On the classification of Regular Lie groupoids

Abstract

We observe that any regular Lie groupoid G over an manifold M fits into an extension K → G → E of a foliation groupoid E by a bundle of connected Lie groups K. If F is the foliation on M given by the orbits of E and T is a complete transversal to F, this extension restricts to T , as an extension KT → GT → ET of an ´etale groupoid ET by a bundle of connected groups KT . We break up the classification into two parts. On the one hand, we classify the latter extensions of ´etale groupoids by (nonabelian) cohomology classes in a new ˇCech cohomology of ´etale groupoids. On the other hand, given K and E and an extension KT → GT → ET over T , we present a cohomological obstruction to the problem of whether this is the restriction of an extension K → G → E over M; if this obstruction vanishes, all extensions K → G → E over M which restrict to a given extension over the transversal together form a principal bundle over a “group” of bitorsors under K.We observe that any regular Lie groupoid G over an manifold M fits into an extension K → G → E of a foliation groupoid E by a bundle of connected Lie groups K. If F is the foliation on M given by the orbits of E and T is a complete transversal to F, this extension restricts to T , as an extension KT → GT → ET of an ´etale groupoid ET by a bundle of connected groups KT . We break up the classification into two parts. On the one hand, we classify the latter extensions of ´etale groupoids by (nonabelian) cohomology classes in a new ˇCech cohomology of ´etale groupoids. On the other hand, given K and E and an extension KT → GT → ET over T , we present a cohomological obstruction to the problem of whether this is the restriction of an extension K → G → E over M; if this obstruction vanishes, all extensions K → G → E over M which restrict to a given extension over the transversal together form a principal bundle over a “group” of bitorsors under K.