2006 Fiscal Year Final Research Report Summary
Non-commutative Geometry and Applications of twisted K-theory to Index theorem
| Project/Area Number |
17540093
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Geometry
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| Research Institution | Keio University |
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Principal Investigator |
MORIYOSHI Hitoshi Keio Univ., Faculty of Sci. and Tech., Associate. Prof., 理工学部, 助教授 (00239708)
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| Co-Investigator(Kenkyū-buntansha) |
MAEDA Yoshiaki Keio Univ., Faculty of Sci. and Tech., Prof., 理工学部, 教授 (40101076)
KAMETANI Yukio Keio Univ., Faculty of Sci. and Tech., Associate. Prof., 理工学部, 助教授 (70253581)
TOSE Nobuyuki Keio Univ., Faculty of Economics, Prof., 経済学部, 教授 (00183492)
NATSUME Toshikazu Nagoya Inst. Tch., Faculty of eng., Prof., 工学部, 教授 (00125890)
ONO Kaoru Hokkaido Univ., Grad school of Sci., Prof., 大学院理学研究科, 教授 (20204232)
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| Project Period (FY) |
2005 – 2006
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| Keywords | Non-commutative Geometry / Index Theorem / K-theory / Cyclic cohomology / Eta invariant / Foliation / Contact structure / Sasakian manifold |
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| Research Abstract |
In the present research we study "Twisted K-theory" and "Twisted Group C*-algebra" and derived the relevant Index Theorem. Twisted K-theory and Twisted Group C^*-algebra have interesting behanior for manifolds with large funcamental groups. Thus it is also interesting to investigate Index theorem on hyperbolic manifolds. Explicitly our objective in this research is stated as follows :
1)We develop the Marcolli-Mathai Index theorem and derive the Index theorem related to Twisted K-theory and Twisted Group C^*-algebras. Also we derive the topological formula for it.
2)We investigate the Index theorem above on hyperbolic manifolds and study the relation to "Geometric secondary invariants such as the Chern-Simons class and R-torsions.
With respect to 1) we clarified the relation among twisted k-theory, Gerbes and the K-group of the twisted groupoid C*-algebras by Cech 2-cocycles with values in U(1). We also developed the twisted Index theorem due to Marcollli-mathai on foliated manifolds and the relevant topological formula. Due to this formula we obtained various interesting results for foliated bundles with large holonomy groups. For instance, when a foliated manifold admits a leafwise symplectic structure and each leaf is K(1)-manifold, then it deoe not admit a longitudinal Riemannian metric with positive scalar curvature. This implies that a generalization of the Gromov-Lawson conjecture still holds for foliated manifolds. Also we proved that Kaehler submanifolds in K-aspherical complex manifolds have non-negative Todd genus up to multiplication of the parity of dimensions.
With respect to 2) we defied the Morita-Hirzebruch invariant on almost contact manifolds and obtained a geometric formula on the eta invariant for 3-dimensional manifolds. Also we clarified the relation among the index theorem for the Reeb vector fields, the Boot localization formula, secondary classes on foliated manifolds and the rotation number of the vector fields due to Ruelle.
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