Non-Commutative Geometry and the Spectral Flow Index Theorem

本文对低维拓扑bas1S上的Atiyah-Singer指标定理进行了推广。该项目的目标如下：1。在非交换几何的框架下建立了详细的指标定理。并研究了指标定理与解析次不变量如eta不变量、谱流的关系；研究了阐述的指标定理与二级特征类（包括马斯洛夫类）的关系。本文给出了与非交换几何中的atiyah - patodi -单指标定理有关的项目的一个结果。2001年7月应邀在日本秋田市拓扑研讨会上作题为“解析k理论与指标定理”的报告。2001年3月，在庆应义塾大学非交换几何与弦理论研讨会上，我们也在研究题为“Eta不变量，Godbillon-Vey类和指标定理”的项目。我们还开展了非交换Hopf不变量的研究，并于2001年3月在名古屋技术研究所发表了题为“谱流、Teoplitz指数和Hopf不变量”的成果，2002年2月在鹿儿岛大学发表了题为“向量场的A指数定理和Hopf不变量”的成果。

In this research we proceeded to a generalization of the Atiyah-Singer index theorem on the bas1S of Low dimensional Topology. The objective of the project are the followings:1. Establish the elaborated Index Theorem in the framework of Non-commutative Geometry. Also study the relationship between the Index Theorem and the analytic secondary invariants like the eta invariants and the spectral flow;2. Study the relationship between the elaborated Index Theorem and secondary characteristic classes including the Maslov class.Here we state one of the results of the project, which is related to the Atiyah-Patodi-Singe Index Theorem in Non-commutative Geometry. In July 2001, we are invited to give a talk with the title 'Analytic K-theory and the index theorem' at Topology Symposium of Japan, Akita. We are also working on the project are presented with the title 'Eta invariants, the Godbillon-Vey classes and the index theorem' in March 2001 at Workshop on Non-commutative Geometry and String Theory, Keio University. We also developed the research on Non-commutative Hopf invariants, and presented some results at Nagoya Technology Institute with the title 'Spectral flow, the Teoplitz index and the Hopf invariant' in March 2001, and with the title 'A index theorem of vector field and the Hopf invariant' at Kagoshima University in February 2002.