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

• 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
• Project Status: Completed (Fiscal Year 2019)
• Budget Amount: ¥3,640,000 (Direct Cost: ¥2,800,000、Indirect Cost: ¥840,000); Fiscal Year 2019: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000); Fiscal Year 2018: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000); Fiscal Year 2017: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000); Fiscal Year 2016: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
• 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 / Haefliger不変量 / 埋め込みの空間の位相幾何学 / グラフ複体 / Lie環のコホモロジー / 双対構成 / 埋め込みのなす空間 / 分類空間 / 群完備化 / 幾何学的トポロジー / 代数的トポロジー / 配置空間積分 / ジェネリック平面曲線 / 有限型不変量 / ループ空間

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".

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．

Research Products

• [Journal Article] Generalized connected sum formula for the Arnold invariants of generic plane curves (2019)
  • Author(s): Keiichi Sakai, Ryutaro Sugiyama
  • Journal Title: Topology and its Applications Volume: 255 Pages: 86-108
  • DOI: 10.1016/j.topol.2018.12.010
  • Peer Reviewed / Open Access
• [Journal Article] The space of short ropes and the classifying space of the space of long knots (2018)
  • Author(s): Syunji Moriya, Keiichi Sakai
  • Journal Title: Algebraic and Geometric Topology Volume: 18 Issue: 5 Pages: 2859-2873
  • DOI: 10.2140/agt.2018.18.2859
  • NAID: 120007173398
  • Peer Reviewed / Open Access
• [Journal Article] The space of short ropes and the classifying space of the space of long knots (2018)
  • Author(s): Syunji Moriya and Keiichi Sakai
  • Journal Title: Algebraic and Geometric Topology Volume: 印刷中
  • NAID: 120007173398
  • Peer Reviewed
• [Presentation] Lin-Wang type formula for the Haefliger invariant (2019)
  • Author(s): Keiichi Sakai
  • Organizer: Spaces of Embeddings: Connections and Applications
  • Int'l Joint Research
• [Presentation] L-infinity algebras and graph cocycles (2019)
  • Author(s): 境圭一
  • Organizer: ホモトピー論シンポジウム2019
  • Invited
• [Presentation] Lin-Wang type formula for Haefliger's invariant (2018)
  • Author(s): 境 圭一
  • Organizer: トポロジー金曜セミナー
  • Invited
• [Presentation] The space of short ropes and the classifying space of the space of long knots (2018)
  • Author(s): 境 圭一
  • Organizer: 第65回トポロジーシンポジウム
  • Invited
• [Presentation] The space of short ropes and the classifying space of the space of long knots (2017)
  • Author(s): Keiichi Sakai
  • Organizer: The 59th Cascade Topology Seminar
  • Int'l Joint Research / Invited
• [Presentation] The space of short ropes and the classifying space of the space of long knots (2017)
  • Author(s): 境 圭一
  • Organizer: 東京大学トポロジー火曜セミナー
  • Invited
• [Presentation] The space of short ropes and the classifying space of the space of long knots (2017)
  • Author(s): 境 圭一
  • Organizer: 京都大学代数トポロジーセミナー
  • Invited
• [Presentation] The space of short ropes and the classifying space of the space of long knots (2017)
  • Author(s): 境 圭一
  • Organizer: 信州トポロジーセミナー
  • Invited
• [Presentation] Generic閉曲線の一般化された連結和に対するArnold不変量の公式 (2017)
  • Author(s): 境 圭一
  • Organizer: 日本数学会2017年度秋季総合分科会
• [Presentation] The space of short ropes and the classifying space of the space of long knots (2016)
  • Author(s): 境 圭一
  • Organizer: 研究集会「結び目の数学IX」
  • Place of Presentation: 日本大学理学部
  • Year and Date: 2016-12-22
• [Remarks] Website of Keiichi Sakai
  • URL: http://math.shinshu-u.ac.jp/~ksakai/index_j.html
• [Remarks] Webpage of Keiichi Sakai
  • URL: http://math.shinshu-u.ac.jp/~ksakai/index.html