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

在本研究中，我们在低维拓扑的基础上推广了Atiyah-Singer指标定理。设Γ是离散群，σ是值在U(1)中的Γ的2-上循环，本文给出了与非交换几何中的Atiyah-Patodi-Singe指数定理有关的一个结果。然后我们在群代数C(Γ)上按如下方式扭曲乘积：U_gu_h=σ(g，h)U_&lt；gh&gt；其中U_g，U_h是对应于g，h∈Γ的形式酉元。由于余循环条件，我们得到了C(Γ)上的结合积。关于L^2(Γ)上的算子范数，我们取C(Γ)的C^*-闭包与上述乘积。2-余圈σ扭曲的群C^*-代数，记为C^*(Γ，σ)，存在一个指数属于C^*(Γ，σ)的K-群的狄拉克算子.让我们用D^▽表示狄拉克算子。然后，我们得到了指数公式，它表示指数IndD^τ在特征类A^(M/▽)的树上的迹▽(IndD^Γ)和相关线丛的曲率R。作为公式的推论。我们可以证明如下结果：假设一个闭辛流形M是非球面的，则M不允许具有正数量曲率的黎曼度量。这为格罗莫夫-劳森猜想提供了一个实用的解决方案。

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.