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

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.

This paper is an expanded version of sections 3 to 6 from arXiv:2311.17045. The other sections of said article will be expanded on in a forthcoming article