Invariants of low dimensional manifolds from infinite dimensional dynamics in gauge theory and homotopy theory

• Project/Area Number: 19K03493
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Review Section: Basic Section 11020:Geometry-related
• Research Institution: Kyushu University
• Principal Investigator: Sasahira Hirofumi 九州大学, 数理学研究院, 教授 (30466825)
• Project Period (FY): 2019-04-01 – 2024-03-31
• Project Status: Completed (Fiscal Year 2023)
• Budget Amount: ¥4,290,000 (Direct Cost: ¥3,300,000、Indirect Cost: ¥990,000); Fiscal Year 2022: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000); Fiscal Year 2021: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000); Fiscal Year 2020: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000); Fiscal Year 2019: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
• Keywords: Seiberg-Witten理論 / Floer理論 / Seiberg-Witten Floer理論 / 低次元トポロジー / ホモトピー論 / 安定ホモトピー / Seiberg-Witten方程式 / ホモトピー理論 / ゲージ理論

Outline of Research at the Start

本研究は３次元多様体、４次元多様体と呼ばれる図形の分類や性質を調べることを目的としている。その方法として、ゲージ理論とよばれる理論に由来する偏微分方程式を用いる。その偏微分方程式からうまく３次元、４次元多様体の情報を取り出すことが重要である。本研究では、Floer理論やホモトピー論というものを用いて情報を取り出す。

Outline of Final Research Achievements

I studied the Seiberg-Witten Floer stable homotopy type which is an invariant of closed 3-manifolds. The invariant had been defined only for 3-manifolds with first Betti number zero and computed only for a few types of Seifert fibered spaces. I have studied to extend the invariant to a class of 3-manifolds and to establish a way to compute for more 3-manifolds.

Academic Significance and Societal Importance of the Research Achievements

シンプレクティック幾何学や結び目理論などに関するFloer理論では、ホモロジー的不変量からホモトピー的不変量への精密化が盛んに研究され、成果が出つつある。本研究ではSeiberg-Witten Floer理論において、その研究の流れを進めることができた。

Research Products

• [Int'l Joint Research] ミシガン州立大学(米国)
• [Int'l Joint Research] テキサス大学(米国)
• [Int'l Joint Research] Michigan state university(米国)
• [Int'l Joint Research] Michigan state university(米国)
• [Int'l Joint Research] ミシガン州立大学(米国)
• [Int'l Joint Research] Massachusetts institute of technology(米国)
• [Journal Article] Unfolded Seiberg?Witten Floer spectra, II: Relative invariants and the gluing theorem (2023)
  • Author(s): Khandhawit Tirasan、Lin Jianfeng、Sasahira Hirofumi
  • Journal Title: Journal of Differential Geometry Volume: 124 Issue: 2 Pages: 231-316
  • DOI: 10.4310/jdg/1686931602
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] Twisted Donaldson invariants (2021)
  • Author(s): T. Kato, H. Sasahira, H.Wang
  • Journal Title: Mathematical Proceedings of the Cambridge Philosophical Society Volume: 171 Issue: 3 Pages: 515-568
  • DOI: 10.1017/s0305004121000013
  • Peer Reviewed / Int'l Joint Research
• [Presentation] Seiberg-Witten Floer stable homotopy type and its applications to Corks and the intersection forms of 4-manifolds (2023)
  • Author(s): 笹平 裕史
  • Organizer: 変換群論の新潮流
  • Invited
• [Presentation] Surgery exact triangle for Seiberg-Witten Floer stable homotopy type (2023)
  • Author(s): 笹平 裕史
  • Organizer: Gauge theory in Kyoto
  • Int'l Joint Research / Invited
• [Presentation] 第一Betti数が正の３次元多様体のSeiberg-Witten-Floer安定ホモトピー型とトポロジーへの応用 (2021)
  • Author(s): 笹平 裕史
  • Organizer: ４次元トポロジー
• [Presentation] Freed-Uhlenbeck 第１０章の解説 (2021)
  • Author(s): 笹平 裕史
  • Organizer: 微分トポロジー
  • Invited
• [Presentation] Seiberg-Witten-Floer安定ホモトピー型 (2019)
  • Author(s): 笹平 裕史
  • Organizer: 幾何学シンポジウム
  • Invited
• [Presentation] Seiberg-Witten-Floer stable homotopy type for 3-manifolds with positive first Betti number (2019)
  • Author(s): 笹平 裕史
  • Organizer: 第1回日独友好幾何学研究集会
  • Int'l Joint Research / Invited
• [Book] サイバーグ-ウィッテン方程式 (2024)
  • Author(s): 笹平 裕史
  • Total Pages: 144
  • Publisher: サイエンス社