A holonomy groupoid is a (topological/Lie-) groupoid naturally associated with a foliation of a manifold . It is, in some sense, the smallest de-singularization of the leaf-space quotient of the foliation, which is, in general, not a manifold itself. Every foliation groupoid of has this de-singularization property, but the holonomy groupoid is, in some sense, minimal with respect to this property.
Explicitly, given a foliation on a manifold , the holonomy groupoid of has as objects the points of . Given points in the same leaf, a morphism between them is the equivalence class of a path in the leaf from to , where two paths are identified if they induce the same germ of a holonomy transformation between small transversal sections through and . If and are not in the same leaf, then there is no morphism between them.
The monodromy groupoid of the foliation is obtained from this by further dividing out the homotopy between paths in a leaf.
Let be a foliated manifold with . Let be a leaf of ; let be two points; and let and be transversal sections through and respectively (i.e., and are sub-manifolds transversal to the leaves of ).
called the holonomy transformation of the path with respect to and , as follows:
If there exists a single foliation chart of that contains the image of , then there exists a sufficiently small open neighborhood of in the space for which there exists a unique smooth map satisfying the following conditions:
For every , the point lies in the same plaque in as . Observe that is a diffeomorphism onto its image.
Then define to be the germ of this diffeomorphism at :
In general, the image of is not contained inside any single foliation chart , but as it is a compact subspace of , there exist finitely many foliation charts and numbers such that
is contained in for .
Then arbitrarily choose transversal sections through for and define
This definition is independent of the choice of ’s, ’s and ’s. It only depends on the initial and final transversal sections and .
Two homotopic paths with the same endpoints induce the same holonomy. (Note, however, that the converse is not true. Two paths with the same endpoints inducing the same holonomy may not be homotopic.)
If are two transversal sections through and two transversal sections through , then
Its objects are the points of .
There are no morphisms between two points on different leaves.
The morphisms between two points on the same leaf are homotopy-classes of paths lying in the leaf joining those points.
The holonomy groupoid is defined analogously, where instead of identifying two paths if they are homotopic, they are identified if they induce the same holonomy as described above.
The holonomy groupoid appears in
and was studied extensively in