Study of Non-Commutative Geometry focusing on the Index theorem, and low-dimensional maniflod theory,

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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.