Noncommutative Geometry and Applications of twisted Ktheory to Index theorem
Project/Area Number 
17540093

Research Category 
GrantinAid for Scientific Research (C)

Allocation Type  Singleyear Grants 
Section  一般 
Research Field 
Geometry

Research Institution  Keio University 
Principal Investigator 
MORIYOSHI Hitoshi Keio Univ., Faculty of Sci. and Tech., Associate. Prof., 理工学部, 助教授 (00239708)

CoInvestigator(Kenkyūbuntansha) 
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)

Project Period (FY) 
2005 – 2006

Project Status 
Completed (Fiscal Year 2006)

Budget Amount *help 
¥3,400,000 (Direct Cost: ¥3,400,000)
Fiscal Year 2006: ¥1,100,000 (Direct Cost: ¥1,100,000)
Fiscal Year 2005: ¥2,300,000 (Direct Cost: ¥2,300,000)

Keywords  Noncommutative Geometry / Index Theorem / Ktheory / Cyclic cohomology / Eta invariant / Foliation / Contact structure / Sasakian manifold 
Research Abstract 
In the present research we study "Twisted Ktheory" and "Twisted Group C*algebra" and derived the relevant Index Theorem. Twisted Ktheory 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 MarcolliMathai Index theorem and derive the Index theorem related to Twisted Ktheory 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 ChernSimons class and Rtorsions. With respect to 1) we clarified the relation among twisted ktheory, Gerbes and the Kgroup of the twisted groupoid C*algebras by Cech 2cocycles with values in U(1). We also developed the twisted Index theorem due to Marcolllimathai 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 GromovLawson conjecture still holds for foliated manifolds. Also we proved that Kaehler submanifolds in Kaspherical complex manifolds have nonnegative Todd genus up to multiplication of the parity of dimensions. With respect to 2) we defied the MoritaHirzebruch invariant on almost contact manifolds and obtained a geometric formula on the eta invariant for 3dimensional 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.

Report
(3 results)
Research Products
(10 results)