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

本文研究了“扭曲k理论”和“扭曲群C*-代数”，并推导了相关的指标定理。扭曲k理论和扭曲群C^*-代数对于具有大基群的流形具有有趣的行为。因此，研究双曲流形上的指标定理也是一个有趣的问题。我们的研究目标明确地表述如下：1)发展了marcoli - mathai指数定理，并推导了与扭曲k理论和扭曲群C^*-代数相关的指数定理。并推导了它的拓扑公式。2)研究了上述双曲流形上的指数定理，并研究了它与几何次不变量（如chen - simons类和r -扭转）的关系。关于1)，我们用值在U(1)内的Cech 2-环澄清了扭曲群C*-代数的扭曲k-理论、Gerbes和k群之间的关系。我们还发展了叶状流形上由marcolli -mathai引起的扭曲指数定理和相应的拓扑公式。利用这个公式，我们得到了具有大完整群的叶状束的各种有趣的结果。例如，当一个叶形流形承认一个叶形辛结构并且每个叶形都是K(1)-流形时，那么它就不承认一个具有正标量曲率的纵向黎曼度规。这意味着Gromov-Lawson猜想的推广仍然适用于叶状流形。同时证明了k -非球复流形中的Kaehler子流形直到维的奇偶性乘法都是非负的Todd格。对于2)，我们对几乎接触流形的Morita-Hirzebruch不变量进行了挑战，得到了三维流形的不变量的几何公式。此外，我们还阐明了Reeb向量场的指标定理、Boot定域公式、叶状流形上的二次类与向量场的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.