2004 Fiscal Year Final Research Report Summary
Study of Non-Commutative Geometry focusing on the Index theorem, and low-dimensional maniflod theory,
| Project/Area Number |
14540089
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Geometry
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| Research Institution | Keio University |
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Principal Investigator |
MORIYOSHI Hitoshi Keio Univ., Faculty of Science and Technology, Associate Prof., 理工学部, 助教授 (00239708)
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| Co-Investigator(Kenkyū-buntansha) |
MAEDA Yoshiaki Keio Univ., Faculty of Science and Technology, Prof., 理工学部, 教授 (40101076)
KAMETANI Yukio Keio Univ., Faculty of Science and Technology, Associate Prof., 理工学部, 助教授 (70253581)
TATE Tatsuya Nagoya Univ., Graduate School of Mathematics, Associate Prof., 多元数理研究科, 助教授 (00317299)
NATSUME Toshikazu Nagoya Institute of Technology, Faculty of Technology, Prof., 工学部, 教授 (00125890)
ONO Kaoru Hokkaido University, Department of mathematics, Prof., 大学院・理学研究科, 教授 (20204232)
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| Project Period (FY) |
2002 – 2004
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| Keywords | NONCOMMUTATIVE GEOMETRY / THE INDEX THEOREM / SCALAR CURVATURE / CYCLIC COHOMOLORY / K-THEORY / ETA INVARIANT / FOLIATION / C^* ALGEBRA |
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| Research Abstract |
In this research we proceeded to a generalization of the Atiyah-Singer index theorem on the basis of Low-dimensional Topology. Here we state one of the results of the project, which is related to the Atiyah-Patodi-Singe Index Theorem in Non-commutative Geometry.
Let Γ be a discrete group and σ a 2-cocycle of Γ with values in U(1). We then twist the product on the group algebra C(Γ) in the following way : U_gU_h = σ(g,h)U_<gh> where U_g, U_h are the formal unitary elements corresponding to g, h ∈ Γ. Due to the cocycle condition we obtain an associative product on C(Γ). With respect to the operator norm on L^2(Γ) we take the C^*-closure of C(Γ) with product above. It is called group C^*-algebra twisted by a 2-cocycle σ and denoted by C^* (Γ,σ).
There exists a Dirac operator whose index belongs to the K-group of C^* (Γ,σ). Let us denote the Dirac operator by D^▽. We then obtain the Index formula which express the trace τ(IndD^▽) of the index IndD^▽ in trems of characteristic classes A^^^(M/Γ) and a curvature R of the associated line bundle. As a corollary of the formula. we can prove the following result :
Suppose that a closed symplectic manifold M is aspherical, then M does not admit a Riemanninan metric of positive scalar curvature. This yields a prtial solution to the Gromov-Lawson conjecture.
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