Non-Commutative Geometry and the Spectral Flow Index Theorem

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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.