2025/08/27 by Francis Bischoff, Bischoff, Francis, Álvaro del Pino +3
Mathematics · #Advanced Topics in Algebra #Rings, Modules, and Algebras #Homotopy and Cohomology in Algebraic Topology
paper · pdf · doi:10.48550/arxiv.2508.20241
We introduce and classify singular foliations of bk+1-type, which formalize the properties of vector fields that are tangent to a submanifold W ⊂ M to order k. When W is a hypersurface, these structures are Lie algebroids generalizing the bk+1-tangent bundles introduced by Scott. We prove that singular foliations of bk+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.