Diffeomorphism and homeomorphism groups of 4-manifolds and gauge theory for families

2022 Fiscal Year Final Research Report

• Project/Area Number: 19K23412
• Research Category: Grant-in-Aid for Research Activity Start-up
• Allocation Type: Multi-year Fund
• Review Section: 0201:Algebra, geometry, analysis, applied mathematics,and related fields
• Research Institution: The University of Tokyo (2020-2022); Institute of Physical and Chemical Research (2019)
• Principal Investigator: Konno Hokuto 東京大学, 大学院数理科学研究科, 助教 (20845614)
• Project Period (FY): 2019-08-30 – 2023-03-31
• Keywords: 族のゲージ理論 / Seiberg-Witten方程式 / 4次元多様体 / 微分同相群 / 多様体のモジュライ空間

Outline of Final Research Achievements

Mainly on the gauge theory for families, I constructed new frameworks of gauge theory related to diffeomorphism groups comprehensively and systematically, and was able to provide many geometric applications. In the comparison with the homeomorphism groups and diffeomorphism groups of 4-manifolds, which was the central theme, we obtained various results in variety. Furthermore, in the final year, we achieved homological instability in 4-dimension as a new and important application of family gauge theory. This captures the difference between other dimensions and 4-dimension at the level of the moduli space of manifolds. These are expected to be fundamental in future studies of gauge theory for families and diffeomorphism groups of 4-manifolds.

Free Research Field

幾何学

Academic Significance and Societal Importance of the Research Achievements

多様体のトポロジーにおいて，その対称性を記述する群である微分同相群は基本的な興味の対象である．各次元の多様体の微分同相群の研究は現在も急速に発展しつつある．他方，4次元多様体の分類論が他の次元と比較し特異的であることは，多様体のトポロジーにおける共通認識となっている．この研究の結果は，多様体の分類論で既に生じていた4次元の特異性を，微分同相群のレベルでの問題設定と解決の両面から確立して来たものと位置づけることができる．