标题： 低余维数角流形上的弗雷德霍姆奇异性与共法向角循环 显示英文标题

标题： Fredholm anomalies on manifold with corners of low codimensions and conormal corner cycles

摘要： 给定一个任意余维数的带角的连通流形$X$，存在一种基本且可计算的同调理论，称为共法同调，其定义基于面和它们的共法丛的方向，其循环在几何上对应于角的循环，这些共法同调群记为$H^{cn}_*(X)$。 利用我们之前的研究，我们定义了一个指标同态$$K^0(^bT^*X)\stackrel{Ind_{ev,cn}^X}{\longrightarrow}H_{ev}^{cn}(X)$$，对于$X$一个余维数小于或等于三的带角流形，这里称为偶共法指标同态。 在$X$为紧致且连通的，且$D$是相关$X$的$b-$计算中的椭圆$b-$伪微分算子的情况下，根据我们的前期工作和其他作者的工作，我们知道，通过添加一个恒等算子，$D$可以（通过计算中的正则化算子）扰动为一个弗雷德霍姆算子，当且仅当$Ind_{ev,cn}^X([\sigma_D])$（其中$[\sigma_D]\in K^0(^bT^*X)$是主符号类）在偶正规同调群$H_{ev}^{cn}(X)$中消失。 本文的主要结果是显式计算了偶数和奇数的共法指标同态$Ind_{ev/odd,cn}^X(\sigma)\in H_{ev/odd}^{cn}(X)$对于$X$一个余维小于或等于三的角流形。共法角循环$Ind_{ev/odd,cn}^X(\sigma)$的系数用$X$最大余维面的一些悬停阿蒂亚-辛格指标以及用$X$非最大余维面的一些悬停阿蒂亚-帕托迪-辛格指标来表示。作为推论，我们给出了闭合角流形在余维小于或等于三的情况下弗雷德霍姆扰动性质的障碍的完整特征，这取决于上述面的指标，这也使我们能够用相应的拓扑指标来给出这样的特征。 显示英文摘要

摘要： Given a connected manifold with corners $X$ of any codimension there is a very basic and computable homology theory called conormal homology defined in terms of faces and orientations of their conormal bundles, and whose cycles correspond geometrically to corner's cycles, these conormal homology groups are denoted by $H^{cn}_*(X)$. Using our previous works we define an index morphism $$K^0(^bT^*X)\stackrel{Ind_{ev,cn}^X}{\longrightarrow}H_{ev}^{cn}(X)$$ for $X$ a manifold with corners of codimension less or equal to three and called here the even conormal index morphism. In the case that $X$ is compact and connected and $D$ is an elliptic $b-$pseudodifferential operator in the associated $b-$calculus of $X$ we know, by our previous works and other authors works, that, up to adding an identity operator, $D$ can be perturbed (with a regularizing operator in the calculus) to a Fredholm operator iff $Ind_{ev,cn}^X([\sigma_D])$ (where $[\sigma_D]\in K^0(^bT^*X)$ is the principal symbol class) vanishes in the even conormal homology group $H_{ev}^{cn}(X)$. The main result of this paper is the explicit computation of the even and odd conormal index morphisms $Ind_{ev/odd,cn}^X(\sigma)\in H_{ev/odd}^{cn}(X)$ for $X$ a manifold with corners of codimension less or equal to three. The coefficients of the conormal corner cycles $Ind_{ev/odd,cn}^X(\sigma)$ are given in terms of some suspended Atiyah-Singer indices of the maximal codimension faces of $X$ and in terms of some suspended Atiyah-Patodi-Singer indices of the non-maximal codimension faces of $X$. As a corollary we give a complete caracterization to the obstruction of the Fredholm perturbation property for closed manifolds with corners of codimension less or equal to three in terms of the above mentioned indices of the faces, this allows us as well to give such a characterization in terms of the respective topological indices.