Noncommutative Geometry and equivariant index theorem for twisted group actions

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The objective in this project of research is a generalization of the Atiyah-Singer Index Theorem from the viewpoint of Noncommutative Geometry, in situations such as foliated manifolds and manifolds with boundary. We focused on tow notions, the twisted K-theory and the group C*-algebras twisted by cocycles, and sought for Atiyah-Singer type index theorems related to them. We first established a Atiyah-Singer type index theorem for a twisted group C*-algebras of the fundamental group on K-aspherical kaehler manifolds, and obtained certain inequalities on the arithmetic genera of complex manifolds, which is related to the vanishing theorem due to Green and Lazarsfeld. Second we formulated the Atiyha-Patodi-Singer index theorem in the framework of Noncommutative Geometry and extended the theorem in the cases of covering space with infinite degree and foliated manifold with boundary. This formulation provide us with an interpretation of the eta invariant including the higher one, as a pairing of relative cyclic cohomology and relative K-theory, which makes even clearer the geometric significance of eta invariants.