Development of the index theorem on foliated manifolds

• Project/Area Number: 25400085
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Geometry
• Research Institution: Nagoya University
• Principal Investigator: Moriyoshi Hitoshi 名古屋大学, 多元数理科学研究科, 教授 (00239708)
• Co-Investigator(Renkei-kenkyūsha): NATSUME TOSHIKAZU 名古屋工業大学, 工学系研究科, 教授 (00125890) ; MAEDA YOSHIAKI 慶應義塾大学, 理工学部, 教授 (40101076) ; MITSUMATSU YOSHIHIKO 中央大学, 理工学部, 教授 (70190725) ; ONO KAORU 京都大学, 数理解析研究所, 教授 (20204232) ; MIYAZAKI NAOYA 慶應義塾大学, 経済学部, 教授 (50315826) ; TAKAKURA TATSURU 中央大学, 理工学部, 教授 (30268974) ; TATE TETSUYA 名古屋大学, 大学院多元数理科学研究科, 准教授 (00317299)
• Project Period (FY): 2013-04-01 – 2016-03-31
• Project Status: Completed (Fiscal Year 2015)
• Budget Amount: ¥4,940,000 (Direct Cost: ¥3,800,000、Indirect Cost: ¥1,140,000); Fiscal Year 2015: ¥1,560,000 (Direct Cost: ¥1,200,000、Indirect Cost: ¥360,000); Fiscal Year 2014: ¥1,690,000 (Direct Cost: ¥1,300,000、Indirect Cost: ¥390,000); Fiscal Year 2013: ¥1,690,000 (Direct Cost: ¥1,300,000、Indirect Cost: ¥390,000)
• Keywords: 指数定理 / 非可換幾何 / 葉層多様体 / Godbillon-Vey 不変量 / K理論 / 巡回コホモロジー / Godbillon-Vey不変量

Outline of Final Research Achievements

First, we extended the index theorem to fractal sets such as the Cantor set and the Sierpinski gasket. Second, by exploiting the framework of Noncommutative Geometry we generalized the Atiyah-Patodi-Singer index theorem to a Galois covering of compact manifold with boundary, which gives a formula for the pairing between K-group and cyclic cohomology. Third, we clarified the relation of the Dixmier-Douady class and the Godbillon-Vey class, which respectively appears as a characteristic class for Gerbe and foliated circle bundles. It turned out that they are connected via the Cheeger-Chern-Simons invariant. As a byproduct we succeeded to describe the universal central extension of circle diffeomorphism group in terms of the Calabi invariant.

Research Products

• [Int'l Joint Research] University of Rome(Italy)
• [Int'l Joint Research] University od Colorado(米国)
• [Journal Article] A note on the higher Atiyah-Patodi-Singer index theorem on Galois coverings (2016)
  • Author(s): Alexander Gorokhovsky, Hitoshi Moriyoshi and Paolo Piazza
  • Journal Title: Journal of Noncommutative Geometry Volume: 10
  • Peer Reviewed / Int'l Joint Research / Acknowledgement Compliant
• [Journal Article] The Calabi invariant and central extensions of diffeomorphism groups (2016)
  • Author(s): Hitoshi Moriyoshi
  • Journal Title: Geometry and Topology of Manifolds Volume: none
  • Peer Reviewed / Acknowledgement Compliant
• [Journal Article] カントール集合上の指数定理 (2015)
  • Author(s): Hitoshi Moriyoshi
  • Journal Title: Geometry and Something 報告集 Volume: none
• [Journal Article] A note on the higher Atiyah-Patodi-Singer index theorem on Galois coverings (2015)
  • Author(s): Alexander Gorokhovsky, Hitoshi Moriyoshi and Paolo Piazza,
  • Journal Title: Journal of Noncommutative Geometry Volume: 10
  • Peer Reviewed
• [Presentation] フラクタル集合上の指数定理 (2016)
  • Author(s): Hitoshi Moriyoshi
  • Organizer: 香川セミナー
  • Place of Presentation: 香川大学，高松
  • Year and Date: 2016-02-06
  • Invited
• [Presentation] カントール集合上の指数定理 (2015)
  • Author(s): Hitoshi Moriyoshi
  • Organizer: 福岡大学微分幾何研究会
  • Place of Presentation: 福岡大学，福岡
  • Year and Date: 2015-11-01
  • Invited
• [Presentation] The Calabi invariant and group extensions of diffeomorphisms (2015)
  • Author(s): Hitoshi Moriyoshi
  • Organizer: Colloquium at Institute of Mathematics, VAST
  • Place of Presentation: Hanoi, Vietnam
  • Year and Date: 2015-04-02
  • Invited
• [Presentation] The Godbillon-Vey index theorem in type III operator algebras (2014)
  • Author(s): H. Moriyoshi
  • Organizer: The Second China-Japan Conference on Noncommutative Geometry and K-Theory
  • Place of Presentation: Naha, Okinawa, Japan
  • Year and Date: 2014-10-04 – 2014-10-07
  • Invited
• [Presentation] The Calabi invariant and group extensions of Diffeomorphisms (2014)
  • Author(s): H. Moriyoshi
  • Organizer: 第１０回日中友好幾何学研究集会
  • Place of Presentation: Fudan University, Shanghai, China
  • Year and Date: 2014-09-06 – 2014-09-12
  • Invited
• [Presentation] The Calabi invariant and extensions of diffeomorphism groups (2014)
  • Author(s): H. Moriyoshi
  • Organizer: Workshop Geometry and Dynamics of Foliations
  • Place of Presentation: Madrid, Spain
  • Year and Date: 2014-09-01 – 2014-09-05
• [Presentation] The Calabi invariant and (2014)
  • Author(s): H. Moriyoshi
  • Organizer: 服部晶夫先生研究報告会
  • Place of Presentation: 東京大学大学院数理科学研究科
  • Year and Date: 2014-06-27 – 2014-06-28
• [Presentation] Secondary cyclic cocycle for Connes’ Chern character on a Cantor set,
  • Author(s): H. Moriyoshi
  • Organizer: China-Japan Conference on Noncommutative Geometry and K-Theory,
  • Place of Presentation: Changchun, China
  • Invited
• [Presentation] On a cyclic volume cocycle in Fractal Geometry,
  • Author(s): H. Moriyoshi
  • Organizer: Further development of Atiyah-Singer index theorem and K-theory,
  • Place of Presentation: Kyoto, Japan
  • Invited
• [Presentation] Quantum volume form on Cantor sets,
  • Author(s): H. Moriyoshi
  • Organizer: 非可換幾何湯谷研究集会
  • Place of Presentation: 愛知県新城市
• [Presentation] A generalization of the Atiyah-Singer index theorem in Fractal Geometry,
  • Author(s): H. Moriyoshi
  • Organizer: Topological phases in spintronics
  • Place of Presentation: Sendai, Japan
  • Invited