2015 Fiscal Year Final Research Report
Development of the index theorem on foliated manifolds
| Project/Area Number |
25400085
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Multi-year Fund |
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| Section | 一般 |
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| Research Field |
Geometry
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| Research Institution | Nagoya University |
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Principal Investigator |
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| 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)
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| Project Period (FY) |
2013-04-01 – 2016-03-31
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| 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 |
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| 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.
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| Free Research Field |
位相幾何，非可換幾何
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