C AHIERS DE
TOPOLOGIE ET GÉOMÉTRIE DIFFÉRENTIELLE CATÉGORIQUES
R ONALD B ROWN
O SMAN M UCUK
The monodromy groupoid of a Lie groupoid
Cahiers de topologie et géométrie différentielle catégoriques, tome 36, no4 (1995), p. 345-369
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THE MONODROMY GROUPOID OF A LIE GROUPOID by
Ronald BROWN and Osman MUCUKCAHIERS DE TOPOLOGIE ET
GEOMETRIE DIFFERENTIELLE CATEGORIQUES
Volume XXXVI-4 (1995)
Resume: Dans cet
article,
on montre que, sous des condi- tionsgénérales,
Puniondisjointe
des recouvrements universels des étoiles d’ungroupoide
de Lie a la structure d’ungroupoide
de Lie dans
lequel la projection possède
unepropriete
de mon-odromie pour les extensions des
morphismes
locauxreguliers.
Ceci
complete
un rapport 46ta,iU6 de r6sultats announces par J.Pradines.
Introduction
The notion of
monodromy groupoid
which we describe here a.rose from thegrand
scheme of J. Pradines in theea.rly
1960s togeneralise
thestandard construction of a
simply
connected Lie group from a Lie al-gebra
to acorresponding
construction of a. LiegrOllpOld
from a Liealgebroid,
a notion first definedby
Pradines. These results werepub-
lished a.s
[20, 21, 22, 23].
The recent surveyby
Mackenzie[17] puts
theseresults in context.
The construction
by
Pra.dines involved severalsteps.
One was the passage from the infinitesimal Lie
algebroid
to alocally
Lie
groupoid.
This we will not deal "1ith here.Next was the passage from the
locally
Liegroupoid
to a Liegroupoid.
In the ca.se of groups, this is a
simple, though
notentirely trivia.l, step,
and ispart
of classicaltheory. However,
in thegroupoid
case, instead of thelocally
Lie structureextending directly,
there is a,groupoid.lying
over the
original
one and which is minimal withrespect
to theproperty
that the Lie structureglobalises
to it. Thisgroupoid
may he called theholonomy groupoid
of thelocally
Liegroupoid.
This new result is themain content of Theor eme 1 of the first note
[20].
Its construction isgiven
in detail(but
in thetopological case)
in[2] (see
section 5below).
Finally,
there is a need to obtain a maximal Liegroupoid analogous
to the universal
covering
group in the group case, and in some senselocally isomorphic
to anyglobalisation
of thelocally
Lie structure. Thisgroupoid
may be called themonodi-omy gT’oupoid,
or star universal cov-ering groupoid. Any glohalisattion
of thelocally
Lie structureoriginally given
is sandwiched between theholonomy
andmonodromy groupoids by
star universalcovering morphisms.
A feature of universal
covering
groups is the classicallVlonodromy Principle.
This is animportant
tool forextending
localmorphisms
onsimply
connectedtopological
groups, and is formula.ted forexample
inChevalley [8], p.46,
so as to be useful also forconstructing
maps onsimply
connectedtopological
spaces. The statementby Chevalley
ismore
neatly expressed
in thelanguage
ofgroupoids
a,s follows:if B
is asimply
connected space, then thetopological groupoid
B xB,
with
multiplication (y, z)(x, y) = (x, z),
has tlzeproperty
thatif f :
W ->H is any local
morphism frorn
an open connected subset 1/1/o f
B x Bcontaining
thediagonal,
to agroupoid H, then f
extends to amorph,ism
on B x B.
The word
monodromy
is alsowidely
used for situations ofparallel transport
aroundloops, yielding morphisms
from a fundamental group to asymmetric
group whoseimage
is called themonodromy
group(see Encyclopaedic Dictionary
ofMathema.tics, Iyanaga
and Kawad[14]).
It is notable that Poinca,r6’s 19.5 paper which defined the fundamen- tal group is concerned at that
point
with themonodromy
ofcomplex
functions of many variables.
Pradines
required
for his results theextendibility
of localmor phisms,
and realised that these uses of
monodromy
had a common basis andgeneral formulation,
as atheory
of extensions of localmorphisms
ondifferentiable
groupoids,
now called Liegroupoids.
Hisresult,
whichwe call the
Monodromy Theorem,
see section 6below,
was stated asTh6or6me 2 in Pradines
[20].
The note
[20] gives
no indica,tions of the constructions of thesegroupoids
or of theproofs.
In the years1981-85,
Pradines outlined to the first author the constructions of theholonomy
andmonodromy groupoids
and theproofs
of thetheorems,
and anincomplete
sketch waswritten up as Brown
[3].
The results onholonomy
were worked up in thetopological
case in the thesis of M.A.-F.E.-S.Aof, [1],
and a furtherrefined
result,
under still weakerassumptions,
ispublished
in Aof andBrown
[2].
The main results of this paper can be summarised
roughly
as follows.Recall that the stars of a
groupoid
are the fibres of the source map.Theorem A Let G be a Lie
groupoid of differentiability
class r > 1 and whose spaceof objects
isparacompact. Suppose
the starsGx
atthe vertices
of
G arepath
connected. Let IIG denote thedisjoint
unionof
the universal coversof
the starsGx for
allobjects
xof G,
and letp : IIG -> G denote the
projection.
Then there is atopology
on IIGmaking
it a Liegroupoid
and such that p restricts on each star to tlze universalcovering
map.Theorem B Under the
assumptions of
TheoremA,
letf :
V -> H be alocal
morphism from a neighbourhood
Vof
the identitiesof
G to a Liegroupoid
H. Then V contains aneighbourhood
Wof
the identitiesof
Gsuch that the restriction
of.f
to 111lifts
to a’m,oTphism of
Liegroupoids
IIG -> H.
We call IIG the
monodrorrzy groupoid
of the Liegroupoid
G.The two theorems are
proved concurrently,
and ar egiven
in moregenerality
in section 6. The structure of theproof
isexplained
in section1.
Note that in the
locally
trivial case, Mackenzie[16] gives
a non-trivial direct construction of thetopology
on IIG and proves also that this IIG satisfies amonodromy principle
on theglobalisability
of continuous localmorphisms
on G.In the case G is a, connected
topological
groupsatisfying
the usuallocal conditions of
covering
spacetheory,
themonodromy groupoid
IIGis the universal
covering
group, while if G is thegroupoid
X xX,
for X a
topological
space, themonodromy groupoid is, again
undersuitable local
conditions,
the fundamentalgroupoid 7rlX.
Thuspart
of the interest of these results is ingiving
a widerperspective
to thefundamental
groupoid
of a space.However,
the construction for the fundamentalgroupoid
of a space isessentially
contained in thelocally
trivial case dealt with in Mackenzie
[16].
In a
companion
paper, Brown and Mucuk[7],
wegive
the basis ofone of Pradines’ intended
applications
ofholonomy by showing
how afoliation on a
paraco1l1pact
manifoldgives
rise to alocally
Liegroupoid,
and how the existence of non-trivial
holonomy
of a foliationgives
ex-amples
ofnon-extendibility
of alocally
Liegroupoid.
It is notable how well the formaltheory
of theholonomy groupoid
fits with the intuition.Acknowledgements
We would like to thank: Jean Pradines for his generous
sharing
of ideasin this area, for
permission
to use hisunpublished method,
and foracting
as External Examiner of Aof’sthesis;
Kirill Mackenzie for con-versa,tions,
andletters;
and both of them forhelpful
comments on earlierdrafts. The second author is
grateful
to theMinistry
of Education ofTurkey
forsupport
of his studies for thisresearch,
which formed a part of his PhD wor k atBangor (Nlucuk [18]),
and to the examiner s(Kirill
Mackenzie and Mark
Lawson),
forhelpful
comments.1 Outline of the proof
In the
following
sta,tements, we omit certain local conditions which arein fact necessary for the
validity
of theproof.
The
topology
on HG is constructed via agroupoid M(G, 1/11)
whichis defined to
satisfy
amonodromy principle, namely
theglobalisability
of local
morphisms
defined on an open subset W of G such thatOG
CHf C G. At this first
stage, M(G, W )
is not a Liegroupoid
but hasthe weaker structure of a "star Lie
groupoid" . Also,
theMonodromy Principle
which it satisfies is weak, in that smoothness of themorphism
is not involved. This We
explain
in section 3.However, using
an extra"locally
sectionable"condition,
we con-struct the structure of Lie
groupoid
onM(G, Tiff).
Thisstage
r elies onmethods of
holonomy,
which we outline in section 5. The method is describedcompletely by
Aof and Brown in[2].
Next, we relate this
groupoid M(G,W)
toIIG,
which isdefined, following
Mackenzie[16],
so that its stars are the universal covers of thestars of G. We
give
conditions under which there is anisomorphism
ofgroupoids M(G,W) ->
IIG(theorem 4.2).
This identifies the stars ofM(G, W)
as the universal covers of the stars of G. It also allows theimposition
of atopology
on lIG from thetopology
onM(G, W).
Thecombination of these two
approaches gives,
in section6,
thegeneral
formulation and
proofs
of Theorems A and B of the Introduction.2 Star Lie groupoids
Let G be a
groupoid.
We writeOG
for the set ofobjects
ofG,
and alsoidentify OG
with the set of identities of G. An element ofOG
ma.y be written as x or1x
as convenient. We writea, B:
G ->OG
for the sourceand
target
maps,and,
asusual,
writeG(x,y)
fora-’ (x)
nB-1(y),
ifX, y E
OG.
Theproduct hg
of two elements of G is defined if andonly
if cxh =
¡3g,
and so theproduct
map 7:(h, g) H hg
is defined on thepullba.ck Gax(3G
of 0: and0.
Thedifference
map 8: G Xa G - G isgiven by 8(g, h)
=gh-1,
and is defined on the doublepullback
of Gby
a.
If x E
OG,
and W CG,
we writeWx
for W na-lx,
and callWx
the star of W at x. The
following
definition isgiven
in Pradines[20]
but
replacing
the term "star"by "a-";
a similar remarka,pplies
toDefinition 1.4. The reason for our
change
ofterminology
is that notation other than a is often used for the source map.In order to cover both the
topological
and differentiable cases, we use the term C’’ manifold for r >-1,
where the case r = -1 dealswith the case of
topological
spaces and continuous ma,ps, with no localassumptions,
while the case r > 0 deals as usual with Cr manifolds and C’’ maps. Of course, aC’
map isjust
a continuous map. We then abbreviate C’’ to smooth. The terms Lie group or Liegroupoid
will theninvolve smoothness in this extended sense.
Definition 2.1 A
locally
.star Liegroupoid
is apair ( G, 11l) consisting
of a
groupoid
G and a, smooth manifold W such that:(i) OG
C 14/ CG ;
(ii)
W is thetopological
sum of thesubspaces vTlx
= W na-1 x,
x E
OG;
(iii) if 9
EG,
then the set W nWg
is open in W a.nd theright
translation
Rg: Wg-1
n W -> W nWg,
w -> wg, is adiffeomorphism.
Remark. In
[2],
theassumption
is also made that Wgenerates
G.Since this
assumption
is notrequired
in some of thefollowing results,
we omit it from the
definition, although
it will hold in theexamples
westudy.
A star Lie
groupoid
is alocally
star Liegroupoid
of the form(G. G),
and the notation
(G, G)
is thenusually
abbreviated to G. Thestronger concept
of a Liegroupoid
will be definedlater,
and we will find that aLie
groupoid
G is a star Liegroupoid
if andonly
if G istopologically
thesum of the spaces
G(x, y )
for all x, y EOG . Also,
a star Lie group is notnecessarily
atopological
group, but is morea.nalogous
to what is calledin the literature a
semi-topological
group, in which all that is assumed is that left andright
translations are continuous. A Liegroupoid
G may beretopologised
as thetopological
sum of its stars to become a star Liegroupoid.
The
following simple proposition
shows that anylocally
star Liegroupoid
is extendible to a star Liegroupoid.
This isessentially Propo-
sition 1 of section 1 of Pra.dines
[20].
For the convenience of thereader,
we
repeat
theproof
a.sgiven
inProposition
5.2 of Aof and Brown[2].
Proposition
2.2(star extendibility)
Let(G, W)
be alocally
star Liegroupoid.
Then G may begiven
the structureof
star Liegroupoid
suchthat
for
all x EOG, Wx
is an open subsetof Gx.
Proof We define charts for G to be the
right
translationsfor g
EG(x,y)
and x, y EOG . Suppose
thath, g
EGx
andWyg
meetsWzh.
Then there are elements it EWy and v
EWz
such that 2cg = vh.So
(Rh)-l Rg
maps the openneighbourhood Wv-1u
n W of uin Wy
tothe open
neighbourhood
W nWu-1v
of v inWz.
So these charts definea smooth structure as
required.
0We shall use
Proposition
2.2 in the situation of thefollowing proposi- tion,
whose last statement, oncovering
ma,ps, will be used several times later. This statementis,
in the case G is a. group,closely
related to aresult of section 1 of
Douady-Lazard [10].
This result was asta,rting
point
for Pradines and us;however,
theassumptions
made there are different.Proposition
2.3 Let(G, W)
be alocally
star Liegroupoid,
and let i : W -> G be the inclusion. Let p : M -> G be amorphism of groupoids
which is the
identity
onobjects,
and supposegiven
i : W -> M such that(a) pi = i,
and(b) i(ww’) - (iw)(iw’)
wheneverw, w’, ww’ belong
to W.Let W
= i(W)
have the smooth structure inducedby
ifrom
that on W.Then
(M, W) is a locally
star Liegroupoid. Further, if
M isgiven
theextended star Lie
structzire,
then each map on stars px :Mx -> Gx
is acovering
map.Proof We have to prove that the conditions for
(M, W)
to be alocally
star Lie
groupoid
are satisfied. This is trivial for the first two conditions.For the
third,
let m E M. In order for W n W m to benon-empty,
wemust have m =
6-lii
where v -iv, u
= i’l.l for u, v E W . Since the smooth structure on W is also induced from that on Wby
the inverseof the restriction of p
mapping
W ->W,
therequired
condition on(M, W)
follow from that on(G, W).
Let g
EG(x, y).
ThenWy,
which is open inGy,
is takenby
theright
translation
Rg: Gy -> Gx
to the openneighbourhood Wyg of g
inGx.
Further
Since p
is theidentity
onobjects, j3h
= y. For ea,ch h EP-’(g),
therestriction p
I lvyh, Wyg
is a,diffeomorphism
since it is thecomposite
of .the
diffeomorphisms Wyh -> Wy -> 1,Vyg.
Finally,
the setsWyh
for all h such thatph
= g aredisjoint.
Forsuppose m E
vVyh n Wyk,
whereph
=pk
= g. Then there are ti, z, E X/such that m =
(iu)h
=(iv)k. Taking
p of thisgives
ug = vg. Hence u = v and so h = k.The sum is hence
topological
because each of the setsWy h
a.re open,by
definition of thetopology
in terms of charts. So pI Mx, Gx
is acovering
map. MFor our purposes, the use of this result is that the easy Lie structure to obta,in on the
monodromy groupoid
is that of star Liegroupoid.
We shall find the
following
result useful. It is due to Nla,ckenzie(private communication, 1986).
We firstgive
a, definition.Definition 2.4 Let G be a star Lie
groupoid.
A subset W of G iscalled star connected
(resp.
starsimply connected)
if for each x EOG,
the star
Wx
of 1;11 at x is connected(resp. simply connected).
So a starLie
groupoid
G is star connected(resp. star si1nply connected)
if eachstar
Gx
of G is connected(resp. simply connected).
Proposition
2.5 Let G be a star Liegroupoid which,
as agroupoid,
is
generated by
the subset W.Suppose
thatOG
C W and that W is star connected. Then G is star connected. A similar result holds with connectedreplaced by path-connected.
Proof
Let g
EG(x, y).
Then g = wn...1VI, where w, E 1;V nG(Xr-1,Xr),
with xo = x and Xn = y. Let 9r = wr .... w1.By
inductionover r we have
that gr
is in the same component a,s 9r-1 becausethey
both
belong
to1Vxr9r-1
which isconnected,
because of the homeomor-phism given by
theright
transla.tionwhere we take wo =
1x.
Hence g = gn a.nd gi are in the samecomponent
ofGx.
But gl and1x
are in14/z
which is connected. SoGx
is connected.That
is,
G is sta.r connected. Theproof
forpath-connectivity
is similar.o
Corollary
2.6Suppose further
that G is a Lie group, W ispath
con-nected,
and p : M ->G,
i : W -> M aregiven
a.s inProposition 2.3,
with M
generated by a(W).
Then the group structure on Mgives
M thestructure
of
Lie group such that p is acovering
map and amorphism of
Lie groups.
Proof The
assumptions imply
that M -> G is acovering
ma.p,by Proposition 2.3,
and that M isconnected, by Proposition
2.5. It isnow standard that the group structure on G lifts to M with the
given
identity
of M as theidentity
for this group structure. But theright
multiplication
on M with the first group structure is a lift of theright
multiplication
ofG,
and so coincides with that obtained from the newstructure. Hence the two structures
coincide,
a.nd the group M obtainsthe structure of Lie group as
required.
MIf G is a non-connected
topological
group, and p: G -> G is a univer- salcovering
map on eachcomponent
ofG,
then there is an obstruction inH3(TT0G, TT1(G, e))
to the group structure on Gbeing
liftable to atopological
group structure on G. This result andgeneralisations
of itare due to R.L.
’Ta.ylor [2.5].
Proofsly
moderngroupoid
methods aregiven
in Mucuk[18]
and Brown and R4ucuk[6].
3 The weak monodromy principle
The construction here is a
generalisation
to thegroupoid
case of a con-struction for groups in
Douady
and Lazard[10].
Let G be a star Liegroupoid,
and let W be any subset of Gcontaining X
=OG,
and suchthat W = W-1. Then W obtains the structure of
pregroupoid:
thismeans that W is a reflexive
graph,
in the sense that it has the struc- ture of maps aw,OW:
Vf1 -X,
i.: X -> Vf1 with awt =owt
=1x,
andfurther there is a
partial multiplication
on W in which if vu is definedthen
Bwu =
awv,(tBwu)u
=u(tawu)
= u, and ea.ch it E W has aninverse
u-1
such that uu-1 =tBwu,
u-1u = tawu. There is also anassociativity
condition. For further discussion ofthis,
see forexample
Crowell and
Smythe [9].
For our purposes, we do not needthis,
sincewe know
already
that W is embeddable in agroupoid.
There is a standard construction
M(W ) associating
to apregroupoid
W a
morphism
i: W ->M(W)
to agroupoid M(W)
and which is uni-versal for
pregroupoid morphisms
to agroupoid.
First form the freegroupoid F(W)
on thegraph W ,
and denote the inclusion W ->F(W ) by
u ->[u].
Let N be the normalsubgroupoid (Higgins [13],
Brown[4])
of
F(W ) generated by
the elements[vu]-1[v][u]
for all u, v E W suchthat vu is defined and
belongs
to W . ThenM(W)
is defined to bethe
quotient groupoid (loc. cit.) F(W)/N.
Thecomposition
W ->F(W) -> M(W)
is writteni,
and is therequired
universalmorphism.
In the ca,se W is the
pregroupoid arising
from a subset W of agroupoid G,
there is aunique morphism
ofgroupoids
p:M(W) ->
Gsuch
that pi
is the inclusion i: W -> G. It follows that i isinjective.
Clearly,
p issur jective
if andonly
if T11generates
G. In this case, wecall
M(W)
themonodromy groupoid
of(G, W)
and write itM(G, W).
We now resume our
assumption
that G is a star Liegroupoid,
andwe assume W is open in G. Let W =
T(W).
It follows fromProposition
2.3 that the
pair (M,
W), where M =M(W),
inherits thP structure of2.3 that the
pair (M, W),
where M ==M(W),
inherits the structure oflocally
sta.r Liegroupoid. Further,
this structure is extendible to make M into a star Liegroupoid,
such tha.t each ma.p on stars px :Ali,
->Gx
is a
covering
map.We can now state a
Monodromy Principle,
which we call "weak"because it involves no
continuity
ordifferentiability
conditions on maps.Theorem 3.1
(Weak Monodromy Principle)
Let G be a Liegroicpoid
and let W be an open subset
of
Gcontaining OG. Suppose
that G is starsimply
connected and W is star conn ected. Let H be cr,groupoid
overOG
and let
O:
14/ - H be amorphism of pregroupoids
which is theidentity
on
OG. Then
extendsuniquely
to agroupoid morphism, 0:
G - H.Proof
By Proposition 2.5, M(G, T/Tf)
is starconnected,
andby Propo-
sition
2.3,
p:M(G, W)
-> Gis,
when restricted to stars, acovering
mapof connected spaces. Since G is star
simply connected,
it follows that p is anisomorphism.
The result now follows from the universalproperty
forM(G, W).
A useful
special
case of Theorem 3.1 is thefollowing.
Let q: E ->OG
be a function a.nd let the
symn2etry groupoid Sq
of q he thegroupoid
over
OG
ofbijections Ex -> Ey
for all fibresEx
=q-1 (x)
of q, and allx, y E
OG.
Corollary
3.2 Let X be a connected andsimply-connected
space, letW be a connected
neigbourhood of
thediagonal of X
x X such that each sectionWx = ly
E X :(x, y)
EW}
is connected. Let0:
W -Sq
bea
morphism of pregroupoids. Suppose
eo E E isgiven.
Then there isa
unique function v: X -> E
such that?Pqeo =
eo andvy = O(x, y)x
whenever
O(x, y) is defined.
Proof This follows from the above Weak
Monodromy Principle, by
setting
forallxEX.
0Corollary
3.2 is stated inChevalley [8]
as theMonodromy Principle.
Note that in 3.1 and 3.2 there is no
topology given
on H orSq
and thereare no
assumptions
ofcontinuity
ordifferentiability
of0.
4 The star universal
coverof
astar Lie
groupoid
Let X be a
topological
space, and suppose that eachpath component
of X admits a
simply
connectedcovering
space. It is standard that ifTT1X
is the fundamentalgroupoid
ofX, topologised
as in Brown andDanesh-Naruie
[5],
and x EX,
then thetarget
map(3: (TT1X)x
- X isthe universal
covering
ma,p of X based at x(see
also Brown[4], Chapter 9).
Let G be a star Lie
groupoid.
Thegroupoid
IIC is defined as follows.As a
set,
IIG is the union of the stars(TT1Gx)1x.
Theobject
set of ITG isthe same as that of G. The function a: IIG ---7 X maps all of
(TT1Gx)1x
to x,
while,6:
IIG -> X is on(TT1Gx)1x
thecomposition
of the twotarget
maps
As
explained
in Mackenzie[16], p.67,
there is amultiplication
on TIGgiven by ’concatenation’,
i.e.where the + inside the bra.cket denotes the usual
composition
ofpaths.
Here a is assumed to be a
path
inG,
fromlx
toa(1),
whereB(a(1))
= y,say, so that b is a
path
inGy,
and for each t E[0,1],
theproduct b(t)a(1)
is defined in
G, yielding
apath b(a(1))
froma(1)
tob(1)a(1).
It isstraightforward
to prove tha.t in this way IIG becomes agroupoid,
andthat the final map of
paths
induces amorphism
ofgroupoids
p: IIG -> G.If each star
Gj,
admits asimply
connected cover at1,,
then we maytopologise
each(IIG)x
so that it is the universal cover ofGx
based at1x,
and then IIG becomes a. star Liegroupoid,
as remarked above(see
also Mackenzie
[16]).
We call IIG the star universal cover of G. Forexample,
if G = X xX,
then IIG is the fundamentalgroupoid 1rlX.
(In general,
one could call IIG thefundamental groupoid
ofG,
but thismight
causeconfusion.)
Let X be a
topological
spaceadmitting
asimply
connected cover.A subset !7 of X is called
liftable
if U is open,path-connected
and theinclusion U -> X maps each fundamental group of U
trivially.
If U isliftable,
and q: Y -> X is acovering
map, then for any y E Y and x E Usuch that qy = x, there is a
unique
map i: U - Y such that ix = y andqi
is the inclusion U -> X. Thisexplains
the termliftable.
We also need a result on
covering
maps and star Liegroupoids.
Proposition
4.1 Letq: H - G be a morphism of
star Liegroupoids
which is the
identity
onobjects
and such that on eachstar,
q:Hx -> Gx
is a
covering
mapof
spaces. Let TV be aneighbourhood of OG
in Gsatisfying
thefollowing
condition:(*)
W is starpath-connected
andW2
is contained in a starpath-
connected
neighbourhood
Vof OG
such thatfor
all x EOG, Vx
isliftable.
Then the inclusion i: W -> G
lifts
to a Liepregroupoid morphism
W - H.
Proof Let 1: W -> G
and j :
V -> G be the inclusions. For each x EG,
the inclusion
Vx -> Gx
liftsuniquely
tojx: Vx
->Hx mapping Ijc
tolx,
and this defines a liftix: 14/x
->Hx.
We prove tha.t the union of these mapsj:
W -> H is apregroupoid morphism.
Let u, v E W be such that uv E 1/1/. Let au = x, av = y. Since W is star
path-connected,
there arepaths
a, b inWx, Wy,
from u, vto
1x, ly, respectively.
Since uv isdefined, (3u
= y. The concatenation b o a is then apath
from 1.l1’ tolx.
SinceVf12
CV, b o a
is apath
inVx.
Hencej(b
oa)
is a lift of b o a to H.By uniqueness
ofpath liftings, j(b
oa) = ( jb)
o( ja). Evalua.ting
a.t0,
andusing i(uv)
=i(uv), gives i(uv) = (iu)(iv).
Hence i is apregroupoid morphism.
11Theorem 4.2
Suppo.se
that G is a star connected star Liegroupoid
andW is an open
neighbourhood of OG satisfying
the condition(*).
Thenthere is an
isomorphism
over Gof
star Liegroupoids M(G, W)
->IIG,
and hence the
morphism M(G, l11)
- G is a star universal,covering
map.
Proof
By Proposition 4.1,
the inclusion i: W -> G liftsuniquely
to amorphism
of star Liepregroupoids
i’: W -> IIG such thatpi’
= i. Theuniversal
property yields
amorphism 0: M(G, W) -
IIG of star Liegroupoids.
Hence for each x E X we have a commutativediagram
But px
and qx
arecovering
maps, qx is universal andM(G, W)x
isconnected. Hence
0.,: M(G, W)x
->(IIG)x
is anisomorphism. Hence O
is an
isomorphism.
0This theorem has a useful
corollary
in the group case .Corollary
4.3 Let G be apath-connected
Lie group and let W be apath-
connected
neighbourhood of
theidentity
eof
G such thatW2
is contained in aliftable neighbourhood of
e. Then themorphism M(G, HI)
- G isa universal
covering
map, and henceM(G, W)
obtains the structureof
Lie group.
Corollary
4.4(Weak monodromy principle)
Letf :
W -> H be a mor-phism from
the star Liepregroupoid
W to the star Liegroupoid H,
whereW
satisfies
the condition(*).
Thenf
determinesuniquely
anzo?-I)hz*sm f’:
IIG -> Hof
star Liegroupoids
such thatf’ j’ = f .
5 Locally Lie groupoids and holonomy groupoids
The a.im now is to obtain a. similar
type
of result for Liegroupoids
andLie
morphisms.
We recall that thelocally
trivial case is handled inMackenzie
[16].
Our aim is the
locally
sectionable case(see
Definition5.1).
Thetechnique
ofusing
here theholonomy groupoid
construction was out- linedby
Pradines to the first author in1982 (see
Brown[4]),
and forPradines
[20]
was ahappy application
of theholonomy
construction which he hadalready
found.We need a Lie version of the
globalisability
theoremproved
in Aof-Brown
[2]
for alocally topological groupoid.
The aim of this section is togive
the modifications necessary for the Lie case. Noessentially
new ideas are involved in the
proofs,
but we do make asimple
butcrucial deduction which clarifies the relation between
extendibility
andholonomy,
and inparticula.r gives
a useable condition forextendibility.
One of the
key
differences between the cases r = -1 or 0 and r > 1 is that for r > 1, thepullba.ck
of Cr ma.ps need not be a, smooth sub- manifold of theproduct,
and sodifferentiability
of maps on thepullback
cannot
always
be defined. We thereforea,dopt
thefollowing
definition of Liegroupoid.
Mackenzie[16] (pp. 84-86)
discusses theutility
of variousdefinitions of differentiable
groupoid.
A Lie
groupoid
is atopological groupoid
G such that(i)
the space of arrows is a smoothmanifold,
and the space ofobjects
is a smooth submanifold of
G,
(ii)
the source and.target
mapsa,B,
a,re smooth maps and are submer-sions,
(iii)
the domain G X a G of the difference map is a smooth submanifold of G xG,
and(iv)
the differencemap 6
is a smooth map.The term
locally
Liegroupoid (G, flV)
is defined later.The
following
definition is due to Ehresmann[11].
Definition 5.1 Let G be a
groupoid
and let X =Oc
be a, smoothmanifold. An admissible local section of G is a function s : U - G from
an open set in X such that 1.
as (x) = x for all x E U;
2.
Bs(U)
is open inX ,
and3.
Bs
maps Udiffeomorphically
toBs(U).
Let W be a subset of G and let W have the structure of a smooth manifold such that X is a submanifold. We say that
(a, B, W )
islocally
sectionable if for each w E W there is an admissible local section s : U ->
G of G such that
(i) sa(w)
= w,(ii) s(U)
C W and(iii)
s is smooth asa function from U to W. Such an s is called a smooth admissible local section.
The
following
definition is due to Pradines[20]
under the name"morceau de
groupoide différentiables".
Recall tha.t if G is agroupoid
then the difference map 8 is 8 : G XQ G -t
G, (g, h)
->gh-1.
Definition 5.2 A
locally
Liegroupoid
is apair (G, W) consisting
of agroupoid
G and a smooth manifold Hf such that:G1) OG C W C G;
G2)
W =W-1;
G3)
the setvV(8) = (W
Xavff)
n8-1(T;V)
is open in W xa W and the restriction of 6 toW(d)
issmooth;
G4)
the restrictions to W of the source andtarget
maps a and(3
aresmooth and the
triple (a, B, W)
islocally sectionable;
G5)
Wgenerates
G as a.groupoid.
Note
that,
in thisdefinition,
G is agroupoid
but does not need tohave a
topology.
Thelocally
Liegroupoid (G, Hf)
is said to be extend2:bl.e if there can be found atopology
on Gmaking
it a Liegroupoid
and forwhich W is an open submanifold. The main result of Aof and Brown
[2]
is a version of Th6or6me 1 of Pradines[20]
and was stated in thetopological
ca.se. In the smooth case it states:Theorem 5.3
(Pradines [20],
Aof and Brown[2]) (Globalisability
the-orem)
Let(G, W)
be alocally
Liegroupoid.
Then there is a Liegroupoid H,
amorphism 0 :
H - Gof groupoids
and ane-mbedding
i : : W -> Hof
W to an openneighborhood of OH
such that thefollowing
conditionsare
satisfied.
i) 0
is theidentity
onobjects, ’oi
=idw, 0-’(W)
is open inH,and
therestriction Ow : Ø-1(W) --+
Wof 0
issmooth;
ii) if
A is a Liegroupoid and
A - G is amorphism of groupoids
such that:
a) E
is theidentity
onobjects;
b)
the restrictionEW : E-1(W) ->
Wof E
is smooth andE-1(W)
isopen in A and
generates A;
c)
thetriple (aA, (3 A, A)
islocally sectionable,
then there is a
unique morphism E’:
A -> Hof
Liegroupoids
such thatOE’ =E
andE’a
=iEa for
a EE-1(W).
The