The Holonomy Groupoids of Singularly Foliated Bundles
arXiv:2006.14271 · doi:10.3842/SIGMA.2021.043
Abstract
We define a notion of connection in a fibre bundle that is compatible with a singular foliation of the base. Fibre bundles equipped with such connections are in plentiful supply, arising naturally for any Lie groupoid-equivariant bundle, and simultaneously generalising regularly foliated bundles in the sense of Kamber-Tondeur and singular foliations. We define hierarchies of diffeological holonomy groupoids associated to such bundles, which arise from the parallel transport of jet/germinal conservation laws. We show that the groupoids associated in this manner to trivial singularly foliated bundles are quotients of Androulidakis-Skandalis holonomy groupoids, which coincide with Androulidakis-Skandalis holonomy groupoids in the regular case. Finally we prove functoriality of all our constructions under appropriate morphisms.
Changes from V1: Definition 4.7 corrected to Definitions 4.7 and 4.8. Theorem 4.9 expanded to Theorem 4.10, and detail added to proof. Minor typos fixed. Changes from V2: Minor editing changes, Example 4.6 revised and generalised. Changes from V3: coordinate representations of prolongations revised. Changes from V4: hypotheses of Theorem 4.15 changed. Changes from V5: Published version