@
Applied General Topology

c© Universidad Politécnica de Valencia

Volume 4, No. 1, 2003

pp. 79–89

Holonomy, extendibility, and the star universal
cover of a topological groupoid

Osman Mucuk and İlhan İçen

Abstract. Let G be a groupoid and W be a subset of G which
contains all the identities and has a topology. With some conditions on
G and W, the pair (G,W) is called a locally topological groupoid. We
explain a criterion for a locally topological groupoid to be extendible
to a topological groupoid. In this paper we apply this result to get a
topology on the monodromy groupoid MG which is the union of the
universal covers of Gx’s.

2000 AMS Classification: 22A05, 55M99, 55R15.
Keywords: Locally topological groupoids, holonomy groupoid, extendibility.

1. Introduction

A groupoid is a small category in which each morphism is an isomorphism.
Thus a group is a particular example of a groupoid. There is considerable
evidence (see for example [9]) that the extension from groups to groupoids is
widely useful in mathematics, and is one way of encoding many of the intuitions
and methods of Sophus Lie which are difficult to encode in the language purely
of group theory. For this encoding, we need the notion of topological (and of
Lie groupoid) and so it is important to examine the extent to which standard
constructions on topological groups are available for topological groupoids. The
book [9] gives considerable information on this. In this paper we give an ex-
position of the construction of an analogue of the classical universal cover of
a connected topological group, and which we call the monodromy groupoid,
following Pradines [12].

The ideas for this are taken from [10,6] but we use a result from [6] to give
a more direct proof of the construction than in [6], although in this way we do
lose some power, notably the monodromy principle as given in [6].

We again emphasise the use of the holonomy groupoid construction, as first
developed by Pradines in [12], which however contains no details. Full details



80 O. Mucuk and İ. İçen

were first given in [1], as explained there. We feel it important to stress the
construction of Pradines as expressing well the intuitive idea of non trivial
holonomy as dealing with an ‘iteration of local procedures which returns to
the starting position but not the starting value’. Thus we see the concept
of groupoid as adding to the concept of group an extra notion of ‘position’,
through the set or space of objects, and of ‘transition’, through the arrows
between objects. This extension has proved to be generally powerful.

Let G be a groupoid and W a subset of G containing all the identities in G.
Suppose that W has a topology. For certain conditions on W the pair (G,W)
is called a locally topological groupoid. The topology on W does not in general
extend to a topological groupoid structure on G which restricts to that on W ,
but there is a topological groupoid H, called the holonomy groupoid, with a
morphism H → G such that H contains W as a subspace and H has a universal
property. The full details of this result are given by Aof and Brown in [1].

A locally topological groupoid (G,W) is called extendible if there is a topol-
ogy on G such that G is a topological groupoid with this topology and W is
open in G. A locally topological groupoid is not in general extendible. It is
proven by Brown and Mucuk in [7] that the charts of a foliated manifold may
be chosen so that they give rise to a locally topological groupoid which in gen-
eral is not extendible. We have also examples of locally topological groupoids,
due to Pradines and explained in [1], which are not extendible. A full account
of the monodromy groupoids was given in [10] and published in [6].

Let G be a topological groupoid in which each star Gx has a universal cover.
Then the monodromy groupoid GM is constructed by Mackenzie in [9] as the
union over x in OG of the universal covers based at 1x of the stars Gx. In the
locally trivial case in [9], the groupoid MG is given a topology such that MG
is a topological groupoid with this topology.

Let G be a locally sectionable topological groupoid and W an open subset
containing all the identities. In this paper we use a criterion obtained from
holonomy to prove that the monodromy groupoid MG has a structure of topo-
logical groupoid such that each star (MG)x is a universal cover of Gx. In
[6] the groupoid associated with a pregroupoid is used to verify a monodromy
property for MG, namely extendibility to MG of a local morphism on G.

2. Groupoids and topological groupoids

A groupoid G on OG is a small category in which each morphism is an
isomorphism. Thus G has a set of morphisms, which we call just elements of
G, a set OG of objects together with functions α,β : G → OG, �: OG → G such
that α� = β� = 1OG, the identity map. The functions α, β are called initial
and final point maps respectively. If a,b ∈ G and β(a) = α(b), then the product
or composite ba exists such that α(ba) = α(a) and β(ba) = β(b). Further, this
composite is associative, for x ∈ OG the element �(x) denoted by 1x acts as
the identity, and each element a has an inverse a−1 such that α(a−1) = β(a),



Holonomy and extendibility 81

β(a−1) = α(a), aa−1 = (�β)(a), a−1a = (�α)(a). The map G → G, a 7→ a−1,
is called the inversion.

In a groupoid G for x,y ∈ OG we write G(x,y) for the set of all morphisms
with initial point x and final point y. For x ∈ OG we denote the star {a ∈
G: α(a) = x} of x by Gx and the costar {a ∈ G: β(a) = x} of x by Gx. In
G the set OG is mapped bijectively to the set of identities by �: OG → G. So
we sometimes write OG for the set of identities. Let G be a groupoid and W a
subset of G such that OG ⊆ W . We say G is generated by W if each element
of G may be written as a product of elements of W .

Let G be a groupoid. A subgroupoid of G is a pair of subsets H ⊆ G and
OH ⊆ OG such that α(H) ⊆ OH, β(H) ⊆ OH, 1x ∈ H for each x ∈ OH and H
is closed under the partial multiplication and the inversion in G.

A morphism of groupoids H and G is a functor, that is, it consists of a pair
of functions f : H → G and Of : OH → OG preserving all the structures.

Definition 2.1. A topological groupoid is a groupoid G on OG, together with
topologies on G and OG, such that the maps which define the groupoid struc-
ture are continuous, namely the initial and final point maps α,β : G → OG,
the object inclusion map �: OG → G, x 7→ �(x), the inversion G → G, a 7→ a−1
and the partial multiplication Gα ×β G → G, (b,a) 7→ ba, where the pullback

Gα ×β G = {(b,a) ∈ G×G: α(b) = β(a)}
has the subspace topology from G×G.

A morphism of topological groupoids f : H → G is a morphism of groupoids
in which both maps f : H → G and Of : OH → OG are continuous.

Note that in this definition the partial multiplication Gα×β G → G, (b,a) 7→
ba and the inversion map G → G,a 7→ a−1 are continuous if and only if the
map δ : G ×α G → G, (b,a) 7→ ba−1, called the groupoid difference map, is
continuous, where the pullback

G×α G = {(b,a) ∈ G×G: α(b) = α(a)}
has the subspace topology from G×G. Again if one of the maps α,β and the
inversion are continuous, then the other map is continuous.

Let X be a topological space. Then G = X × X is a topological groupoid
on X, in which each pair (y,x) is a morphism from x to y and the groupoid
composite is defined by (z,y)(y,x) = (z,x). The inverse of (y,x) is (x,y) and
the identity at 1x is the pair (x,x).

Note that in a topological groupoid, G, for each a ∈ G(x,y) right trans-
lation Ra : Gy → Gx,b 7→ ba and left translation La : Gx → Gy, b 7→ ab are
homeomorphisms.

A groupoid G in which each star Gx has a topology such that for each
a ∈ G(x,y) the right translation Ra : Gy → Gx,b 7→ ba (and hence also the
left translation La : Gx → Gy,b 7→ ab) is a homeomorphism, is called a star
topological groupoid



82 O. Mucuk and İ. İçen

3. Locally topological groupoids and Extendibility

The following definition is due to Ehresmann [8].

Definition 3.1. Let G be a groupoid and let X = OG be a topological space.
An admissible local section of G is a function σ : U → G from an open set in
X such that ασ(x) = x for all x ∈ U; βσ(U) is open in X, and βσ maps U
homeomorphically to βσ(U).

Let W be a subset of G containing OG and let W have the structure of a
topological space. We say that (α,β,W) is locally sectionable if for each w ∈ W
there is an admissible local section σ : U → G of G such that (i) σα(w) = w,
(ii) σ(U) ⊆ W and (iii) σ is continuous as a function from U to W . Such a σ
is called a continuous admissible local section.

The following definition is due to Pradines [12] under the name “morceau de
groupoide différentiables”.

Definition 3.2. A locally topological groupoid is a pair (G,W) consisting of a
groupoid G and a topological space W such that:

i) OG ⊆ W ⊆ G;
ii) W = W−1;
iii) the set W(δ) = (W ×α W) ∩ δ−1(W) is open in W ×α W and the

restriction of δ to W(δ) is continuous;
iv) the restrictions to W of the source and target maps α and β are con-

tinuous, and the triple (α,β,W) is locally sectionable;
v) W generates G as a groupoid.

Note that in this definition, G is a groupoid but does not need to have a
topology. The locally topological groupoid (G,W) is said to be extendible if
there can be found a topology on G making it a topological groupoid and for
which W is an open subset. In general, (G,W) is not extendible, but there is a
holonomy groupoid Hol(G,W) and a morphism ψ : Hol(G,W) → G such that
Hol(G,W) admits the structure of topological groupoid and is the “minimal”
such overgroupoid of G. The construction is given in detail in [1] and is outlined
below.

It is easiest to picture locally topological groupoids (G,W) for groupoids G
such that α = β, so that G is just a bundle of groups. Here is a specific such
example of a locally topological groupoid [1], which is not extendible.

Example 3.3. Let F be the bundle of groups p : R × R → R, where R is
the set of real numbers and p is the first projection. The usual topology on
R×R gives F the structure of a topological groupoid in which each p−1(x) is
isomorphic as an additive group to R.

Let N be the subbundle of F given by the union of the sets {(x, 0)} if
x < 0 and {x}× Z if x ≥ 0, where Z is the set of integers. Let G be the
quotient bundle F/N and let q : F → G be the quotient morphism. Then the
source map α : G → R has α−1(x) isomorphic to R for x < 0 and to R/Z for



Holonomy and extendibility 83

x ≥ 0. Let W ′ be the subset R× (−1
4
, 1

4
) of F. Then q maps W ′ bijectively to

W = q(W ′); let W have the topology in which this map is a homeomorphism.
It is easily checked that (G,W) is a locally topological groupoid. Suppose this
locally topological groupoid is extended to a topological groupoid structure on
G. Let s′ be the section of p in which x 7→ (x, 1

8
), and let s = qs′. Then s is an

admissible section of α but t = 9s is not. However t(0) = q(0, 1
8
). Let U be an

open neighbourhood of ( 1
8
, 0) in R2 such that U is contained in W ′. Then p(U)

is contained in W and is a neighbourhood of t(0). But t−1q(U) is contained in
[0,∞), so that t is not continuous. This gives a contradiction, and shows that
the locally topological groupoid (G,W) is not extendible. By contrast, if we
proceed as before but replace N by N1, which is the union of the sets {(x, 0)}
for x ≤ 0 and {x}×Z for x > 0, then the resulting locally topological groupoid
(G1,W1) is extendible.

Example 3.4. There is a variant of the last example in which F is as before,
but this time N is the union of the groups {x}×(1 + |x|)Z for all x ∈ R. If one
takes W ′ as before, and W is the image of W ′ in G = F/N, then the locally
topological groupoid (G,W) can be extended to give a topological groupoid
structure on G. However, now consider W as a differential manifold. The dif-
ferential structure cannot be extended to make G a differential groupoid with
W as submanifold. The reason is analogous to that given in the previous ex-
ample, namely that such a differential structure would entail the existence of a
local differentiable admissible section whose sum with itself is not differentiable,
thus giving a contradiction.

Example 3.5. ([7]) Let X be a paracompact foliated manifold. Then there is
an equivalence relation, written RF , on X determined by the leaves. So RF is
a subspace of X ×X and becomes a topological groupoid on X with the usual
multiplication (z,y)(y,x) = (z,x), for (y,x), (z,y) ∈ RF .

For any subset U of X we write RF (U) for the equivalence relation on U
whose classes are the plaques of U. If Λ = {(Uλ,φλ)} is a foliated atlas for X,
we write W(Λ) for the union of the sets RF (Uλ) for all domains Uλ of charts
of Λ. Let W(Λ) have its topology as a subspace of RF and so of X ×X.

In [7] it is proved that the pair (RF ,W ′), where W ′ derives from a refinement
of Λ, is a locally topological groupoid. Some special foliated manifolds are given
in which the locally topological groupoid (RF ,W ) is not extendible.

There is a main globalisation theorem for a locally topological groupoid due
to Aof-Brown [1], and a Lie version of this is stated in Brown-Mucuk [6]; it
shows how a locally topological groupoid gives rise to its holonomy groupoid,
which is a topological groupoid satisfying a universal property. This theorem
gives a full statement and proof of a part of Théorème 1 of [12].

Theorem 3.6. (Globalisation Theorem) Let (G,W) be a locally topological
groupoid. Then there is a topological groupoid H, a morphism φ : H → G of
groupoids, and an embedding i : W → H of W to an open neighbourhood of
OH, such that:



84 O. Mucuk and İ. İçen

i) φ is the identity on objects, φi = idW , φ−1(W) is open in H, and the
restriction φW : φ−1(W) → W of φ is continuous;

ii) (universal property) If A is a topological groupoid and ζ : A → G is a
morphism of groupoids such that:

a) ζ is the identity on objects;
b) The restriction ζW : ζ(W) → W of ζ is continuous and ζ−1(W)

is open in A and generates A;
c) The triple (αA,βA,A) has enough continuous admissible local sec-

tions,
then there is a unique morphism ζ′ : A → H of topological groupoids
such that φζ′ = ζ and ζ′a = iζa for a ∈ ζ−1(W).

The groupoid H is called the holonomy groupoid Hol(G,W) of the locally
topological groupoid (G,W); its essential uniqueness follows from the condition
(ii) above.

We outline the proof of which full details are given in [1]. Some details of
part of the construction are needed for Proposition 3.7.

Proof. (Outline) Let Γ(G) be the set of all admissible local sections of G. Define
a product on Γ(G) by

(ts)x = (tβsx)(sx)

for two admissible local sections s and t. If s is an admissible local section
then write s−1 for the admissible local section βsD(s) → G,βsx 7→ (sx)−1.
With this product Γ(G) becomes an inverse semigroup. Let Γc(W) be the
subset of Γ(G) consisting of admissible local sections which have values in W
and are continuous. Let Γc(G,W) be the subsemigroup of Γ(G) generated by
Γc(W). Then Γc(G,W) is again an inverse semigroup. Intuitively, it contains
information on the iteration of local procedures.

Let J(G) be the sheaf of germs of admissible local sections of G. Thus
the elements of J(G) are the equivalence classes of pairs (x,s) such that s ∈
Γ(G),x ∈D(s), and (x,s) is equivalent to (y,t) if and only if x = y and s and
t agree on a neighbourhood of x. The equivalence class of (x,s) is written [s]x.
The product structure on Γ(G) induces a groupoid structure on J(G) with X
as the set of objects, and source and target maps [s]x 7→ x, [s]x 7→ βsx. Let
Jc(G,W) be the subsheaf of J(G) of germs of elements of Γc(G,W). Then
Jc(G,W) is generated as a subgroupoid of J(G) by the sheaf Jc(W) of germs
of elements of Γc(W). Thus an element of Jc(G,W) is of the form

[s]x = [sn]xn · · · [s1]x1
where s = sn · · ·s1 with [si]xi ∈ Jc(W),xi+1 = βsixi, i = 1, . . . ,n and x1 =
x ∈D(s).

Let ψ : J(G) → G be the final map defined by ψ([s]x) = s(x), where s is
an admissible local section. Then ψ(Jc(G,W)) = G. Let J0 = Jc(W) ∩ ker ψ.
Then J0 is a normal subgroupoid of Jc(G,W); the proof is in [1] Lemma
2.2. The holonomy groupoid Hol = Hol(G,W) is defined to be the quotient



Holonomy and extendibility 85

Jc(G,W)/J0. Let p : Jc(G,W) → Hol be the quotient morphism and let
p([s]x) be denoted by 〈s〉x. Since J0 ⊆ ker ψ there is a surjective morphism
φ : Hol → G such that φp = ψ.

The topology on the holonomy groupoid Hol such that Hol with this topology
is a topological groupoid, is constructed as follows. Let s ∈ Γc(G,W). A partial
function σs : W → Hol is defined as follows. The domain of σs is the set of
w ∈ W such that βw ∈D(s). A continuous admissible local section f through
w is chosen and the value σsw is defined to be

σsw = 〈s〉βw〈f〉αw = 〈sf〉αw.

It is proven that σsw is independent of the choice of the local section f and
that these σs form a set of charts. Then the initial topology with respect to
the charts σs is imposed on Hol. With this topology Hol becomes a topological
groupoid. The proof is in Aof-Brown [1]. �

From the construction of the holonomy groupoid we easily obtain the fol-
lowing extendibility condition, which is proved in [6].

Proposition 3.7. The locally topological groupoid (G,W) is extendible to a
topological groupoid structure on G if and only if the following condition holds:

(1) if x ∈ OG, and s is a product sn · · ·s1 of local sections about x such
that each si lies in Γc(W) and s(x) = 1x, then there is a restriction
s′ of s to a neighbourhood of x such that s′ has its image in W and is
continuous, i.e. s′ ∈ Γc(W).

Proof. The canonical morphism φ : H → G is an isomorphism if and only if
ker ψ ∩ Jc(W) = ker ψ. This is equivalent to ker ψ ⊆ Jc(W). We now show
that ker ψ ⊆ Jc(W) if and only if the condition (1) is satisfied.

Suppose ker ψ ⊆ Jc(W). Let s = sn · · ·s1 be a product of admissible local
sections about x ∈ OG with si ∈ Γc(W) and x ∈Ds such that s(x) = 1x. Then
[s]x ∈ Jc(G,W) and ψ([s]x) = s(x) = 1x. So [s]x ∈ ker ψ, so that [s]x ∈ Jc(W).
So there is a neighbourhood U of x such that the restriction s | U ∈ Γc(W).

Suppose the condition (1) is satisfied. Let [s]x ∈ ker ψ. Since [s]x ∈
Jc(G,W), then [s]x = [sn]xn · · · [s1]x1 where s = sn · · ·s1 and [si]xi ∈ Jc(W),
xi+1 = βsixi, i = 1, . . . ,n and x1 = x ∈ D(s). Since s(x) = 1x, then by (1),
[s]x ∈ Jc(W). �

In effect, Proposition 3.7 states that the non-extendibility of (G,W) arises
from the holonomically non trivial elements of Jc(G,W). Intuitively, such an
element h is an iteration of local procedures (i.e. of elements of Jc(W)) such
that h returns to the starting point (i.e. αh = βh) but h does not return to
the starting value (i.e. ψh 6= 1).

The following result, which is given as Corollary 4.6 in [6], gives a circum-
stance in which this extendibility condition is easily seen to apply.



86 O. Mucuk and İ. İçen

Corollary 3.8. Let Q be a topological groupoid and let p : M → Q be a
morphism of groupoids such that p : OM → OQ is the identity. Let W be an
open subset of Q such that

(1) OQ ⊆ W ;
(2) W = W−1;
(3) W generates Q;
(4) (αW ,βW ,W ) is continuously locally sectionable;

and suppose that ı̃ : W → M is given such that pı̃ = i : W → Q is the inclusion
and W ′ = ı̃(W) generates M.

Then M admits a unique structure of topological groupoid such that W ′ is
an open subset and p : M → Q is a morphism of topological groupoids mapping
W ′ homeomorphically to W .

Proof. It is easy to check that (M,W ′) is a locally topological groupoid. We
prove that condition (1) in Proposition 3.7 is satisfied (with (G,W) replaced
by (M,W ′)).

Suppose we are given the data of (1). Clearly, ps = psn · · ·ps1, and so ps
is continuous, since G is a topological groupoid. Since s(x) = 1x, there is a
restriction s′ of s to a neighbourhood of x such that Im(ps) ⊆ W . Since p maps
W ′ homeomorphically to W , then s′ is continuous and has its image contained
in W . So (1) holds, and by Proposition 3.7, the topology on W ′ is extendible
to make M a topological groupoid. �

Remark 3.9. It may seem unnecessary to construct the holonomy groupoid
in order to verify extendibility under condition (1) of Proposition 3.7. However
the construction of the continuous structure on M in the last corollary, and
the proof that this yields a topological groupoid, would have to follow more or
less the steps given in Aof and Brown [1] as sketched above. Thus it is more
sensible to rely on the general result. As Corollary 3.8 shows, the utility of (1)
is that it is a checkable condition, both positively or negatively, and so gives
clear proof of the non-existence or existence of non-trivial holonomy.

4. The Star Universal Cover of a Topological Groupoid

Let X be a topological space and suppose that each path component of X
admits a simply connected covering space. It is standard that if π1X is the
fundamental groupoid of X, topologised as in Brown and Danish-Naruie [3]
and x ∈ X, then the target map β : (π1X)x → X is the universal covering
map of X based at x. Let G be a topological groupoid. The groupoid MG
is defined as follows. As a set, MG is the union of the stars (π1Gx)1x. The
object set is the same as that of G. The initial point map α: MG → X maps
all of (π1Gx)1x to x, while the final point map β : MG → X is on (π1Gx)1x the
composition of the two target maps

(π1Gx)1x
β→ Gx

β→ X.



Holonomy and extendibility 87

As explained in [9] there is a groupoid multiplication on MG defined by con-
catenation, i.e.

[b] ◦ [a] = [ba(1) + a]
where the + inside the bracket denotes the usual composition of the paths.
Here a is assumed to be a path in Gx from 1x to a(1), where β(a(1)) = y, say,
so that b is a path in Gx, and for each t ∈ [0, 1], the product b(t)a(1) is defined
in G, yielding a path b(a(1)) from a(1) to b(1)a(1). It is straightforward to
prove that in this way MG becomes a groupoid, and that the final maps of
paths induces a morphism of groupoids p: MG → G. If each Gx admits a
simply connected cover at 1x then we may topologise each (MG)x so that it is
the universal cover of Gx based at 1x, and then MG becomes a star topological
groupoid, which means each star (MG)x has a topology such that each right
translation (and hence each left translation) is a homeomorphism

We call MG the star universal cover of G.
If X is a topological space which has a simply connected cover and G =

X×X, then MG = π1(X). If G is a topological group, then MG is a universal
cover of G.

Theorem 4.1. Let G be a locally sectionable topological groupoid in which each
star Gx is path connected and has a simply connected cover. Let W be an open
subset of G containing OG such that W = W−1 and W generates G. Suppose
that each star Wx = W ∩ Gx is connected and simply connected. Then the
groupoid MG constructed above may be given a structure of topological groupoid
such that each star (MG)x is a universal cover of Gx and W is isomorphic to
an open subset W̃ of MG.

Proof. To get a topology on MG as required we use Corollary 3.8. For this we
first define a map ı̃: W → MG as follows: Let u ∈ W(x,y), where W(x,y) =
W ∩ G(x,y), then u ∈ Wx. Since Wx is path connected, there is a path a in
Wx from 1x to u. Here note that 1x ∈ Wx since OG ⊆ W . Define ı̃(u) to be the
unique homotopy class of a in Wx. Note that since Wx is simply connected, ı̃
is well defined.

The map ı̃: W → MG is injective. For if u,v ∈ W such that ı̃(u) = ı̃(v),
then we have p̃ı(u) = p̃ı(v) and so u = v.

Let W̃ denote the image of W under the map ı̃: W → MG. Thus W̃ has
a topology such that the map ı̃: W → W̃ is a homeomorphism. Note that by
assumption the pair (G,W) satisfies the conditions 1-4 of Corollary 3.8. So
to apply Corollary 3.8 to the pair (MG,W̃), we only need to prove that the
subset W̃ generates MG as a groupoid. We prove this in the following Lemma.

Lemma 4.2. The subset W̃ generates MG as a groupoid.

Proof. For this let [a] ∈ MG(x,y). So a is a path from 1x to g ∈ G(x,y). Let
S ⊆ [0, 1] be the set of s ∈ [0, 1] such that as = a|[0,s], the restriction of a
to [0,s], can be written as = an ◦ · · · ◦ a1 for some n and Im ai ⊆ W . Since



88 O. Mucuk and İ. İçen

S ⊆ [0, 1], S is bounded above by 1, and so u = sup S exists. Then we prove
the following:

i) u ∈ S
ii) u = 1.

To prove (i), let a(u) ∈ G(x,xu). Then the map f : [0, 1] → Gxu defined by
f(t) = a(t)(a(u))−1 is continuous and f(u) = 1xu ∈ W . Hence there is an � > 0
such that f([u− �,u + �]) ⊆ W . Hence the composition map

δW ◦ (f ×f) : [u− �,u + �] × [u− �,u + �] → W ×α W → G

(t1, t2) 7→ a(t1)(a(t2))−1

is continuous, where δW is the restriction to W ×α W → G of the groupoid
difference map δ : G ×α G → G, (b,a) 7→ ba−1. Hence there is an �′ > 0 such
that �′ < � and

δW (f ×f)([u− �′,u + �′] × [u− �′,u + �′]) ⊆ W (?)

Since u = sup S, there is an element s ∈ S such that u − �′ < s. Hence as
can be written as an ◦ · · · ◦ a1 for n with Im ai ⊆ W and so we have that
au = an+1 ◦·· ·◦a1 where an+1(t) = a(t)(a(s))−1 for t ∈ [s,u]. By (?) we have
that Im an+1 ⊆ W . Hence u ∈ S.

To prove (ii) suppose that u < 1. Since u ∈ S, we have au = an ◦ · · · ◦ a1
for some n such that Im ai ⊆ W . Let ai(1) = gi ∈ G(xi−1,x) with x0 = x and
xn = y. Hence we have a(u) = gn ◦ ·· · ◦g1 and the path a can be divided into
small paths as

a = a(u + �) + a(u) + (an ◦ · · · ◦a1)
where Im ai ⊆ W . Since the map

[u, 1] → Gxn, t 7→ a(t)(a(u))
−1

is continuous there is an � > 0 such that a(t)(a(u))−1 ∈ W for t ∈ [u,u + �].
Hence au+� = an+1 ◦ (an · · ·a1) with an+1(t) = a(t)((a(u))−1 for t ∈ [u,u + �].
Hence we have that au+� ∈ S, which is a contradiction. This proves that
u = 1. �

Hence by Corollary 3.8, the groupoid MG has a unique structure of topo-
logical groupoid such that W̃ is open in MG and p: MG → G is a morphism
of the topological groupoids. �

Acknowledgements. We are grateful to Ronald Brown for introducing us
to this area and for his helpful encouragement. We would also like to thank
the referee for several helpful comments.



Holonomy and extendibility 89

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Received January 2002 Revised September 2002

O. Mucuk

Erciyes University,
Faculty of Science and Art,
Department of Mathematics,
Kayseri,
Turkey.

E-mail address : mucuk@erciyes.edu.tr

İ. İçen

İnönü University,
Faculty of Science and Art,
Department of Mathematics,
Malatya,
Turkey.

E-mail address : iicen@inonu.edu.tr


	Holonomy, extendibility, and the star universal cover of a topological groupoid. By O. Mucuk and I. Içen