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

## TL;DR

This paper introduces and classifies singular foliations of $b^{k+1}$-type, generalizing tangent structures near submanifolds, and constructs associated groupoids to classify these foliations up to local isomorphism.

## Contribution

It formalizes $b^{k+1}$-type singular foliations, encodes them via jets of distributions, and constructs topological groupoids for classification, including obstructions to extending foliations.

## Key findings

- Classification of $b^{k+1}$-type singular foliations
- Construction of $k$-th order foliation groupoids
- Identification of characteristic classes as obstructions

## Abstract

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.