The Index Theorem and Analytic Secondary Invariants in Symplectic Geometry

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The objective of the project are the followings:1. Establish the elaborated Index Theorem in the framework of Noncommutative Geometry. Also study the relationship between the Index Theorem and the analytic secondary invariants like the eta invarinats and the spectral flow;2. Apply the elaborated Index Theorem to Symplectic Geometry and study the Maslov class from the viewpoint of secondary classes.Here we state one of the results of the project, which is related to the Atiyah-Patodi-Singe Index Theorem in Noncommutative Geometry. Let X be a compact even-dimensional manifold with boundary Y. We equip X with a Riemaniann metric and assume that X is isometric to the product space Y×(-1,0) in a neighborhood of Y. We then denote by X the complete manifold obtained by attaching the half cylinder Y×[0,+∞] to X. To understand the Atiyah-Patodi-Singer Index Theorem in a framework of Noncommutative Geometry, we first introduce a notion of group quasi-action and understand X as the quotient with … More respect to a quasi-action of R. Next we construct a short exact sequence of CィイD1*ィエD1-algebras involved with kernel functions on X. We then define the index of operators on X as elements in a relative K-group. The short exact sequence constructed above is also interesting itself since it yields the Wiener-Hopf extension for CィイD1*ィエD1R even in the simplest case. Given the K-theoretic definition of index, we construct a relative cyclic cocycle that is related to the eta invariant of Y. This description makes clear the role of the integral on the L-polynomial and the eta invariant appeared in the Atiyah-Patodi-Singer Index Theorem, which are a priori depending on the choice of Riemannian metric on X. In short, the eta invariant appears as the transgression form connecting the local invariant with the index of an R-invariant operator on the cylinder Y×R. We also developed the research toi obtain the result that clearify the relation between the eta invarinats and the spectral flow for type II von Neumann algebras. Less