Symmetries of spatial graphs by 3-manifold topology

2018 Fiscal Year Final Research Report

• Project/Area Number: 16K05163
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Geometry
• Research Institution: Kindai University
• Principal Investigator: IKEDA Toru 近畿大学, 理工学部, 教授 (00325408)
• Project Period (FY): 2016-04-01 – 2019-03-31
• Keywords: ３次元多様体 / デーン手術 / 空間グラフ / 対称性

Outline of Final Research Achievements

(1) We showed that a link in the 3-sphere is the fixed point set of a cyclic group action on a 3-manifold obtained by Dehn surgery, and gave a condition for a spatial graph in the 3-sphere to have a symmetry given by an involution with fixed point set being a closed surface.
(2) We proved that an orientation-reversing periodic diffeomorphism on a 3-manifold with a reduced fixed point set has a surgery description in either the 3-sphere, the circle-bundle over the 2-sphere, or the 3-torus.
(3) We gave a condition for an abstract graph with a symmetry given by a finite subgroup of the orthogonal group O(4) to admit a spatial embedding which is setwise invariant under the linear action on the three-dimensional sphere.

Free Research Field

３次元多様体論

Academic Significance and Societal Importance of the Research Achievements

３次元多様体論の基盤となる重要な事実の一つとして，向き付け可能閉３次元多様体が３次元球面内の枠付き絡み目で記述されることが知られており，これに基づいて多くの理論が展開されている。本研究により枠付き絡み目に反映される３次元多様体の幾何学的性質を新たに示し，３次元多様体論の視界を広げることに貢献できた。また，先行研究に比べて汎用性の高いグラフ対称性の実現方法を提案したため，複雑な空間グラフや様々なグラフ対称性を扱うための研究の手段を提供することができた。