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
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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.
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