Coho mology of discrete groups and characteristic classes of flat bundles

• Project/Area Number: 20340008
• Research Category: Grant-in-Aid for Scientific Research (B)
• Allocation Type: Single-year Grants
• Section: 一般
• Research Field: Geometry
• Research Institution: Hokkaido University
• Principal Investigator: AKITA Toshiyuki 北海道大学, 大学院・理学研究院, 准教授 (30279252)
• Co-Investigator(Kenkyū-buntansha): OHMOTO Toru 北海道大学, 大学院・理学研究院, 准教授 (20264400) ; WATANABE Tadayuki 北海道大学, 大学院・理学研究院, 助教 (70467447) ; YOSHIDA Tomoyuki 北海道大学, 大学院・理学研究院, 教授 (30002265)
• Co-Investigator(Renkei-kenkyūsha): KURIBAYASHI Katsuhiko 信州大学, 理学部, 教授 (40249751) ; YAGITA Nobuaki 茨城大学, 教育学部, 教授 (20130768)
• Project Period (FY): 2008 – 2011
• Project Status: Completed (Fiscal Year 2011)
• Budget Amount: ¥12,480,000 (Direct Cost: ¥9,600,000、Indirect Cost: ¥2,880,000); Fiscal Year 2011: ¥2,990,000 (Direct Cost: ¥2,300,000、Indirect Cost: ¥690,000); Fiscal Year 2010: ¥2,990,000 (Direct Cost: ¥2,300,000、Indirect Cost: ¥690,000); Fiscal Year 2009: ¥2,990,000 (Direct Cost: ¥2,300,000、Indirect Cost: ¥690,000); Fiscal Year 2008: ¥3,510,000 (Direct Cost: ¥2,700,000、Indirect Cost: ¥810,000)
• Keywords: コホモロジー / 特性類 / 線型表現 / 写像類群 / 分岐被覆 / 離散群 / 群のコホモロジー / 平坦束 / Mackey 函手 / トポロジー / 分類空間 / Mackey函手 / 幾何学

Research Abstract

We construct a functorial framework to deal with all finite transformation groups on closed surfaces (Galois covers on closed surfaces) simultaneously. Characteristic classes associated with such transformation groups, Mumford-Morita-Miller classes and Newton classes, are shown to be natural transformations of functors. These characteristic classes are shown to satisfy Riemann-Roch type formulae.

Research Products

• [Journal Article] 1-loop graphs and configuration space integral for embedded spaces (2012)
  • Author(s): Keiichi Sakai, Tadayuki Watanabe
  • Journal Title: Mathematical Proceedings of the Cambridge Philosophical Society Volume: 152 Issue: 3 Pages: 497-533
  • DOI: 10.1017/s0305004111000429
  • NAID: 120005228644
  • Peer Reviewed
• [Journal Article] Periodicity for Mumford-Morita-Miller classes of surface symmetries (2011)
  • Author(s): Toshiyuki Akita
  • Journal Title: Publications of the Research Institute for Mathematical Sciences Volume: 47 Issue: 4 Pages: 897-909
  • DOI: 10.2977/prims/55
  • NAID: 110008747020
  • Peer Reviewed
• [Journal Article] Periodicity for Mumford-Morita-Miller classes of surface symmetries (2011)
  • Author(s): Toshiyuki Akita
  • Journal Title: Publ.RIMS Kyoto Univ. 印刷中
  • NAID: 110008747020
  • Peer Reviewed
• [Journal Article] Periodicity for Mumford-Morita-Miller classes for surface symmetries (2011)
  • Author(s): Toshiyuki Akita
  • Journal Title: Publ.RIMS Kyoto Univ. Volume: 印刷中
  • Peer Reviewed
• [Journal Article] 1-loop graphs and configuration space integral for embedded spaces (2011)
  • Author(s): Keiichi Sakai, Tadayuki Watanabe
  • Journal Title: Math.Proc.Cambridge Phil.Soc. Volume: 印刷中
  • NAID: 120005228644
  • Peer Reviewed
• [Journal Article] On Kontsevich's characteristic classes for higher dimensional sphere bundles. II. Higher claases (2009)
  • Author(s): Tadayuki Watanabe
  • Journal Title: Journal of Topology Volume: 2 Issue: 3 Pages: 624-660
  • DOI: 10.1112/jtopol/jtp024
  • Peer Reviewed
• [Journal Article] On Kontsevich's characteristic classes for higher dimensional sphere bundles. I. The simplest class (2009)
  • Author(s): Tadayuki Watanabe
  • Journal Title: Mathematische Zeitschrift Volume: 262 Issue: 3 Pages: 683-712
  • DOI: 10.1007/s00209-008-0396-4
  • Peer Reviewed
• [Journal Article] Thom polynomial and around, RIMS Kokyuroku Bessatsu B11 (2009)
  • Author(s): Toru Ohmoto
  • Journal Title: New Trends in Combinatorial Representation Theory (eds. S. Arikiand M. Okado) Pages: 75-86
  • Peer Reviewed
• [Journal Article] 特異点の数え上げと同変Chern類 (2009)
  • Author(s): 大本亨
  • Journal Title: 数学 Volume: 61 Pages: 21-39
  • Peer Reviewed
• [Journal Article] On Kontsevich's characteristic classes for higher-dimensional sphere bundles. II. Higher classes. (2009)
  • Author(s): Tadayuki Watanabe
  • Journal Title: J.Topol. 2 Pages: 624-660
  • Peer Reviewed
• [Journal Article] On mod p Riemann-Roch formulae for mapping class groups (2008)
  • Author(s): Toshiyuki Akita
  • Journal Title: Advanced Studies in Pure Mathematics Volume: 52 Pages: 111-118
  • Peer Reviewed
• [Journal Article] A formula for the Euler characteristics of evendimensional triangulated manifolds (2008)
  • Author(s): Toshiyuki Akita
  • Journal Title: Proceedings of the American Mathematical Society Volume: 136 Issue: 07 Pages: 2571-2573
  • DOI: 10.1090/s0002-9939-08-09148-x
  • Peer Reviewed
• [Journal Article] Nariya Kawazumi, Integral Riemann-Roch formulae forcyclic subgroups of mapping classgroups (2008)
  • Author(s): Toshiyuki Akita
  • Journal Title: Mathematical Proceedings of the Cambridge Philosophical Society Volume: 144 Issue: 2 Pages: 411-421
  • DOI: 10.1017/s0305004107001016
  • Peer Reviewed
• [Journal Article] Generating functions of orbifold Chern classes I : Symmetric Products (2008)
  • Author(s): Toru Ohmoto
  • Journal Title: Mathematical Proceedings of the Cambridge Philosophical Society Volume: 144 Issue: 2 Pages: 423-438
  • DOI: 10.1017/s0305004107000898
  • Peer Reviewed
• [Journal Article] Surface symmetries, homology representations, and group cohomology (2008)
  • Author(s): Toshiyuki Akita
  • Journal Title: 数理解析研究所講究録 Volume: 1581 Pages: 103-108
  • NAID: 120001442850
• [Journal Article] Integral Riemann-Roch formulae for cyclic subgroups of mapping class groups (2008)
  • Author(s): Toshiyuki Akita, Nariya Kawazumi
  • Journal Title: Math.Proc.Cambridge Phil.Soc. 144 Pages: 411-421
  • Peer Reviewed
• [Journal Article] On mod p Riemann-Roch formulae for mapping class groups (2008)
  • Author(s): Toshiyuki Akita
  • Journal Title: Advanced Studies in Pure Math. 52 Pages: 111-118
  • Peer Reviewed
• [Journal Article] Generating functions of orbifold Chern classes I : Symmetric Products (2008)
  • Author(s): Toru Ohmoto
  • Journal Title: Math.Proc.Cambridge Phil.Soc. 144 Pages: 423-438
  • Peer Reviewed
• [Journal Article] Integral Ricmann-Roch formulae for cyclic subgroups of mappina class groups (2008)
  • Author(s): Toshiyuki Akita. Nariva Kawazumi
  • Journal Title: Mathcmatical Proceedings of the Cambridge Philosophical Society 144 Pages: 411-421
  • Peer Reviewed
• [Journal Article] A formula for the Euler characteristics of even dimensional manifolds (2008)
  • Author(s): Toshiyuki Akita
  • Journal Title: Proceedings of the American Mathematical Society 136 Pages: 2571-2573
  • Peer Reviewed
• [Journal Article] On mod p Riemann-Roch formulae for mapping class groups (2008)
  • Author(s): Toshiyuki Akita
  • Journal Title: Advanced Studies in Pure Mathematics 52 Pages: 111-118
  • Peer Reviewed
• [Journal Article] Vassiliev type invariants for generic mappings, revisited
  • Author(s): Toru Ohmoto
  • Journal Title: Contemporary Mathematics, AMS Volume: (印刷中)(掲載決定)
  • Peer Reviewed
• [Journal Article] A note on Chern-Schwartz-MacPherson class
  • Author(s): Toru Ohmoto
  • Journal Title: Proceedings of the 5th Franco-Japanese Singularities, Strasbourg Volume: (印刷中)(掲載決定)
  • Peer Reviewed
• [Presentation] Surface symmetries and equivariant characteristic classes (2012)
  • Author(s): Toshiyuki Akita
  • Organizer: Topology of Singularities and Related Topics III (JSPS-VAST Japan-Vietnam Bilateral Joint Projects)
  • Place of Presentation: University of Dalat, Dalat, ベトナム
  • Year and Date: 2012-03-29
• [Presentation] Surface symmetries and equivariant characteristic classes (2012)
  • Author(s): Toshiyuki Akita
  • Organizer: Topology of Singularities and Related Topics III
  • Place of Presentation: ダラット大学(ベトナム)(招待講演)
  • Year and Date: 2012-03-29
• [Presentation] Singular Chern class and degree formula (2012)
  • Author(s): Toru Ohmoto
  • Organizer: 第7回代数・解析・幾何セミナー
  • Place of Presentation: 鹿児島大学理学部
  • Year and Date: 2012-02-15
• [Presentation] Singular Chern class and degree formula (2012)
  • Author(s): Toru Ohmoto
  • Organizer: The 7th Kagoshima Algebro-Analysis-Geometry Seminar
  • Place of Presentation: 鹿児島大学理学部(招待講演)
  • Year and Date: 2012-02-15
• [Presentation] Obstructions forπ1 of the space of pseudo-isotopies of disks in non-stable range (2011)
  • Author(s): Tadayuki Watanabe
  • Organizer: The 4th East Asian Conference on Algebraic Topology
  • Place of Presentation: 東京大学大学院数理科学研究科
  • Year and Date: 2011-12-09
• [Presentation] Obstructions for π1 of the space of pseudo-isotopies of disks in non-stable range (2011)
  • Author(s): Tadayuki Watanabe
  • Organizer: The 4th East Asian Conference on Algebraic Topology
  • Place of Presentation: 東京大学大学院数理科学研究科(招待講演)
  • Year and Date: 2011-12-09
• [Presentation] Surface symmetry and cohomology of finite groups (2011)
  • Author(s): Toshiyuki Akita
  • Organizer: Seminarios de Topologia
  • Place of Presentation: ICMC-USP, Sao Carlos, ブラジル
  • Year and Date: 2011-09-02
• [Presentation] Surface symmetry and cohomology of finite groups (2011)
  • Author(s): Toshiyuki Akita
  • Organizer: Seminarios de Topologia, ICMC-USP
  • Place of Presentation: サンパウロ大学(ブラジル)(招待講演)
  • Year and Date: 2011-09-02
• [Presentation] Thom polynomial associated to the Milnor number of ICSC. (2011)
  • Author(s): Toru Ohmoto
  • Organizer: Banach Center Conference, Workshop on Singularities in Geometry and Application
  • Place of Presentation: ポーランド
  • Year and Date: 2011-05-21
• [Presentation] Thom polynomial associated to the Milnor number of ICSC (2011)
  • Author(s): Toru Ohmoto
  • Organizer: Banach Center Conference, Workshop on Singularities in Geometry and Application
  • Place of Presentation: バナッハッセンター(ポーランド)(招待講演)
  • Year and Date: 2011-05-21
• [Presentation] Mackey functors associated with surface symmetries (2011)
  • Author(s): Toshiyuki Akita
  • Organizer: 第6回代数・解析・幾何セミナー
  • Place of Presentation: 鹿児島大学理学部
  • Year and Date: 2011-02-15
• [Presentation] Chern class for singular spaces (2011)
  • Author(s): Toru Ohmoto
  • Organizer: JSPS-VAST Japan-Vietnam Bilateral Joint Projects Topology of singularities and related topics II
  • Place of Presentation: 東北大学大学院理学研究科
  • Year and Date: 2011-01-06
• [Presentation] Vassiliev type invariants revisited (2010)
  • Author(s): Toru Ohmoto
  • Organizer: The 11th International Workshop on Real and Complex Singularities
  • Place of Presentation: Instituto de Ciencias Matematicase de Computacao, ブラジル
  • Year and Date: 2010-07-27
• [Presentation] (2010)
  • Author(s): 大本亨
  • Organizer: The 11th International Workshop on Real and Complex Singularities
  • Place of Presentation: Instituto de Ciencias Matematicas e de Computacao(ブラジル)
  • Year and Date: 2010-07-27
• [Presentation] Cohomology of finite groups and characteristic classes (2009)
  • Author(s): 秋田利之
  • Organizer: 京都大学数学教室談話会
  • Place of Presentation: 京都大学
  • Year and Date: 2009-12-16
• [Presentation] Kontsevich's graph characteristic classes of smooth bundles (2009)
  • Author(s): Tadayuki Watanabe
  • Organizer: Quantum algebra related to various topological field theories in geometries
  • Place of Presentation: 京都大学理学部
  • Year and Date: 2009-02-16
• [Presentation] (2009)
  • Author(s): 渡邉忠之
  • Organizer: Quantum algebra related to various topological field theories in geometries
  • Place of Presentation: 京都大学理学部
  • Year and Date: 2009-02-16
• [Presentation] Surface symmetries and group cohomology (2008)
  • Author(s): 秋田利之
  • Organizer: 福岡微分幾何研究会
  • Place of Presentation: 福岡大学セミナーハウス
  • Year and Date: 2008-11-23
• [Remarks]
  • URL: http://www.math.sci.hokudai.ac.jp/~akita/