Construction of the covering monopole map and its applications

• Project/Area Number: 17H02841
• Research Category: Grant-in-Aid for Scientific Research (B)
• Allocation Type: Single-year Grants
• Section: 一般
• Research Field: Geometry
• Research Institution: Kyoto University
• Principal Investigator: Kato Tsuyoshi 京都大学, 理学研究科, 教授 (20273427)
• Co-Investigator(Kenkyū-buntansha): 上 正明 京都大学, 理学研究科, 教授 (80134443)
• Project Period (FY): 2017-04-01 – 2021-03-31
• Project Status: Completed (Fiscal Year 2022)
• Budget Amount: ¥14,690,000 (Direct Cost: ¥11,300,000、Indirect Cost: ¥3,390,000); Fiscal Year 2020: ¥3,900,000 (Direct Cost: ¥3,000,000、Indirect Cost: ¥900,000); Fiscal Year 2019: ¥3,380,000 (Direct Cost: ¥2,600,000、Indirect Cost: ¥780,000); Fiscal Year 2018: ¥3,640,000 (Direct Cost: ¥2,800,000、Indirect Cost: ¥840,000); Fiscal Year 2017: ¥3,770,000 (Direct Cost: ¥2,900,000、Indirect Cost: ¥870,000)
• Keywords: サイバーグ・ウィッテン理論 / 被覆モノポール写像 / 有限伝播性を持つユニタリ作用素 / 非コンパクト空間 / ファイバー束の微分構造 / シンプルタイプ予想 / バウアー・古田理論 / 有限伝播性ユニタリ作用素 / 分類空間 / サイバーグ・ウイッテン理論 / 非可換幾何学 / Bauer-Furuta理論 / K理論 / ファイバー束 / 調和振動子 / 有限伝搬ユニタリ作用素 / 特性類 / Bauer-Fruta理論 / 無限群 / ゲージ理論 / 無限次元Bott周期性

Outline of Final Research Achievements

1. We found out the covering monopole degree for the infinite cyclic covering case. 2. We determined homotopy type of the group of finitely propagated unitary operators. 3. We verified non-existence of complete Riemannian metrics with certain properties on the exotic R^4 by deforming SW moduli spaces on K3 surface. 4. We presented an example of a compact topological fiber bundle whose total space, the fiber and the base are all smoothable but is non-smoothable as a fiber bundle by applying the family version of SW theory. 5. We introduced mod2 version of the simple type conjecture in SW theory and verified it for some classes of smooth 4-manifolds.

Academic Significance and Societal Importance of the Research Achievements

4次元多様体上の微分構造に対して、さまざまなアプローチによる研究を遂行した。特に、族の4次元多様体上の微分構造や、被覆空間上の微分構造に関して、新しい視点を与えた。また、SW不変量の情報を落としたところで、代数トポロジーでこれまで深く研究されてきたホモトピー論を適用することが本格的に可能であることを示したことで、今後の代数トポロジーと微分トポロジーのより深い融合研究への突破口を与えた。

Research Products

• [Int'l Joint Research] Brandeis University/Vanderbildt University/Pensilvania State University(米国)
• [Int'l Joint Research] Seoul National University(韓国)
• [Int'l Joint Research] Regensburg University(ドイツ)
• [Int'l Joint Research] ペンシルバニア州立大学(米国)
• [Int'l Joint Research] 復旦大学/華東師範大学(中国)
• [Int'l Joint Research] ECNU(中国)
• [Int'l Joint Research] Fudan University/East China Normal University(中国)
• [Int'l Joint Research] Seoul National University(韓国)
• [Int'l Joint Research] Vanderbilt University/Pennsylvania State University(米国)
• [Journal Article] Upper bounds for virtual dimensions of Seiberg-Witten moduli spaces (2023)
  • Author(s): T.Kato, D.Kishimoto, N.Nakamura, K.Yasui
  • Journal Title: arXiv Volume: 2111(15201) Pages: 1-20
• [Journal Article] Homotopy types of spaces of finite propagation unitary operators on Z (2022)
  • Author(s): T.Kato, D.Kishimoto, M.Tsutaya
  • Journal Title: Homotopy, homology and applications (accepted) Volume: 1 Pages: 1-22
  • Peer Reviewed
• [Journal Article] Vector fields on non-compact manifolds (2022)
  • Author(s): T.Kato, D.Kishimoto, M.Tsutaya
  • Journal Title: arXiv Volume: 2211(00512) Pages: 11-11
• [Journal Article] A note on exotic families of 4-manifolds (2022)
  • Author(s): T.Kato, H.Konno, N.Nakamura
  • Journal Title: Proc. AMS (accepted) Volume: 1 Pages: 1-11
  • Peer Reviewed
• [Journal Article] A note on exotic families of 4-manifolds (2021)
  • Author(s): T. Kato, N. Nakamura, H. Konno
  • Journal Title: arXiv Volume: 2101.00367 Pages: 1-3
• [Journal Article] Homotopy type of the unitary group of the uniform Roe algebra on Z^n (2021)
  • Author(s): Tsuyoshi Kato, Daisuke Kishimoto, Mitsunobu Tsutaya
  • Journal Title: Journal of Topology and Analysis Volume: 1 Pages: 1-15
  • Peer Reviewed
• [Journal Article] Homotopy types of spaces of finite propagation unitary operators on Z (2020)
  • Author(s): T. Kato, D. Kishimoto, M. Tsutaya
  • Journal Title: arXiv Volume: 2007.05787 Pages: 1-22
• [Journal Article] Twisted Donaldson invariants (2020)
  • Author(s): T.Kato, H.Sasahira, H.Wang
  • Journal Title: Mathematical Proceedings of the Cambridge Philosophical Society Volume: 1 Pages: 1-52
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] The simple type conjecture for mod 2 Seiberg-Witten invariants (2020)
  • Author(s): T. Kato, N. Nakamura, K. Yasui
  • Journal Title: Journal of the European Mathematical Society Volume: 1 Pages: 1-8
  • Peer Reviewed
• [Journal Article] L2 harmonic forms and the Seiberg-Witten map on non compact four manifolds (2018)
  • Author(s): Tsuyoshi Kato
  • Journal Title: arXiv Volume: 1807.06741 Pages: 1-26
• [Journal Article] Higher Nahm transform in non commutative geometry (2018)
  • Author(s): T. Kato, H. Sasahira and H. Wang
  • Journal Title: arXiv Volume: 1807.08239 Pages: 1-26
  • Int'l Joint Research
• [Journal Article] Induced map on K theory for certain Γ-equivariant maps between Hilbert spaces (2018)
  • Author(s): Tsuyoshi Kato
  • Journal Title: arXiv Volume: 1802.00701 Pages: 1-34
  • Open Access
• [Journal Article] Twisted Donaldson invariants (2018)
  • Author(s): Tsuyoshi Kato, Hirofumi Sasahira and Hang Wang
  • Journal Title: arXiv Volume: 1709.08861 Pages: 1-46
  • Open Access
• [Presentation] Covering monopole map and aspherical 10/8-conjecture (2023)
  • Author(s): T.Kato
  • Organizer: Conference `Geometric Analysis' at Universitat Re Germanygensburg,
  • Int'l Joint Research / Invited
• [Presentation] Higher Nahm transform in non commutative geometry (2018)
  • Author(s): T. Kato
  • Organizer: 2018 Spring operator algebra program at ECNU
  • Int'l Joint Research / Invited
• [Presentation] Higher Nahm transform in non commutative geometry (2018)
  • Author(s): T. Kato
  • Organizer: Workshop on Noncommutative Geometry and Representation Theory/四川大学
  • Int'l Joint Research / Invited
• [Presentation] Twisted Donaldson invariant (2018)
  • Author(s): T. Kato
  • Organizer: Colloquium talk at University of New South Wales
  • Invited
• [Presentation] Twisted Donaldson invariant (2018)
  • Author(s): t. Kato
  • Organizer: Gauge Theory and Applications at Regensburg
  • Int'l Joint Research / Invited
• [Presentation] Non commutative geometry and gauge theory (2018)
  • Author(s): T. Kato
  • Organizer: Geometry, Analysis, Groups at Euler Institute, St Petersburg
  • Int'l Joint Research / Invited
• [Presentation] Twisted Donaldson invariant (2018)
  • Author(s): Tsuyoshi Kato
  • Organizer: Non commutative geometry seminar at Pennsilvania State University
  • Invited
• [Presentation] Twisted Donaldson invariant (2018)
  • Author(s): Tsuyoshi Kato
  • Organizer: Topology/Geometry lectures at Seoul National Univ
  • Invited
• [Presentation] Twisted Donaldson invariant (2018)
  • Author(s): Tsuyoshi Kato
  • Organizer: Gauge theory in Fukuoka
  • Int'l Joint Research
• [Remarks] Gauge theory in Kyoto
  • URL: https://gauge-theory.wixsite.com/kyoto2023
• [Remarks] Gauge theory seminar
  • URL: https://kansai-gauge.squares.net/index.html
• [Remarks] グローバル非可換幾何学セミナー；アジア・環太平洋地域
  • URL: https://globalncgseminar.org/organizers/
• [Remarks] 京都大学教育研究活動データベース
  • URL: https://kyouindb.iimc.kyoto-u.ac.jp/j/cA2iC
• [Funded Workshop] Geometry and Everything (2019)
• [Funded Workshop] East Asian conference on Gauge theory and related topics (2018)
• [Funded Workshop] K theory and index theory (2018)
• [Funded Workshop] Gauge theory in Fukuoka (2018)