标题： $b^k$-代数胚和叶层射流的种类 显示英文标题

标题： $b^k$-algebroids and the variety of foliation jets

摘要： 我们引入并分类了$b^{k+1}$型奇异叶状结构，这些结构形式化了与子流形$W \subset M$在阶数$k$上相切的向量场的性质。 当$W$是超曲面时，这些结构是Scott引入的$b^{k+1}$-切丛的李代数胚推广。 We prove that singular foliations of $b^{k+1}$-type are encoded by $k$-th order foliations: jets of distributions that are involutive up to order $k$, equivalently described as foliations on the $k$-th order neighborhood of $W$. Using this encoding, we construct topological groupoids of $k$-th order foliations and employ the holonomy invariant to show that these groupoids fiber over certain character stacks, yielding Riemann-Hilbert style classifications up to local isomorphism and isotopy. 我们还研究了将一个$k$阶叶状结构扩展为一个$(k+1)$阶叶状结构的问题。 我们证明，这是由一个特征类所阻碍的，该特征类作为相关特征堆栈上的向量丛的一个截面出现。 显示英文摘要

摘要： We introduce and classify singular foliations of $b^{k+1}$-type, which formalize the properties of vector fields that are tangent to a submanifold $W \subset M$ to order $k$. When $W$ is a hypersurface, these structures are Lie algebroids generalizing the $b^{k+1}$-tangent bundles introduced by Scott. We prove that singular foliations of $b^{k+1}$-type are encoded by $k$-th order foliations: jets of distributions that are involutive up to order $k$, equivalently described as foliations on the $k$-th order neighborhood of $W$. Using this encoding, we construct topological groupoids of $k$-th order foliations and employ the holonomy invariant to show that these groupoids fiber over certain character stacks, yielding Riemann-Hilbert style classifications up to local isomorphism and isotopy. We also study the problem of extending a $k$-th order foliation to a $(k+1)$-st order foliation. We prove that this is obstructed by a characteristic class that arises as a section of a vector bundle over the relevant character stack.