Pin(2)-monopole equations and 4-dimensional topology

2023 Fiscal Year Final Research Report

• Project/Area Number: 19K03506
• 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: Fukushima Medical University (2022-2023); Osaka Medical and Pharmaceutical University (2019-2021)
• Principal Investigator: Nobuhiro Nakamura 福島県立医科大学, 公私立大学の部局等, 教授 (10512171)
• Project Period (FY): 2019-04-01 – 2024-03-31
• Keywords: ゲージ理論 / 4次元 / トポロジー

Outline of Final Research Achievements

Considering Pin(2)-monopole equations on families of 4-manifolds, we obtained some applications on topology of diffeomorphism groups of 4-manifolds. We proved a non-vanishing theorem of families Seiberg-Witten invariants for spin families of 4-manifolds. As an application, we construct topological fiber bundles such that their total spaces are smoothable, but they are non-smoothable as fiber bundles. By using spin bordism invariants, we prove the mod 2 version of the simple type conjecture is true under some topological conditions.

Free Research Field

ゲージ理論と4次元トポロジー

Academic Significance and Societal Importance of the Research Achievements

我々の行った族の研究は，Pin(2)モノポール方程式によるものも，スピン多様体に対するものも，族のゲージ理論を実質的に発展させるものである．さらにその応用は微分同相群に対する新たな知見を切り拓くものものとなっている点が意義深い．また mod 2 simple type についての研究は，この四半世紀ほとんど進展が見られなかった simple type 予想を着実に前進させるものである点が意義深い．