Algebraic and category-theoretic structures in low-dimensional topology

• Project/Area Number: 18H01119
• Research Category: Grant-in-Aid for Scientific Research (B)
• Allocation Type: Single-year Grants
• Section: 一般
• Review Section: Basic Section 11020:Geometry-related
• Research Institution: Kyoto University
• Principal Investigator: Habiro Kazuo 京都大学, 理学研究科, 教授 (80346064)
• Project Period (FY): 2018-04-01 – 2022-03-31
• Project Status: Completed (Fiscal Year 2022)
• Budget Amount: ¥4,810,000 (Direct Cost: ¥3,700,000、Indirect Cost: ¥1,110,000); Fiscal Year 2021: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000); Fiscal Year 2020: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000); Fiscal Year 2019: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000); Fiscal Year 2018: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000)
• Keywords: 写像類群 / Johnson準同型 / Yetter-Drinfeld加群 / タングル / 自由群のIA自己同型群 / 群コホモロジー / Hopf代数 / リボン圏 / IA自己同型群 / Hochschild-Serreスペクトル系列 / 3次元多様体 / 量子不変量 / Kontsevich不変量 / 群 / Johnsonフィルトレーション / Hochschild-Mitchellホモロジー / 線形圏 / 量子群 / 圏化

Outline of Final Research Achievements

Within the research period, I studied several topics in the field of low-dimensional topology and group cohomology. With Anderson Vera, I studied double filtrations of the mapping class groups of surfaces and the Johnson homomorphisms. With Yuka Kotorii, I studied ribbon Yetter-Drinfeld modules and quantum invariants of tangles. With Mai Katada, I studied the stable rational cohomology of the IA-automorphism groups of free groups and the Torelli subgroups of the mapping class groups of surfaces.

Academic Significance and Societal Importance of the Research Achievements

写像類群の2重フィルトレーションは、写像類群の構造を調べるための新しい構造を与えている。ribbon Yetter-Drinfeld加群の概念は、Yetter-Drinfeld加群からタングルや結び目の新しい不変量を構成する方法を与えている。自由群のIA自己同型群の安定有理コホモロジーについて我々が与えた予想は、この重要な群のコホモロジー構造についての見通しを与えていると考える。この予想を証明することが、この群のコホモロジーの研究における一つの重要なステップとなるだろうと考える。

Research Products

• [Int'l Joint Research] ブルゴーニュ大学(フランス)
• [Journal Article] The Kontsevich integral for bottom tangles in handlebodies (2021)
  • Author(s): Kazuo Habiro and Gwenael Massuyeau
  • Journal Title: Quantum Topology Volume: -
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] Generalized Johnson homomorphisms for extended N-series (2018)
  • Author(s): Kazuo Habiro, Gwenael Massuyeau
  • Journal Title: Journal of Algebra Volume: 510 Pages: 205-258
  • DOI: 10.1016/j.jalgebra.2018.05.031
  • Peer Reviewed / Int'l Joint Research
• [Presentation] Cyclic nerves of stratified categories (2019)
  • Author(s): Kazuo Habiro
  • Organizer: Johnson homomorphisms and related topics 2019
  • Int'l Joint Research / Invited
• [Presentation] Hochschild-Mitchell homology of stratified linear categories (2018)
  • Author(s): Kazuo Habiro
  • Organizer: Topology Festival, Cornell University
  • Int'l Joint Research / Invited
• [Presentation] CQ3MI Summer School and Workshop (2018)
  • Author(s): Unified quantum invariants of 3-manifolds
  • Organizer: University of Southern California
  • Int'l Joint Research / Invited
• [Remarks] Kazuo Habiro
  • URL: http://www.kurims.kyoto-u.ac.jp/~habiro/
• [Funded Workshop] The 15th East Asian Conference on Geometric Topology (2020)