Chern-Simons perturbation theory and its application to topology

2022 Fiscal Year Final Research Report

• Project/Area Number: 17K05252
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Geometry
• Research Institution: Kyoto University (2021-2022); Shimane University (2017-2020)
• Principal Investigator: Watanabe Tadayuki 京都大学, 理学研究科, 准教授 (70467447)
• Project Period (FY): 2017-04-01 – 2023-03-31
• Keywords: 微分同相群 / 配置空間 / グラフホモロジー / 4次元多様体 / 有限型不変量 / Morse理論 / ホモトピー群 / 可微分多様体

Outline of Final Research Achievements

Kontsevich constructed differential topological invariants of 3-manifolds and families of homology disks by configuration space integrals. We studied Kontsevich's invariants and their applications to topology, and obtained the following results.
(1) We proved that the group of relative diffeomorphisms of the 4-dimensional disk is not contractible, by using Kontsevich's configuration space integral invariant.
(2) We extended Kontsevich's configuration space integral invariant to some closed 4-manifolds equipped with non-trivial local coefficient systems. By using the extension, we found many non-trivial elements of the mapping class groups of some closed 4-manifolds.

Free Research Field

位相幾何学

Academic Significance and Societal Importance of the Research Achievements

4次元円板の相対微分同相の群のトポロジーは、多様体の局所構造に関するカテゴリーの差の根本に関わる基本的な研究対象であるが、その具体的な構造についてはほとんど何もわかっていない状態であった。そのホモトピー型が全く自明でないということを初めて明らかにしたことは学術的意義があると考える。