nLab
holonomy groupoid

Idea

A holonomy groupoid is a (topological/Lie-)groupoid naturally associated with a foliation $ℱ$ of a manifold $X$. It is in some sense the smallest de-singularization of the leaf space quotient $X/ℱ$ of the foliation, which is in general not itself a manifold. Every foliation groupoid of $ℱ$ has this de-singularization property, but the holonomy groupoid is minimal with this property, in some sense.

Explicitly, given a (Riemannian) foliation $F$ on a manifold $X$, the holonomy groupoid of $F$ has as objects the points of $X$. Given points $x,y$ on the same leaf, a morphism between them is the equivalence class of a path in the leaf from $x$ to $y$, where two paths are identified if they induce the same germ of a holonomy transformation on a small transversal subspace between their endpoints. If $x$ any $y$ are not on the same leaf, then there is no morphism between them.

This is naturally a topological groupoid and a Lie groupoid if done right.

The monodromy groupoid of the foliation is obtained from this by furthermore dividing out homotopy between paths in a leaf.

Definition

Foliation holonomy

Let $(X,ℱ)$ be a foliated manifold with $q=\mathrm{codim}(ℱ)$. Let $L$ be a leaf of $ℱ$, let $x,y\in L$ be two points, and $T$ a transverse section through $x$, $S$ a transverse section through $y$ (submanifolds transversal to the leaves of $ℱ$).

To a path $\gamma :[0,1]\to L$ from $x$ to $y$ associate the germ of a diffeomorphism

$$\mathrm{hol}(\gamma )={\mathrm{hol}}^{S,T}(\gamma ):(T,x)\to (S,y)$$hol(\gamma) = hol^{S,T}(\gamma) : (T,x) \to (S,y)

called the holonomy of the path $\gamma$ with respect to $S$ and $T$, as follows.

If there exists a foliation chart $U$ of $ℱ$ that contains the whole path, then there exists an open neighbourhood $A$ of $x$ in $T$ inside $U$ on which there is a smooth map $f:A\to S$ satisfying

1. $f(x)=y$;

2. for every $x\prime \in A$ the point $f(X\prime )$ lies on the same plaque in $U$ as $x\prime$. We can choose $A$ small so that this is a diffeomorphism onto its image. Define then ${\mathrm{hol}}^{S,T}(\gamma )$ to be the germ of this diffoemorphism

$${\mathrm{hol}}^{S,T}(\gamma )={\mathrm{germ}}_x(f).$$hol^{S,T}(\gamma) = germ_x(f) \,.

Generally, if the path does not sit in a single $U$ there is a decompoasition of $\gamma$ into pieces $\{{\gamma }_i\}$ and a sequence $U_1,U_2,\cdots ,U_n$ of foliation charts such ${\gamma }_i$ sits in $U_i$. Choose transversal sections $T_i$ in the double overlaps and define then

$${\mathrm{hol}}^{S,T}(\gamma )={\mathrm{hol}}^{T,T_{i-1}}(\gamma 1_{i-1})\circ \cdots \circ {\mathrm{hol}}^{T_1,T_2}(\gamma 1_2)\circ {\mathrm{hol}}^{S,T_1}(\gamma 1_1).$$hol^{S,T}(\gamma) = hol^{T,T_{i-1}}(\gamma1_{i-1}) \circ \cdots \circ hol^{T_1,T_2}(\gamma1_2) \circ hol^{S,T_1}(\gamma1_1) \,.

This definition is idenpendent of the choices of the $U_i$ and only depends on $T$ and $S$.

Proposition

• Two homotopic paths with the same endpoints induce the same holonomy.

• If $S,S\prime$ are two transversal sections through $x$ and $T,T\prime$ two transversal sections through $y$, then

  $${\mathrm{hol}}^{S\prime ,T\prime }(\gamma )=\mathrm{hol}(\gamma {)}^{T,T\prime }({\mathrm{const}}_y)\circ {\mathrm{hol}}^{S,T}(\gamma )\circ {\mathrm{hol}}^{S,S\prime }({\mathrm{const}}_x).$$hol^{S',T'}(\gamma) = hol(\gamma)^{T,T'}(const_y)\circ hol^{S,T}(\gamma) \circ hol^{S,S'}(const_x) \,.

Holonomy groupoid

Given a foliated manifold $(ℱ,X)$, the monodromy groupoid is the disjoint union of the fundamental groupoids of the leaves of $ℱ$ the groupoid whose objects are the points of $X$, which has no morphisms between points on different leaves and for which the morphisms between points on the same leaf are homotopy-classes relative endpoints between paths in the leaf.

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 transformation, in the above sense.

References

The holonomy groupoid appears in

• Charles Ehresmann, Structures Feuilletées , Proc. 5th Canadian Math. Congress, Univ. of Toronto Press 1963, 1961, pp. 109–172.

and was studied extensively in

• H. E. Winkelnkemper, The graph of a foliation , Ann. Global Anal. Geom. 1 (1983), no. 3, 51–75.

See also

• Marius Crainic, Ieke Moerdijk, Foliation groupoids and their cyclic homology (arXiv:math/0003119)

• Ronnie Brown, Osman Mucuk, Foliations, locally Lie groupoids and holonomy (numdam)

• Iakovos Androulidakis, Georges Skandalis, The holonomy groupoid of a singular foliation (arXiv:math/0612370)