An interpretation of finite type invariants from the viewpoint of the topology of embedding spaces

2019 Fiscal Year Final Research Report

• Project/Area Number: 16K05144
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Geometry
• Research Institution: Shinshu University
• Principal Investigator: Sakai Keiichi 信州大学, 学術研究院理学系, 准教授 (20466824)
• Project Period (FY): 2016-04-01 – 2020-03-31
• Keywords: 埋め込みの空間 / グラフ / 配置空間 / オペラッド / L無限代数

Outline of Final Research Achievements

I have studied the space of long embeddings between Euclidean spaces from the viewpoint of topology. One of the most important results that I have obtained in the duration is the characterization of the classifying space of the space of long knots (i.e. 1-dimensional embeddings) as the space of "short ropes". This is joint work with Syunji Moriya (Osaka Prefecture University). Other results include the construction of the cohomology classes of the space of long embeddings using L-infinity algebras and a geometric methods inspired by the "configuration space integrals".

Free Research Field

位相幾何学

Academic Significance and Societal Importance of the Research Achievements

Short ropeを用いたlong knotの空間の分類空間の特徴づけはMostovoyによる予想の肯定的解決である．結果自体はもちろん重要であるが，証明に用いた手法はGalatiusとRandal-Williamsらが「コボルディズム圏」の研究に使ったものであり，具体的な問題への応用の可能性を示したという意味でも意義深い．また数学にとどまらず諸分野への応用が期待される「結び目理論」について，本研究成果は1つの拡張として"short ropeの族の分類"を考えることの意味を与えており，その点でも重要である．