Algebraic and category-theoretic study of low-dimensional topology

• Project/Area Number: 24540077
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Geometry
• Research Institution: Kyoto University
• Principal Investigator: Habiro Kazuo 京都大学, 数理解析研究所, 准教授 (80346064)
• Project Period (FY): 2012-04-01 – 2016-03-31
• Project Status: Completed (Fiscal Year 2015)
• Budget Amount: ¥4,290,000 (Direct Cost: ¥3,300,000、Indirect Cost: ¥990,000); Fiscal Year 2014: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000); Fiscal Year 2013: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000); Fiscal Year 2012: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000)
• Keywords: 3次元多様体 / Kirby計算 / 量子群 / 圏化 / トレース / 量子不変量 / Hochschild-Mitchellホモロジー / 線形圏 / 2圏 / 巡回ホモロジー / カレント代数 / 絡み目 / 枠付き絡み目 / Kirbyカルキュラス / リボン元 / Borromean手術 / 複体

Outline of Final Research Achievements

I studied several topics on 3-dimensional topology and related algebra.
Jointly with T. Widmer, I studied Kirby calculus for null-homotopic framed links and null-homologous framed links in 3-manifolds. Jointly with A. Beliakova, Z. Guliyev, A. D. Lauda, B. Webster and M. Zivkovic, I studied categorified quantum groups, in particular the ribbon element, the traces and the pivotal structure. Jointly with T. Q. T. Le, for each finite dimensional complex simple Lie algebra g, I constructed an invariant of integral homology 3-spheres which unify the quantum g invariants at various roots of unity.

Research Products

• [Int'l Joint Research] チューリッヒ大学(スイス)
• [Int'l Joint Research] ジョージア工科大学/南カリフォルニア大学/バージニア大学(米国)
• [Int'l Joint Research] グルノーブル大学/ストラスブール大学(フランス)
• [Journal Article] Cyclicity for categorified quantum groups (2016)
  • Author(s): A. Beliakova, K. Habiro, A.D. Lauda, B. Webster
  • Journal Title: J. Algebra Volume: 452 Pages: 118-132
  • DOI: 10.1016/j.jalgebra.2015.11.041
  • Peer Reviewed / Open Access / Int'l Joint Research / Acknowledgement Compliant
• [Journal Article] Trace decategorification of categorified quantum sl(2) (2016)
  • Author(s): A. Beliakova, K. Habiro, A.D. Lauda, M. Zivkovic
  • Journal Title: Math. Ann. Volume: 未定
  • DOI: 10.1007/s00208-016-1389-y
  • Peer Reviewed / Open Access / Int'l Joint Research / Acknowledgement Compliant
• [Journal Article] Unified quantum invariants for integral homology spheres associated with simple Lie algebras (2016)
  • Author(s): K. Habiro, T.Q.T. Le
  • Journal Title: Geometry and Topology Volume: 未定
  • Peer Reviewed / Open Access / Int'l Joint Research / Acknowledgement Compliant
• [Journal Article] Trace as an alternative decategorification functor (2014)
  • Author(s): Anna Beliakova, Zaur Guliyev, Kazuo Habiro, Aaron D. Lauda
  • Journal Title: Acta Mathematica Vietnamica Volume: 39 Pages: 425-480
  • DOI: 10.1007/s40306-014-0092-x
  • Peer Reviewed / Acknowledgement Compliant
• [Journal Article] On Kirby calculus for null-homotopic framed links in 3-manifolds (2014)
  • Author(s): Kazuo Habiro, Tamara Widmer
  • Journal Title: Algebr. Geom. Topol. Volume: 14 Pages: 115-134
  • DOI: 10.2140/agt.2014.14.115
  • Peer Reviewed
• [Presentation] On the category of finitely generated free groups (2016)
  • Author(s): Kazuo Habiro
  • Organizer: Low dimensional topology and number theory VIII
  • Place of Presentation: 九州大学産学官連携イノベーションプラザ(福岡県福岡市）
  • Year and Date: 2016-03-22
  • Invited
• [Presentation] Category of bottom tangles with colored tangles (2015)
  • Author(s): Kazuo Habiro
  • Organizer: Seminaire Quantique
  • Place of Presentation: ストラスブール（フランス）
  • Year and Date: 2015-12-07
  • Invited
• [Presentation] Traces of categorified quantum groups (2015)
  • Author(s): Kazuo Habiro
  • Organizer: Braids, Configuration Spaces, and Quantum Topology
  • Place of Presentation: 東京大学大学院数理科学研究科（東京都目黒区）
  • Year and Date: 2015-09-08
  • Int'l Joint Research / Invited
• [Presentation] Trace decategorification of categorified quantum groups (2015)
  • Author(s): Kazuo Habiro
  • Organizer: Diagram algebra and topology
  • Place of Presentation: ホテルスポーツロッジ糸満(沖縄県糸満市）
  • Year and Date: 2015-08-04
  • Invited
• [Presentation] Quantum groups and topology (2015)
  • Author(s): Kazuo Habiro
  • Organizer: Diagram algebra and topology
  • Place of Presentation: ホテルスポーツロッジ糸満(沖縄県糸満市）
  • Year and Date: 2015-08-03
  • Invited
• [Presentation] Functors on Lagrangian cobordisms associated to ribbon Hopf algebras (2015)
  • Author(s): Kazuo Habiro
  • Organizer: New Developments in TQFT
  • Place of Presentation: オーフス(デンマーク）
  • Year and Date: 2015-07-30
  • Int'l Joint Research / Invited
• [Presentation] Trace and categorification (2015)
  • Author(s): Kazuo Habiro
  • Organizer: Categorification in Algebra, Geometry and Physics (A conference in honor of the 60th birthday of Christian Blanchet)
  • Place of Presentation: カルジェーズ(フランス）
  • Year and Date: 2015-05-06
  • Int'l Joint Research / Invited
• [Presentation] Traces of categorified quantum groups (2015)
  • Author(s): Kazuo Habiro
  • Organizer: Low dimensional topology and number theory VII
  • Place of Presentation: 産学官連携イノベーションプラザ（福岡県福岡市）
  • Year and Date: 2015-03-27
  • Invited
• [Presentation] Trace decategorification of categorified quantum sl(2) (2015)
  • Author(s): Kazuo Habiro
  • Organizer: Winter School on Representation Theory
  • Place of Presentation: 京都大学数理解析研究所（京都府京都市）
  • Year and Date: 2015-01-20
  • Invited
• [Presentation] Kirby calculus for null-homologous framed links in 3-manifolds (2014)
  • Author(s): 葉廣和夫
  • Organizer: 多様体のトポロジーの展望
  • Place of Presentation: 東京大学大学院数理科学研究科（東京都目黒区）
  • Year and Date: 2014-11-30
• [Presentation] Kirby calculus for null-homologous framed links in 3-manifolds (2014)
  • Author(s): Kazuo Habiro
  • Organizer: Knots and Low Dimensional Manifolds
  • Place of Presentation: BEXCO Convention & Exhibition Center II, Busan, 大韓民国
  • Year and Date: 2014-08-22
  • Invited
• [Presentation] Kirby calculus for null-homologous framed links in 3-manifolds (2014)
  • Author(s): Kazuo Habiro
  • Organizer: Modern Trends in Topological Quantum Field Theory
  • Place of Presentation: Erwin Schroedinger International Institute for Mathematical Physics, University of Vienna
  • Invited
• [Presentation] Kirby calculus over K(G,1) (2014)
  • Author(s): Kazuo Habiro
  • Organizer: Low dimensional topology and number theory VI
  • Place of Presentation: Fukuoka Soft Research Park Center
  • Invited
• [Presentation] Kirby calculus for null-homologous framed links in 3-manifolds (2013)
  • Author(s): Kazuo Habiro
  • Organizer: Quantum topology and hyperbolic geometry
  • Place of Presentation: Yasaka Saigon Nhatrang Resort Hotel
  • Invited
• [Presentation] Quantum fundamental groups of 3-manifolds (2013)
  • Author(s): Kazuo Habiro
  • Organizer: The 9th East Asian School of Knots and Related Topics
  • Place of Presentation: 東京大学大学院数理科学研究科
  • Invited
• [Presentation] Kirby calculus for null-homologous framed links in 3-manifolds (2013)
  • Author(s): Kazuo Habiro
  • Organizer: Low Dimensional Topology and Number Theory V
  • Place of Presentation: 福岡ソフトリサーチパークセンター
  • Invited
• [Presentation] On the structure of the mapping class category (2012)
  • Author(s): Kazuo Habiro
  • Organizer: Mapping Class Groups and Quantum Topology
  • Place of Presentation: Institut de Recherche Mathematique Avancee, Strasbourg(フランス)
  • Invited
• [Presentation] Quantum fundamental groups (2012)
  • Author(s): Kazuo Habiro
  • Organizer: Winter Braids III
  • Place of Presentation: University of Grenoble(フランス)
  • Invited
• [Remarks] Kazuo Habiro
  • URL: http://www.kurims.kyoto-u.ac.jp/~habiro/index.html
• [Remarks] 京都大学数理解析研究所 - -
  • URL: http://www.kurims.kyoto-u.ac.jp/ja/list/habiro.html
• [Remarks] Kazuo Habiro
  • URL: http://www.kurims.kyoto-u.ac.jp
• [Remarks] Kazuo Habiro
  • URL: http://www.kurims.kyoto-u.ac.jp/~habiro/
• [Remarks] Kazuo Habiro
  • URL: http://www.kurims.kyoto-u.ac.jp/~habiro/