低余维角流形上的 Fredholm 异常与余法角循环
Fredholm anomalies on manifold with corners of low codimensions and conormal corner cycles
AI总结：
本文针对余维不超过三的带角流形，显式计算偶、奇余法指标态射，并用各面的悬浮 Atiyah-Singer 与 APS 指标完整刻画 Fredholm 扰动障碍。
摘要：
给定任意余维的连通带角流形 $X$，存在一种非常基础且可计算的同调理论，称为余法同调，它依据面及其余法丛的定向来定义，其循环在几何上对应于角循环；这些余法同调群记为 $H^{cn}_*(X)$。利用我们以往的工作，我们对余维数小于或等于三的带角流形 $X$ 定义了指标态射 $K^0(^bT^*X)\stackrel{Ind_{ev,cn}^X}{\longrightarrow}H_{ev}^{cn}(X)$，这里称之为偶余法指标态射。在 $X$ 紧致连通且 $D$ 是与 $X$ 关联的 $b$-演算中的椭圆 $b$-伪微分算子的情形下，根据我们以往的工作以及其他作者的工作，我们知道：在加上一个恒等算子之前，$D$ 可（用该演算中的一个正则化算子）扰动为 Fredholm 算子，当且仅当 $Ind_{ev,cn}^X([σ_D])$（其中 $[σ_D]\in K^0(^bT^*X)$ 是主象征类）在偶余法同调群 $H_{ev}^{cn}(X)$ 中为零。本文的主要结果是，对余维数小于或等于三的带角流形 $X$，显式计算偶与奇余法指标态射 $Ind_{ev/odd,cn}^X(σ)\in H_{ev/odd}^{cn}(X)$。余法角循环 $Ind_{ev/odd,cn}^X(σ)$ 的系数由 $X$ 的最大余维面的某些悬浮 Atiyah-Singer 指标，以及 $X$ 的非最大余维面的某些悬浮 Atiyah-Patodi-Singer 指标给出。作为推论，我们用上述面的指标对余维数小于或等于三的闭带角流形的 Fredholm 扰动性质的障碍给出了完整刻画，这也使我们能够用相应的拓扑指标给出这样的刻画。
英文摘要：
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 $Ind_{ev,cn}^X([σ_D])$ (where $[σ_D]\in K^0(^bT^*X)$ 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 $Ind_{ev/odd,cn}^X(σ)\in H_{ev/odd}^{cn}(X)$ for $X$ a manifold with corners of codimension less or equal to three. The coefficients of the conormal corner cycles $Ind_{ev/odd,cn}^X(σ)$ 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.
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