplectic Geometry, Gauge Theory, and Low-Dimensional Topology Seminar
    • Related Report
      2023 Research-status Report
    • Invited
  • [Presentation] Homological instability for moduli spaces of 4-manifolds2023

    • Author(s)
      Hokuto Konno
    • Organizer
      Gauge Theory in Kyoto
    • Related Report
      2022 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Homological instability for moduli spaces of 4-manifolds2023

    • Author(s)
      Hokuto Konno
    • Organizer
      Conference "Geometric Analysis"
    • Related Report
      2022 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Homological instability for moduli spaces of 4-manifolds2023

    • Author(s)
      Hokuto Konno
    • Organizer
      MIT Geometry and Topology Seminar
    • Related Report
      2022 Research-status Report
    • Invited
  • [Presentation] Homological instability for moduli spaces of 4-manifolds2023

    • Author(s)
      Hokuto Konno
    • Organizer
      Brandeis Topology Seminar
    • Related Report
      2022 Research-status Report
    • Invited
  • [Presentation] Homological instability for moduli spaces of 4-manifolds2022

    • Author(s)
      Hokuto Konno
    • Organizer
      The 7th Japan-China Geometry Conference
    • Related Report
      2022 Research-status Report
    • Invited
  • [Presentation] Cohomological Invariants in Gauge Theory for Families2022

    • Author(s)
      Hokuto Konno
    • Organizer
      Diffeomorphisms and Gauge Theory Seminar
    • Related Report
      2022 Research-status Report
    • Invited
  • [Presentation] Equivariant K-Theory2022

    • Author(s)
      Hokuto Konno
    • Organizer
      Homotopy Computations and uses in Floer Theory Seminar
    • Related Report
      2022 Research-status Report
    • Invited
  • [Presentation] Homological Instability in Dimension 42022

    • Author(s)
      Hokuto Konno
    • Organizer
      FHT Program Seminar
    • Related Report
      2022 Research-status Report
    • Invited
  • [Presentation] Dehn twists and Nielsen realization in dimension 42022

    • Author(s)
      Hokuto Konno
    • Organizer
      リーマン面に関連する位相幾何学
    • Related Report
      2022 Research-status Report
    • Invited
  • [Presentation] Seiberg-Witten Floer homotopy and its applications2022

    • Author(s)
      Hokuto Konno
    • Organizer
      第69回 幾何学シンポジウム 基調講演
    • Related Report
      2022 Research-status Report
    • Invited
  • [Presentation] Seiberg-Witten Floer homotopy2022

    • Author(s)
      Hokuto Konno
    • Organizer
      iTHEMS Math Seminar
    • Related Report
      2022 Research-status Report
    • Invited
  • [Presentation] Gauge theory and the diffeomorphism groups of 4-manifolds2022

    • Author(s)
      Hokuto Konno
    • Organizer
      京都大学 談話会
    • Related Report
      2022 Research-status Report
    • Invited
  • [Presentation] Floer K-theory for knots2022

    • Author(s)
      Hokuto Konno
    • Organizer
      Intelligence of Low-dimensional Topology
    • Related Report
      2022 Research-status Report
    • Invited
  • [Presentation] Nielsen realization, knots, and Seiberg-Witten (Floer) homotopy theory2022

    • Author(s)
      Hokuto Konno
    • Organizer
      トポロジー火曜セミナー
    • Related Report
      2022 Research-status Report
    • Invited
  • [Presentation] Floer K-theory for knots2022

    • Author(s)
      今野北斗
    • Organizer
      Stanford Topology Seminar
    • Related Report
      2021 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Seiberg-Witten Floer K-theory for knots2022

    • Author(s)
      今野北斗
    • Organizer
      CGP Seminar
    • Related Report
      2021 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Floer K-theory for knots2022

    • Author(s)
      今野北斗
    • Organizer
      Topology and Geometry of Low-dimensional Manifolds
    • Related Report
      2021 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Floer K-theory for knots2022

    • Author(s)
      今野北斗
    • Organizer
      Japan-Netherlands Joint Online Seminar
    • Related Report
      2021 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Positive scalar curvature and higher-dimensional families of Seiberg-Witten equations2022

    • Author(s)
      今野北斗
    • Organizer
      Seminar on Seiberg-Witten theory
    • Related Report
      2021 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Floer K-theory for knots2022

    • Author(s)
      今野北斗
    • Organizer
      MIT Geometry and Topology Seminar
    • Related Report
      2021 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Positive scalar curvature and homology cobordism invariants2021

    • Author(s)
      今野北斗
    • Organizer
      関西ゲージ理論セミナー
    • Related Report
      2021 Research-status Report
    • Invited
  • [Presentation] Involutions, knots, and Floer K-theory2021

    • Author(s)
      今野北斗
    • Organizer
      ハンドルセミナー’21
    • Related Report
      2021 Research-status Report
    • Invited
  • [Presentation] Exotic diffeomorphisms of 4-manifolds with boundary2021

    • Author(s)
      今野北斗
    • Organizer
      4次元トポロジー
    • Related Report
      2021 Research-status Report
  • [Presentation] Involutions, knots, and Floer K-theory2021

    • Author(s)
      今野北斗
    • Organizer
      関西ゲージ理論セミナー
    • Related Report
      2021 Research-status Report
    • Invited
  • [Remarks] ホームページ

    • URL

      https://www.ms.u-tokyo.ac.jp/~konno/index.html

    • Related Report
      2023 Research-status Report
  • [Remarks]

    • URL

      http://ithems-members.riken.jp/konno/index.html

    • Related Report
      2022 Research-status Report

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Published: 2021-04-28   Modified: 2024-12-25  

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