$b^k$-algebroids and the variety of foliation jets
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| Format: | Preprint |
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2025
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| author | Bischoff, Francis del Pino, Álvaro Witte, Aldo |
| author_facet | Bischoff, Francis del Pino, Álvaro Witte, Aldo |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2508_20241 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | $b^k$-algebroids and the variety of foliation jets Bischoff, Francis del Pino, Álvaro Witte, Aldo Differential Geometry Geometric Topology 53C12, 14M35, 22A22, 53D17 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. |
| title | $b^k$-algebroids and the variety of foliation jets |
| topic | Differential Geometry Geometric Topology 53C12, 14M35, 22A22, 53D17 |
| url | https://arxiv.org/abs/2508.20241 |