Project/Area Number  09640081 
Research Category 
GrantinAid for Scientific Research (C).

Section  一般 
Research Field 
Geometry

Research Institution  Keio University 
Principal Investigator 
MORIYOSHI Hitoshi Faculty of Science and Technology, Keio University, Associate Professor, 理工学部, 助教授 (00239708)

CoInvestigator(Kenkyūbuntansha) 
山田 浩嗣 北見工業大学, 工学部, 助教授 (50210472)
ONO Kaoru Faculty of Science, Hokkaido University, Professor, 大学院・理学研究科, 教授 (20204232)
神田 雄高 北海道大学, 大学院・理学研究科, 助手 (30280861)
河澄 響矢 北海道大学, 大学院・理学研究科, 助教授 (30214646)
MATSUMOTO Makoto Faculty of Science, Kyushu University, Associate Professor, 大学院・数理学研究科, 助教授 (70231602)
MAEDA Yoshiaki Faculty of Science and Technology, Keio University, Professor, 理工学部, 教授 (40101076)
KAMETANI Yukio Faculty of Science and Technology, Keio University, Assistant Professor, 理工学部, 講師 (70253581)
NATSUME Toshikazu Faculty of Technology, Nagoya Institute of Technology, Professor, 工学部, 教授 (00125890)

Project Fiscal Year 
1997 – 1999

Project Status 
Completed(Fiscal Year 1999)

Budget Amount *help 
¥3,100,000 (Direct Cost : ¥3,100,000)
Fiscal Year 1999 : ¥900,000 (Direct Cost : ¥900,000)
Fiscal Year 1998 : ¥800,000 (Direct Cost : ¥800,000)
Fiscal Year 1997 : ¥1,400,000 (Direct Cost : ¥1,400,000)

Keywords  NONCOMMUTATIVE GEOMETRY / KTEHORY / CYCLIC COHOMOLORY / THE INDEX THEOREM / SYMPLECTIC GEOMETRY / 非可換幾何学 / K理論 / 巡回コホモロジー / 指数定理 / シンプレクティック幾何 / 非可換微分幾何学 / K理論 / Maslov類 / spectral flow 
Research Abstract 
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 AtiyahPatodiSinge Index Theorem in Noncommutative Geometry. Let X be a compact evendimensional 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 AtiyahPatodiSinger Index Theorem in a framework of Noncommutative Geometry, we first introduce a notion of group quasiaction and understand X as the quotient with
… More
respect to a quasiaction of R. Next we construct a short exact sequence of CィイD1*ィエD1algebras involved with kernel functions on X. We then define the index of operators on X as elements in a relative Kgroup. The short exact sequence constructed above is also interesting itself since it yields the WienerHopf extension for CィイD1*ィエD1R even in the simplest case. Given the Ktheoretic 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 Lpolynomial and the eta invariant appeared in the AtiyahPatodiSinger 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 Rinvariant 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
