High dimensional deformations of linear representations and distribution and complexity of essential surfaces

2022 Fiscal Year Final Research Report

• Project/Area Number: 18KK0380
• Research Category: Fund for the Promotion of Joint International Research (Fostering Joint International Research (A))
• Allocation Type: Multi-year Fund
• Review Section: Basic Section 11020:Geometry-related
• Research Institution: The University of Tokyo
• Principal Investigator: KITAYAMA Takahiro 東京大学, 大学院数理科学研究科, 准教授 (10700057)
• Project Period (FY): 2019 – 2022
• Keywords: 3次元多様体 / 位相不変量 / 表現 / 交叉形式

Outline of Final Research Achievements

We studied deformations of linear representations of the fundamental group and corresponding behaviors of topological invariants in order to describe distribution and complexity of subsurfaces essentially decomposing a 3-manifold. On twisted Alexander polynomials we showed for a certain class of groups including 3-manifold groups a generalization of the fact that the Thurston norm is uniformly detected, and discovered an obstruction for two knots to be ribbon concordant. On the Blanchfield form we gave a lower bound on the Gordian distance of knots, and presented another proof for a formula of the topological integral 4-ball genus. From the point of view of arithmetic topology we introduced analogues of algebraic p-adic L-functions associated with universal deformations of representations of a knot group.

Free Research Field

位相幾何学

Academic Significance and Societal Importance of the Research Achievements

本研究は、線形表現のなす空間の幾何学の低次元トポロジーへの応用を基礎付けるとともに、当研究領域の育成を図るものである。本研究によって、表現に付随する位相不変量による3次元多様体のトポロジーの理解が進展し、それら不変量の新たな応用が提示された。また、本研究は4次元トポロジーと数論的トポロジーとも関わる研究へと展開した。レーゲンスブルク大学におけるStefan Friedl氏との共同研究を通じて、当該分野における今後の国際的連携の基盤構築に繋がる学術交流を深めることができた。