202 5. Exact Groups and Related Topicsa pseudogroup. If ,...., is an equivalence relation on X, then the collection
of all partial homeomorphisms f with f(x) ,...., x for all x E dom(f) is a
pseudogroup.
Example 5.6.6 (Groupoid of germs). Let g be a pseudogroup on a locally
compact space X. For f E g and x E dom(f), the germ fx off at x consists
of all f' E g which agree with f on some neighborhood of x. For f, g E Q
and x E dom(f) with f(x) E dom(g), we define the composition gf(x) o fx
of germs in the obvious way. The groupoid of germs is the set G of all
germs equipped with its natural groupoid structure. We introduce the germ
topology on G by declaring that
O(f) = {f x E G : x E dom(f)}
is open for every f E Q. This topology may not be Hausdorff (and we ignore
such examples in this book). One can check that a neighborhood basis of
fx E G is given by { O(f o idu )}u for a neighborhood basis U of x and that
G is an etale locally compact groupoid.
Example 5.6.7 (Holonomy groupoid of a foliation). A foliation (M,F) of
codimension q can be defined by a Haefliger cocycle {Si: Ui ------> ~q}, where
{Ui} is an open covering of M and the s/s are submersions from Ui onto
open subsets of ~q such that there are (necessarily unique) diffeomorphisms
fij: sj(Ui n Uj) ------> si(Ui n Uj) with fij o Sj = si for every i,j. We assume
that the fibers of the submersions are connected. Two cocycles define the
same foliation on M if their common refinement is a cocycle. (See [125] for
more on this subject.)
Let {Si : ui ------> ~q} be a Haefliger cocycle. We define x to be the disjoint
union LJ si(Ui) and Q to be the pseudogroup on X generated by the partial
homeomorphisms fij. The holonomy groupoid of the Haefliger cocycle is the
groupoid G of germs of g.
The holonomy groupoid G defined above depends on the choice of the
Haefliger cocycle, but equivalent Haefliger cocycles define Morita equivalent
groupoids. Hence the Morita equivalence class of the holonomy groupoids is
an invariant of the foliation ( M, F). Before introducing the notion of Morita
equivalence for etale groupoids, we need one more example.
Example 5.6.8. Let G be a groupoid and U = {Ui : i E I} be an open
covering of Q(O) = u ui. (Recall that Q(O) is clopen in G.) The localization
of G over U is the groupoid
Gu= {(i,1,j) EI x G x I: s(r) E Uj and r(r) E Ui}
with G~) = {(i, x, i) : i EI, x E Ui} ~ LJi Ui (disjoint union) and with
s(i,1,j) = (j,s(r),j), r(i,1,j) = (i,r(r),i) and (i,a,j)(j,/3,k) = (i,a/3,k).