The explicit computations of the modular obstruction for G-kernel on a certain AFD factor

设N是群G的正规子群，它的商群Q=G/N。在前人的工作中，利用连续交积给出了模阻塞的定义。根据A.Connes对AFD III_λ因子的离散分解，我们可以考虑群Q_m={(p，t)∈Q×R：α_p=t mod T‘Z}。因此，G上的扩张模阻塞同构于Q_m上的某个Torus值3-上同调，G上的特征不变量也是如此。然后我们得到了III型λ因子的扩展HJR-正合列。这个正合序列是下图的第一列，它显示了它与一个普通的HJR-正合序列之间的关系：[数值公式]我们通过给出群胚的归纳和约化理论及其移维定理证明了上面的图，另一方面，这个环值扩展的模不变量是关于外作用的完全不变量，直到外共轭，对于类型III_λ的AFD-因子，这个离散的扩展的HJR-正合序列帮助我们给出具有给定的模障碍的外作用的模型。设A是SL(2Z)的一个元素。我们考虑A的交叉乘积群G(A)=Z^2×_A Z，我们将3-上同调群H^3(G(A)，T)〓T参数化，明确地给出了H^3(G(A)，T)中与t相关的一些条件下的模型外作用，并给出了从给定的外作用中求元素t的方法.通过构造具有给定不变量的模型作用，我们必须显式地计算群G(A)的HJR-正合序列，例如，群G(G)的2-上同调群特征不变量。

Let N be a normal subgroup of a group G and its quotient group Q=G/N. In the former work, the definition of modular obstruction was given by the use of continuous crossed product. According to a discrete decomposition of AFD III_λ-factor (by A.Connes), we may consider the group Q_m={(p,t)∈Q×R : mod α_p=t mod T'Z}. Consequently the extended modular obstruction on G is isomorphic to a certain Torus valued 3-cohomology on Q_m and the characteristic invariant on G is also so. Then we have the extended HJR-exact sequence for AFD-factor of type III_λ. This exact sequence is the first column in the following diagram which shows the relation between it and an ordinary HJR-exact sequence :【numerical fomula】We proved the above diagram, by giving induction and reduction theory for groupoids and their dimension shifting theorem, On the other hand, this Torus valued extended modular invariant is a complete invariant, up to outer conjugacy, for outer actions on AFD-factor of type III_λ and this discrete extended HJR-exact sequence help us to give a model of outer action with a given modular obstruction.Let A be an element of SL(2 Z). We consider a crossed product group G(A)=Z^2×_A Z by A. We parameterize the 3-cohomology group H^3(G(A),T)〓T. We give explicitly the model outer action with some condition associated with t in H^3 (G(A),T) and we also give the way how to get the element t from a given outer action. By cotistructing a model action with a given invariant, we have to compute HJR-exact sequence for the group G(A) explicitly, for example, 2-cohomology group characteristic invariant for the group G(G).