Structures of the cohomology rings of the Torelli group and Lagrangian mapping class groups

• Project/Area Number: 21740044
• Research Category: Grant-in-Aid for Young Scientists (B)
• Allocation Type: Single-year Grants
• Research Field: Geometry
• Research Institution: Tokyo Institute of Technology
• Principal Investigator: SAKASAI Takuya 東京工業大学, 大学院・理工学研究科, 助教 (60451902)
• Project Period (FY): 2009 – 2011
• Project Status: Completed (Fiscal Year 2011)
• Budget Amount: ¥2,340,000 (Direct Cost: ¥1,800,000、Indirect Cost: ¥540,000); Fiscal Year 2011: ¥650,000 (Direct Cost: ¥500,000、Indirect Cost: ¥150,000); Fiscal Year 2010: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000); Fiscal Year 2009: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000)
• Keywords: 位相幾何 / 写像類群 / トレリ群 / Johnson準同型 / 位相幾何学

Research Abstract

Johnson-Morita theory plays an important role in the study of subgroups of mapping class groups of surfaces called Lagrangian mapping class groups. We determined the images of the rational fifth and sixth Johnson homomorphisms and studied the structures of related Lie algebras(Joint work with Shigeyuki Morita and Masaaki Suzuki). In particular, we determined the abelianization of so-called "the associative case" and showed that the highest rational homology group of the moduli space of Riemann surfaces vanish.

Research Products

• [Journal Article] Lagrangian mapping class groups from a group homological point of view (2012)
  • Author(s): Takuya Sakasai
  • Journal Title: Algebraic & Geometric Topology Volume: 12 Pages: 267-291
  • Peer Reviewed
• [Journal Article] Homology cylinders and sutured manifolds for homologically fibered knots (2012)
  • Author(s): Hiroshi Goda, Takuya Sakasai
  • Journal Title: Tokyo Journal of Mathematics Volume: (印刷中)
  • Peer Reviewed
• [Journal Article] A survey of Magnus representations for mapping class groups and homology cobordisms of surfaces (2012)
  • Author(s): Takuya Sakasai
  • Journal Title: Handbook of Teichmueller theory Volume: 3巻(印刷中)
• [Journal Article] Johnson homomorphisms as fibering obstructions of homologically fibered knots (2011)
  • Author(s): Hiroshi Goda, Takuya Sakasai
  • Journal Title: 数理解析研究所講究録 Volume: 1747 Pages: 47-66
• [Journal Article] Factorization formulas and computations of higher-order Alexander invariants for homologically fibered knots (2011)
  • Author(s): Hiroshi Goda, Takuya Sakasai
  • Journal Title: Journal of Knot Theory and Its Ramifications Volume: 20 Pages: 1355-1380
  • Peer Reviewed
• [Journal Article] Abelian quotients of monoids of homology cylinders (2011)
  • Author(s): Hiroshi Goda, Takuya Sakasai
  • Journal Title: Geometriae Dedicata Volume: 151 Pages: 387-396
  • Peer Reviewed
• [Journal Article] Homology cylinders and sutured manifolds for homologically fibered knots
  • Author(s): Hiroshi Goda, Takuya Sakasai
  • Journal Title: Tokyo Journal of Mathematics Volume: (掲載確定)
  • Peer Reviewed
• [Journal Article] A survey of Magnus representations for mapping class groups and homology cobordisms of surfaces
  • Author(s): Takuya Sakasai
  • Journal Title: Handbook of Teichmueller theory Volume: 3(In press)
• [Presentation] 曲面の写像類群と自由結合代数のシンプレクティック微分リー代数I, II (2012)
  • Author(s): 逆井卓也
  • Organizer: 第6回福岡・札幌幾何学セミナー
  • Place of Presentation: 九州大学西新プラザ
• [Presentation] 曲面の写像類群と自由結合代数のシンプレクティック微分リー代数I, II (2012)
  • Author(s): 逆井卓也
  • Organizer: 第6回福岡・札幌幾何学セミナー
  • Place of Presentation: 九州大学西新プラザ(招待講演)
• [Presentation] 自由結合代数のシンプレクティック微分のなすリー代数のアーベル化とその応用 (2011)
  • Author(s): 逆井卓也
  • Organizer: 東京理科大学理工学部数学科談話会
  • Place of Presentation: 東京理科大学
  • Year and Date: 2011-12-09
• [Presentation] 自由結合代数のシンプレクティック微分のなすリー代数のアーベル化とその応用 (2011)
  • Author(s): 逆井卓也
  • Organizer: 東京理科大学理工学部数学科談話会
  • Place of Presentation: 東京理科大学(招待講演)
  • Year and Date: 2011-12-09
• [Presentation] The Magnus representation and homology cobordism groups of homology cylinders (2011)
  • Author(s): Takuya Sakasai
  • Organizer: GCOE workshop"Circle valued Morse theory and Alexander invariants"
  • Place of Presentation: 東京大学
  • Year and Date: 2011-11-18
• [Presentation] The Magnus representation and homology cobordism groups of homology cylinders (2011)
  • Author(s): Takuya Sakasai
  • Organizer: GCOE workshop "Circle valued Morse theory and Alexander invariants"
  • Place of Presentation: 東京大学(招待講演)
  • Year and Date: 2011-11-18
• [Presentation] 自由リー代数のシンプレクティック微分のなすリー代数の構造について (2011)
  • Author(s): 逆井卓也
  • Organizer: トポロジーとコンピュータ2011
  • Place of Presentation: 名城大学名駅サテライト
  • Year and Date: 2011-11-14
• [Presentation] On a structure of the symplectic derivation Lie algebra of free Lie algebra(ポスター発表) (2011)
  • Author(s): Shigeyuki Morita, Takuya Sakasai, Masaaki Suzuki
  • Organizer: Todai Forum 2011 "Geometry and Dynamics"
  • Place of Presentation: Ecole normale superieure de Lyon(フランス)
  • Year and Date: 2011-10-17
• [Presentation] On a structure of the symplectic derivation Lie algebra of free Lie algebra (2011)
  • Author(s): Takuya Sakasai
  • Organizer: Todai Forum 2011 "Geometry and Dynamics"
  • Place of Presentation: Ecole normale superieure de Lyon(フランス)(ポスター発表)
  • Year and Date: 2011-10-17
• [Presentation] 自由結合代数の微分のなすリー代数のアーベル化 (2011)
  • Author(s): 逆井卓也,鈴木正明
  • Organizer: リーマン面に関連する位相幾何学
  • Place of Presentation: 東京大学
  • Year and Date: 2011-09-03
• [Presentation] 自由結合代数の微分のなすリー代数のアーベル化 (2011)
  • Author(s): 逆井卓也
  • Organizer: リーマン面に関連する位相幾何学
  • Place of Presentation: 東京大学(招待講演)
  • Year and Date: 2011-09-03
• [Presentation] ohnson homomorphisms in knot theory (2010)
  • Author(s): Takuya Sakasai
  • Organizer: Teichmueller theory
  • Place of Presentation: Mathematisches Forschungsinstitut Oberwolfach(ドイツ)
  • Year and Date: 2010-12-03
• [Presentation] Johnson homomorphisms in knot theory (2010)
  • Author(s): Takuya Sakasai
  • Organizer: Teichmueller theory
  • Place of Presentation: Mathematisches Forschungsinstitut Oberwolfach(ドイツ)
  • Year and Date: 2010-12-03
• [Presentation] The Magnus representation and the homology cobordism group of homology cylinders (2010)
  • Author(s): Takuya Sakasai
  • Organizer: Teichmueller theory and its Interactions in Mathematics and Physics
  • Place of Presentation: Centre de Recerca Matematica, Bellaterra(スペイン)
  • Year and Date: 2010-07-01
• [Presentation] The Magnus representation and the homology cobordism group of homology cylinders (2010)
  • Author(s): Takuya Sakasai
  • Organizer: Teichmueller Theory and its Interactions in Mathematics and Physics
  • Place of Presentation: Centre de Recerca Matematica, Bellaterra(スペイン)
  • Year and Date: 2010-07-01
• [Presentation] Homology cylinders and knot theory (2010)
  • Author(s): Takuya Sakasai
  • Organizer: GCOE and Horiba International Symposium "Knots, Contact Geometry and Floer Homology"
  • Place of Presentation: 東京大学
  • Year and Date: 2010-05-24
• [Presentation] Homology cylinders and knot theory (2010)
  • Author(s): Takuya Sakasai
  • Organizer: GCOE and Horiba International Symposium, Knots, Contact Geometry and Floer Homology
  • Place of Presentation: 東京大学
  • Year and Date: 2010-05-24
• [Presentation] agrangian mapping class groups from group homological point of view (2009)
  • Author(s): Takuya Sakasai
  • Organizer: International Conference on Loops, strings and moduli spaces
  • Place of Presentation: 南開大学(中国)
  • Year and Date: 2009-08-03
• [Presentation] Lagrangian mapping class groups from group homological point of view (2009)
  • Author(s): 逆井卓也
  • Organizer: International Conference on Loops, strings and moduli spaces
  • Place of Presentation: 南開大学(中国)
  • Year and Date: 2009-08-03
• [Remarks]
  • URL: http://www.math.titech.ac.jp/~sakasai/
• [Remarks]
  • URL: http://www.math.titech.ac.jp/~sakasai/