Algebraic structures in 3-dimensional topology

• Project/Area Number: 19540077
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Single-year Grants
• Section: 一般
• Research Field: Geometry
• Research Institution: Kyoto University
• Principal Investigator: HABIRO Kazuo Kyoto University, 数理解析研究所, 講師 (80346064)
• Project Period (FY): 2007 – 2008
• Project Status: Completed (Fiscal Year 2008)
• Budget Amount: ¥1,820,000 (Direct Cost: ¥1,400,000、Indirect Cost: ¥420,000); Fiscal Year 2008: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000); Fiscal Year 2007: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
• Keywords: 位相幾何学 / 3次元多様体 / 絡み目 / 量子不変量 / Jacobi図 / Hopf代数 / Chevalley-Eilenberg複体 / Y-clasper / ホモロジー球面 / Kirbyの定理 / 完備化 / 解析関数 / 写像類群 / Torelli部分群 / Lie代数 / ラグランジアン同境 / ホモロジーシリンダー / Moyal-Weyl積

Research Abstract

3次元トポロジーのいくつかのトピックスについて研究を行った。特に主要な結果として、G.Massuyeau氏と共同で、曲面のホモロジーシリンダーのなすモノイドをY_n手術同値関係で割って得られる群を集めて構成されるLie代数が、ある種のJacobiダイアグラムのなすLie代数と同型であることを証明した。この同型により得られる曲面の写像類群のTorelli部分群の表現の代数的構造についても考察を行った。

Research Products

• [Journal Article] Jean-Baptiste Meilhan, Finite type invariants and Milnor invariants for Brunnian links (2008)
  • Author(s): Kazuo Habiro
  • Journal Title: International Journal of Mathematics 19 Pages: 747-766
  • Peer Reviewed
• [Journal Article] A functorial LMO invariant of Lagrangian cobordisms (2008)
  • Author(s): Dorin Cheptea, Kazuo Habiro, Gwenael Massuyeau
  • Journal Title: Geometry & Topology 12 Pages: 1091-1070
  • Peer Reviewed
• [Journal Article] A unified Witten-Reshetikhin-Turaev invariant of integral homology spheres (2008)
  • Author(s): Kazuo Habiro
  • Journal Title: Inventiones Mathematicae 171 Pages: 1-81
  • Peer Reviewed
• [Journal Article] A functorial LMO invariant for Lagrangian cobordisms (2008)
  • Author(s): D. Cheptea, K. Habiro, G. Massuyeau
  • Journal Title: Geometry & Topology 1 Pages: 1091-1170
  • Peer Reviewed
• [Journal Article] Finite type invariants and Milnor invariants for Brunnian links (2008)
  • Author(s): K. Habiro, J. -B. Meilhan
  • Journal Title: International Journal of Mathematics 19 Pages: 747-766
  • Peer Reviewed
• [Journal Article] A unified Witten-Reshetikhin-Turaev invariant for integral homology spheres (2008)
  • Author(s): Kazuo Habiro
  • Journal Title: Inventiones Mathematicae 171 Pages: 1-81
  • Peer Reviewed
• [Journal Article] A functorial LMO invariant for Lagrangian cobordisms (2008)
  • Author(s): Dorin Cheptea, Kazuo Habiro, Gwenael Massuyeau
  • Journal Title: Geometry and Topology(掲載決定) 12 Pages: 1091-1170
  • Peer Reviewed
• [Journal Article] Finite type invariants and Milnor invariants for Brunnian links (2008)
  • Author(s): Kazuo Habiro, Jean-Baptiste Meilhan
  • Journal Title: International Journal of Mathematics(掲載決定) 19 Pages: 747-766
  • Peer Reviewed
• [Journal Article] claspers and Goussarov-Vassiliev finite type invariants (2007)
  • Author(s): Kazuo Habiro, Brunnian links
  • Journal Title: Mathematical Proceedings of the Cambridge Philosophical Society 142 Pages: 459-468
  • Peer Reviewed
• [Journal Article] An integral form of the quantized enveloping algebra of sl_2 and its completions (2007)
  • Author(s): Kazuo Habiro
  • Journal Title: Journal of Pure and Applied Algebra 211 Pages: 265-292
  • Peer Reviewed
• [Journal Article] Brunnian links, claspers and Goussarov-Vassiliev finite type invariants (2007)
  • Author(s): Kazuo Habiro
  • Journal Title: Mathematical Proceedings of the Cambridge Philosophical Society 142 Pages: 459-468
  • Peer Reviewed
• [Presentation] On the Witten-Reshetikhin-Turaev invariant and analytic functions on roots of unity (2009)
  • Author(s): Kazuo Habiro
  • Organizer: Workshop "Low dimensional topology and number theory"
  • Place of Presentation: 福岡ソフトリサーチパークセンター
  • Year and Date: 2009-03-18
• [Presentation] Category of tangles and quantum invariants (2009)
  • Author(s): Kazuo Habiro
  • Organizer: The 6th RIMS-Kyoto University and SNU Joint Symposium on Mathematics
  • Place of Presentation: Seoul National University, Korea
  • Year and Date: 2009-02-17
• [Presentation] 絡み目と3次元多様体のクラスパーに沿った手術について (2008)
  • Author(s): 葉廣和夫
  • Organizer: 日本数学会秋季総合分科会
  • Place of Presentation: 東京工業大学
  • Year and Date: 2008-09-26
• [Presentation] On the unified Witten-Reshetikhin-Turaev invariant for integral homology spheres and its extension to Lagrangian cobordisms (2008)
  • Author(s): Kazuo Habiro
  • Organizer: Workshop "Finite type invariants, fatgraphs and Torelli-Johnson-Morita theory"
  • Place of Presentation: Center for the Topology and Quantization of Moduli Spaces (CTQM), Aarhus University, Denmark
  • Year and Date: 2008-03-29
• [Presentation] 底タングルの圏と量子不変量について (2008)
  • Author(s): 葉廣和夫
  • Organizer: 城崎新人セミナー
  • Place of Presentation: 豊岡市立健康福祉センター
  • Year and Date: 2008-02-19
• [Presentation] コボルディズムの圏と底タングルの圏におけるHopf代数構造 (2007)
  • Author(s): 葉廣和夫
  • Organizer: 信州トポロジーセミナー
  • Place of Presentation: 信州大学理学部
  • Year and Date: 2007-11-19
• [Presentation] Morse理論から得られるTQFTについて (2007)
  • Author(s): 葉廣和夫
  • Organizer: 研究集会「2+1次元トポロジーの新しい流れ」
  • Place of Presentation: 早稲田大学大久保キャンパス
  • Year and Date: 2007-11-14
• [Presentation] 圏B, L, Cにおいて定義される関手について (2007)
  • Author(s): 葉廣和夫
  • Organizer: 研究集会「2+1次元トポロジーの新しい流れ」
  • Place of Presentation: 早稲田大学大久保キャンパス
  • Year and Date: 2007-11-14
• [Presentation] Morse理論から得られるTQFTについて (2007)
  • Author(s): 葉廣和夫
  • Organizer: 大阪市大COE・京大COE若手合同発表会
  • Place of Presentation: 大阪市立大学文化交流センター
  • Year and Date: 2007-11-14
• [Presentation] 底タングルの圏Bとコボルディズムの圏L, Cについて (2007)
  • Author(s): 葉廣和夫
  • Organizer: 研究集会「2+1次元トポロジーの新しい流れ」
  • Place of Presentation: 早稲田大学大久保キャンパス
  • Year and Date: 2007-11-13
• [Presentation] Topological quantum field theories derived from non-semisimple spherical categories (2007)
  • Author(s): Kazuo Habiro
  • Organizer: Talks in Mathematics Physics
  • Place of Presentation: Institut fur Mathematik, Zurich University, Switzerland
  • Year and Date: 2007-06-28
• [Presentation] Non-semisimple TQFTs derived from spherical categories, Seminaire quantique (2007)
  • Author(s): Kazuo Habiro
  • Organizer: Institut de Recherche Mathematique Avancee
  • Place of Presentation: Strasbourg University I, France
  • Year and Date: 2007-06-25
• [Remarks]
  • URL: http://www.kurims.kyoto-u.ac.jp/~habiro/
• [Remarks]
  • URL: http://www.kurims.kyoto-u.ac.jp/ja/list/habiro.html