The Holonomy Groupoids of Singularly Foliated Bundles

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 shown to simultaneously generalise regularly foliated bundles in the sense of Kamber-Tondeur, bundles that are equivariant under the actions Lie groupoids with simply connected source fibres, and singular foliations. We define hierarchies of diffeological holonomy groupoids associated to such bundles, which arise from the parallel transport of germs of local conservation laws on the base that take values in the total space. In particular, for any singular foliation with "enough" local conservation laws, our construction recovers the holonomy groupoid defined by Androulidakis and Skandalis as a special case. Finally we prove functoriality of all our constructions under appropriate morphisms.