Non-commutative Geometry and Applications of twisted K-theory to Index theorem

本文研究了扭K-理论和扭群C ~*-代数，并导出了相应的指标定理。扭K-理论和扭群C^*-代数对于具有大泛函群的流形有着有趣的性质。因此，研究双曲流形上的指数定理也是一个有趣的问题。本文的主要工作如下：1）发展了Marcoli-Mathai指标定理，并导出了与扭群C^*-代数和扭K-理论相关的指标定理。2）研究了双曲流形上的指数定理，并研究了它与Chern-Simons类、R-挠等几何第二不变量的关系。对于1），我们利用U（1）中的Cech 2-上圈阐明了扭k-理论、Gerbes和扭群胚C*-代数的K-群之间的关系。我们还发展了由Marcollli-mathai提出的关于叶状流形的扭指数定理和相应的拓扑公式。由于这个公式，我们得到了各种有趣的结果，叶状丛与大holonomy群。例如，当一个叶流形具有叶辛结构且每一叶都是K（1）-流形时，则它不具有正数量曲率的纵向黎曼度量。这意味着Gromov-Lawson猜想的推广仍然适用于叶状流形。证明了K-非球面复流形中的Kaehler子流形在维数宇称乘下具有非负的托德亏格，并对2）定义了几乎切触流形上的Morita-Hirzebruch不变量，得到了三维流形上eta不变量的一个几何公式.阐明了向量场的Reeb指标定理、靴子局部化公式、叶流形上的第二类以及向量场的Ruelle旋转数之间的关系。

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.