Math. Nachr. 74, 148-166 (1076)

Topological groupoids : 11. Covering morphisms and G-spaces

By R. BROWN (Bangor), G. DANESH-NARUIE (Tehran) and J. P. L. HARDY (Bangor)

(Eingegengen am 4.7.1974)

6 1. Introduction

The theory of covering groupoids plays an important role in the applications of groupoids (cf. [ll]), and in this theory there are two key results. One is that if Q is a groupoid there is an equivalence between the category Or, (Q) of operations of Q on sets (or Q-sets aa they are called) and the category t70vlQ of covering groupoids of Q. (This result seems to have been stated first in this form in [9], ulthough the constructions involved had been used previously.) The other is t h a t if Q is a transitive groupoidi), there is a bijection between the equivalence olassee of transitive covering groupoids of Q and the conjugacy classes of sub- groups of Q. (This result is an abstract formulation of known results on covering H;)acea and the fundamental groug-it is due to HIWINS [ll], page 110.)

The object of this paper is to prove topological versions of these results. For the first reeult there is no problem. We define the category XtfoulU of

i~o~~ological covering morphisms of the topological groupoid Q ; we follow EHRES- M A " [7] in defining the category ZOr, (a) of &paces; and we prove the equi- viilence of these categories. This allows us to give a number of useful examples in t.liese categories.

The second problem presents difficulties which are related to the fact that a. I riuisitive Q-spaoe need not be a homogeneous space of Q. However we present a ctorresponding result for the locally trivial case.

Jn general this paper is independent of [6]; however, we will not repeat any id' ihe results which appmr there.

Rom? of the results of this paper appeared in [6] and [lo]. During part of. ltth research the second author waa supported by an S. R. C. Research Grant 11/RG/2574,. and the third author was supported by an S. R. C. Research Student- r h l p

6 2. General Cme

ltooall that a morphism q: H-tG of groupoids is a covering morphism if for onvh y€Ob (H), the restriction of q mapping St,y-St& is a bijection. Let

a ) 'Itoneitive groupaide we deo d e d oonnected groupoib in the litaretare. 1 . --

144 Brown/Daneeh-Neruie/Hardy, Topologioal groupoids

# F o b (H) be given by the pullback diagram of sets. QZOb (H) - -0b (H)

l o b (q) -0b (U)

1 a. B

I fq : H -Q is a covering morphism, we have a lifting funotion at: U ZOb (H) 4 H assigning to the pair (8, y) the unique element h of St,y such that q(h) = g ; clearly 84 is inverse to (q, 8): H + a z O b (a). So we can state: q: a& is a covering morphism if and only if (q, 8): H+UROb (R) is a bijection. It is in terms of this function (q, Y) that the notion of topological covering morphism is most con- veniently phraaed.

Definition. A 33-morphism q : H -4 of topological groupoids is a topological covering morphhm if the function (q, 8): H+UZOb (a) is a homeomorphism.

In such a cam the inverse to (q, a') is written aq and called the lifting function. Notice that the identity Q-B is a topological covering morphism. Also the

composite of topological covering morphisms is again a topological covering morphism-this is one part of the following proposition

Proposition 1. Suppose given a cornmututive diagram of m q h h of twpological groupoias

in which q b a topological mering morphbm. Then p is a topological covering morphim if and only if r h a topological covering morphicnn.

The proof is given in 5 4. Let T3/Q be the category of topological groupoids over G; the full subcategory

of this on the topological covering morphisms is written TlPoulB. By Proposition 1 , the morphisms of this category are topological covering morphisms.

The basic examples of topological covering morphisms come from the relation between these and Q-spaces, whose definition is due to EHRESMAKN [7].

Definition. A (left) Q-space is a triple (S; p, q) where S is a topological space, p : S -0b (U) is a continuous function, and q : Q 2s -S, (a, 8 ) H a 8, is a continuous action with G ZS given by the pull-back diagram

azs .s

a! -- Ob (a) a'

Brown/Denesh-Naruie/Hardy, Topological groupoide 145

The action Q must satiafy the usual axioms (1) p(a - 8 ) =aa ( 2 ) b . (a-8)=(ba) - 8

whenever these expreesions are defined. (3) lp(,(,)g = a

By an abuse of language, we also say that S is a (left) G-space via p. A morphism of (left) Q-spaces (S, p, q) - (S', p', 9') consists of a continuous

function f: S -S' such thatp'f=p and f (a - 8 ) = a .!(a) whenever a 8 is defined. So we have a category S o p (Q) of Q-spaces.

Theorem 2. The cdegozies c7IpovlQ and T o p (G) are equivalent.

Proof. We define functors r: L7IpovlG -302) (a), @: S o p (G) - c 7 h ~ l Q and natural equivalences IW r l ,@Fr 1.

Let q : H - Q be a topological covering morphism, and let 8,: GZOb (H) -H be the continuous inverse of (q, 8). Let Q = ~ s , : GZOb (H)-+Ob (61). Then Q is continuous and it is clear that r(q) = (Ob (H); Ob (q), 9) is a G-space. A map of covering morphisms induces a morphism of 0-spaoea, so we have a functor as required.

Now let (8; p; q) be a (left) Q-space. Then we make GZS into a topological groupoid with object space S as follows. Theinitial and final maps are defined by #(a, 8 ) = 8 , a (a ,8)=0. i ;6( (b , t ) , (a,g))=(b-a,e); u(.9)=(lP,, 8); a(a ,s )=(a-1 ,0-8) . The verificmtion that QZS is a topological groupoid is straight forward. The jxojection q : G z L 3 - 4 is a 33-morphism and it is clearly a topological covering morphism since (q, 8) : Q ZS -c Q 2s is the identity. Once again we have a functor 0 aa required.

Clearly I!D= 1. The natural equivalence 0rr 1 is given by the fact that if q : H -0 is a topological covering morphism, then sq: Q Z Ob (H) -c H is an iso- morphism of topological groupoids.

Remark. It is clear from the above proof that there is a general result. Let Ip be an arbitrary category with pull-backs, and let 0 be a groupoid object in Ip.

A covering groupoid of G is a morphism q : H - G of groupoid objects in Ip such that (q, 8') : H -Q Ob (H) is an isomorphism in g. Then the categories Ip- Op (Q) and g- IpovIG are equivalent.

We can now give examples of coverings and G-0paces by giving either first, and can also translate concepts from one category into the other.

Example 1. If G is a topological groupoid, then Ob (a) is a left G-space via the identity, the action being given by a - z = y for zEOb (a) andaEQ(z, y). Note that this action derives from the identity covering morphism 1 : G 4.

If (S; p, q) is a G-space, then the orbik of S are the equivalence classes under the equivalence relation 8 - t if and only if t =a - s for some a in Q. These orbits can be identified with the (abstract) components of the groupoid Q%S defined 40 mth. Na.ohr. Bd. 74

146 Brown/Danesh-Naruie/Hardy, Topological groupoida

in the proof of Theorem 2. In particular, G operates transitively on S (i.e S has only one orbit) if and only if the groupoid GZ’s is transitive.

The set of orbits of S under this (left) sction of 0 is written Q\S, and this set is given the identification topology with respect to the projection 8 -@8. How- ever, unlike in the m e of groups, this projection need not be an open map.

Example 2. Let f: S-T be an identification map which is not an open map. Define a topological groupoid Q by Ob (a) =S, and Q(a, a’) =[zI if&) =k f(a’), and otherwise G(s, 8’) ={(a, a’)} ; thus Q haa its topology as a subset of S xS. The unique composition in G which makes it a groupoid also makes it a topological groupoid. Then S, the object space of Q, is a left G-space, and the orbit space G\S may bo identified with T. So in this case S-Q\S is not open.

If (S; p, p) is a Q-space, and aES, then the atability grow (or isotrqy group) of a is Q,={aEG: a - 8 = 8 } . Clearly the covering morphism p: Q%S-tG maps t h o object group (QZS) {a} isomorphically to G,.

A topological covering morphism q : H + G is regulccr if for any zEOb (Q) and a<G{z}, then either all or none of the elements of q - l ( c r ) lie in object groups of I1 It is easily verified that p is regular if and only if for each yEOb (a), the groiii) p(H{y]) is normal in U{pz}. Similarly, a G-space (S; p, p) is regular if for all a in A’, the stability group Q, of 8 is normal in @@8}.

In a similar way to left G-spaces we define a right &qmce (p, p; s), where p:S-.Ob(Q) is continuous, p: S%G--8, ( a , a ) H 8 - a J is continuouswithS%Ggivei by the pullback

sza -.S I k G -- Ob (Q)

a The axioms are that p(a - a) = 8’a, a ld,) =a, (a - a ) - b = a - (ub) whenever they run defined. The space of orbits of the action is again well-defined, and for a riplit Q-space S is written S/G. We can transfer left Q-spaces to right Q-spaces, in the same way as is don(, fiir

actions of groups, by the rule 8 - a = a-’ . 8.

We could also take our above definition of covering morphism to be II ItJ/.

couering and define a right-covering q : H 4 to be a morphism such that ( ; I , f/) I H -0b (H)%G is a homeomorphism. However it is easy to see tha ta~~-mor~~ l i i c i i i is a right-covering if and o d y if it is a left-covering.

Example 3. Let p : X - Y be a covering map of topological spaces, wl1lwc~ A’ Y are locally path-connected and semi-locally l-connected. In [4] u “ l l h c l topology” is described on the fundamental groupoids nX, nY so that they bcwc~ri~@ topological groupoids, and it is proved that np : nX --nY is a topological o o v c i ~ ~ l ~ ~ y morphism. So X obtains the structure of a nY-space.

Another example of covering morphisms comes from the morphism grou Iicdd (X, a) used in [3].

Brown/Denesh-Naruie/Hardy, Topological groupoids 147

l ~ t X, Q be topological groupoids. The set of morphisms X - 4 is written A l ( X , Q); we give this the compact-open topology (as a set of functions from the r i m e of elements of X to the space of elements of Q). We put a topology on the proupoid ( X , Q ) so that it becomes a topological groupoid with object space

If A, B are sets, let P(A, B) denote the space of functions A - B with the c~oiiipact-open topology. The elements of (X, Q) can be taken as pairs (f, 8) such tlwt f(z)= 8'8(z) for each sCOb (X) (121 p. 196). Thus (X, a) can be identified with the pull-back in the diagram

M(X, a).

( X , Q ) - I

a.1

I L I I ~ this defines the topology on (X, a). The structure maps of (X, a) me: (i) 8': ( S , a) - M ( X , Q) aa in the above diagram; (ii) a: (X, Q) -+M(X, G) is given by ($,Q) H g , where g(a) = (88~) o ( f a ) o (M'a)-j, a € X; (iii) the identity u : M ( X , Q) - -(XJ G) is given by u(f)=(f, /), where/: ZH l,(=), zCOb (X); (iv) the inverse u : (A', Q) - (X, a) is given by (f, 6) - (a(f, 6), 8-1). Clearly all these structurefunctione two continuous, and so (X, Q) is a topological groupoid.

Proposition 3. Let i : A - X be a morphiem of topological growpoi&s sw;h that Oh (i) is a heomorphiam. Then i*: ( X , G ) - ( A , Q ) is a topological covering

Proof. Consider (i*, a'): (X, @ - ( A , Q ) % M ( X , Q). Now (A, Q ) = M ( A , Q)Z k P ( 0 b (A), a). So we define an inverse to (i*, 8) to be the composite

M ( A , G)%F(Ob (A), Q ) % M ( X , Q) %P(Ob (A), G ) % M ( X , Q) Ob(i)- 1 --P(0b (X), Q ) % M ( X , Q) .

\Ye can combine left and right actions in a way which has important appli- w18ions.

Let G, H be topological groupoids. A G - H-bispace is a quintuple (qJ 1y; S; p , q) wliiwo S is a topological space, (8; p, q) is a left G-space, (q, y ; S) is a right H-space # I N l

q(a =&) P(8 .b ) = P W a - (s - b) = (a.s).b

wlmover s€S, aEQ, b € H and a - 8 , s - b are defined. Notice that if yEOb(H) then I h uction of Q makes q-Q) a left G-space, while if Z€ Ob (Q) then the action of H lrrirltoti ~ - I (z) a right H-space.

,

411'

13y an abuse of language we also say S is a a - H-bispace via p - q. A etandmd example of G -Q-bispace is the groupoid Q itself via a - a' with

148 Brown/Denesh-Naruie/IEerdy, Topological groupoids

left and right actions given by composition in G. In particular, if zEOb (Q) then St#= al-i(z) is a left Q-space via al. These particular actions are important in what follows.

Example 4. One of the constructions in the proof of Theorem 2 generalises to Q - H-bispaces. Let S be a Q - H-bispace via p - q. Then we define a topological groupoid Q?S%H to have object space S and elements the triples (g,8, h) in Q XS X H such that p(8) = al(g), q(s) = a@). The initial and final maps are given byal(g,s,h)=s, ~(g,8,h)=g.s.h;compositionis(g8,8',h').(g,s,h)=(g'.g,8,h'.h); the unit and inverse functions are u(s) = (l,s, 1) and a(g, 8, h) = (g-1, g - 8 - h, h- 1)

respectively. Now let a! z H be the subgroupoid of Q X H of pairs (8, h) such that there is an 8 in S with al(g)=p(8), a (h)=q(s) ; let r : Q%SEH--Q%'rr be the pro- jection morphism. Then (r, al ) : QzS%H -QHH %S is (8, s, h) w (g, h, s), and is a homeomorphism. So r is a topological covering morphism.

An important use of Q - H-bispaces is in constructing left actions of 0 011

spaces of H-orbits. Suppose S is a Q - H-bispace via p -q. Although the left aotion of Q on S defines a left action of Q on the set SIH, as is easy to check, there is a difficulty in proving continuity of this action due to the fact that a pull-beck of identification maps need not be an identification map; this difficulty can bv overcome in the useful special case given by the following theorem.

Theorem 4. ff S is a 0' - H-bkpoce in which H ia a tolpologicrr2 growp and Ob (U) i s HAIJSDORFF, then the action of Q on S determines on the orbit space SIH l h ~ structure of a left Q - q m e .

The proof is given in 8 4. Let Q be a topologiortl groupoid, let sEOb (Q) and let D be any subgrou]~ of'

Q{z }. SinceQ is aQ -Q-bispace, i t follows by restriction that St# is a Q -D-bispltcw, We define Q, to be the space (St#)/D of left cosets of D. It is easily verified t l ~ t Q, :8 a Ti-space if and only if D is closed in S t s , and we will see later (Proliw sition 14) that this implies QD is HAIJSDORFF.

Corollary 6. If Ob (Q) is HAIJSDORFF, and D is a subgroup of Q then left lndh plication g i v a QD the structure of a left Q-space.

This follows easily from Theorem 4. Let Q be a topological groupoid and 8 a transitive left Q-space viap: S - 011 ( t i ) ,

Let sES, and let D be the group of stability of s. Then the mapping tp,: Sl&(r)** -S, h c , h 8 , is surjective and defines a continuous bijection tp; : QD -S. In gw1Wd tp; will not be a homeomorphism-however, if tp; is a homeomorphism for title I in S then p; will be a homeomorphism for each 8 in S, and in this case wo t w l l A' 4 homogenema space of Q .

If Q is a topological ~ o u p , there are useful conditions for a G - ~ l i u . ~ l o la homogeneous. For example, if Q is a topological group which is com]ilt~lv fitid satisfies the second axiom of countability, while 8 is a HAIJSDORFF, IIOII-II)FIY(Y Q-space, then S is homogeneous (cf. [l] and [14]). For the groupoid CWO, H'P later a rather different type of condition for a Q-space to be homogenoonn,

Brown/Daneeh-Naruie/Hardy, Topological groupoids 149

Let f : H - Q be any morphism of topological groupoids, and let y c O b ( H ) ; the characteristic group off at y is the subgroup f ( H { y } ) of Q{f(y)} . Now Corollary 6 can be restated in a way which shows it to be the basic existence theorem in the theory of topological covering morphisms.

Theorem 6. Let Q be a transitive topological grougmid with HAUSDORFF object space. Let x€Ob (a), and k t D be a subgroup of Q(x). Then there &te a iopologicul covering m r p h i m g: H - 4 , such that H is transitive and there b a y in Ob ( H ) such that the &racte&tk-group of q at y is D. Further q : H -Q 8at i s fh the follouring univer8al property:

i f r : K-Q is a: topological covering rnor(phisnz, zCOb ( K ) ia such that r ( z ) = x and the characte&tic group of r at z c W i n s D, then there is a; un@e tolpologkal oovering m o r p h k 6 : H -K such that s ( y ) = z and rs=q.

The proof is given in 8 4. A special case of this theorem is when D is the trivial subgroup, so that

Q,=St#. Then Q j?St&-B is called a universal topological covering rnorphism of Q, since it covers any other topological covering morphism of Q.

8 3. The locally3iivial case

An important class of topological groupoids introduced by EHRESMANN is !hut of locally trivial groupoids. These arise in nature because of their close rvrnnection with principal bundles. From our point of view they are convenient 1)ooause one can construct continuous lifts of morphisms (Proposition 7).

Definition. A topological groupoid Q is locclwy trivial if for each x,COb (Q) !hare is an open neighbourhood U, of x, and a continuous function A,: 27,-B ati011 that A,(x)EQ(x,, x) for all x in U,. (There is clearly no loss in generality in runuming A,(x,) = is,.)

The following example shows that local triviality is a restriction. IExsmple 6. Let Q be a non-trivial topological group with the discrete topology,

n t d lot S be Q with the indiscrete topology. Multiplication in the group turns S I i i l , ~ a left B-space. The topological groupoid BZh’ satisfies: (UgS) (8, t ) haa r~rcit~ly one element for all s, tCS. However G%S is not locally trivial, since the iiIrrrCity mapping S -Q is not continuous.

Ha jtpose now given a diagram of morphisms of groupoids, H

1q F - Q

f H’hicrh q is a covering morphism. Suppose further that P is transitive and that .

#f 0 1 ) (N), zE Ob ( P ) satisfy q ( y ) = f ( z ) . Then a necessary and sufficient condition

150 Brown/Denesh-Nanrie/Hardy, Topologioal groupoida

for f to lift to a morphism f * : P + H such that f * ( z ) = y is that f(F{z)) is a subgroup of g(H{y}) ([ll], page 107). However that proof of sufficiency involves choosing a tree in F, and so cannot be expected to go over to the topological cam without additional assumptions.

Suppose given a commutative diagram H

such that F, G, H are topological groupoids, f is a c73-morphism, q is a topolo- gical covering morphism, and f * is a 3-morphism.

Proposition 7. If F is locally trivial then f * is continuow, i.e. ia a c 7 3 - m p h i s m . The proof is given in 0 4. This proposition enables algebraic results given in [2], [ l l] to be translated

into the topological case as follows. Let q: H-tQ, q': H'-Q be topological covering morphisms such that H , H'

are transitive let ycOb ( H ) , y'EOb (H' ) be such that q(y)=q'(y'), and let C =

Corollary 8. If H i8 locally trivkd, then tAere is a unique topological coverinfl morphiam r : H + H such that q'r=q and r(y)=y' i f and only i f CSC'. Furthvr i f H also is locally trivial then r is a topological isomorphism i f and only i f C = C".

Corollary 9. Let H be a locally trivial t o p o l o g ~ l covering groupoid of Q swlr that H ( x , y) has one element for all objects x , y of H . Then H is a universal topologiciii covering groupoid of Q.

To complete the story we need criteria for a topological covering of Q to hn locally trivial-this is most conveniently phrased in terms of G-spaces as in I,lw next proposition. Theorem 12 then gives a useful condition for the existencn of' locally trivial topological coverings of 0.

Let f : X - Y be a map of topological spaces. We say f is a submersion if l i i r

each xEX there is an open neighbourhood U of f ( x ) and a continuous func+iciii A : U - X such that Af(x)=x and f A = l o .

=!dH{y}), C'=qW'{y'}).

Notice that a submersion is necessarily an open mapping. Proposition 10. Let G be a topological groupid and ( S ; p , ~ ) a left G-uprrn

( j ) Q 57 S i s locally trivial. (ii) For each sES, the function q?,: StGp(8) -8, a + + a ' 8 is a submersion. (iii) For each 8 in S the orbit Q, of s is open in S, and the prq'ection q?#: Sl&(u) .* (44

The proof is given in 8 4. A consequenoe of Proposition 10 is that if S is a transitive Q-space and t / R &

is locally trivial, then S is homogeneous. Also by taking S=Ob (Q) we Rut' ih#{

Then the following conditions are equivalent.

i s a locally trivial principal &,-bundle.

Brown/Danesh-Naruie/Hardy, Topological groupoide 151

Proposition 10 oontains a result of EHRESMANN [7], that if Q is locally trivial and sE Ob (G), then a : S t g -c Ob (G) is a locally trivial principal G{x}-bundle.

A result in a different direction is the following. Proposition 11. Let Q be a transitive, lucally trivial topological grmpoid. ( i ) If q : H-cQ is a topological covering morphism such that H i s transitive, then

(ii) If (S; p, p) is a transitive left #-space then p : S -0b (c) is an open map. By results of $ 2 either of (i) or (ii) implies the other-the proof of (ii) is given

in $4 . The condition of local triviality cannot be dropped. For example, if G, S

are aa in Example 5 then St,;# is a (G%S)-space via 8; also a : -S is continuous and bijective. But a is not a homeomorphism, and so is not open.

Let H be a topological group. A subgroup D of H is said to begood if the right action of D on H makes H a locally trivial, principal D-bundle; a necessary and sufficient condition for this is that the projection H -HID is a submersion ([8], page 48, Corollary 8.3). For example, any closed subgroup of a LIE group is good.

Recall that if G is a topological groupoid, s,,COb (a), and D is a subgroup of G{Zn}, then GD is the space of left cosets of D; and that if Ob (a) is HAUSDORFF then 0, is a left G-space. We say D is a good subgroup of 0 if i t is a good subgroup

Theorem 12. Let G be a locally trivial topological groupoid such that Ob (a) is

The proof will be given in Q 4. This theorem can be combined with previous results and the classification

theorem for the abstract cme to yield a clmsification theorem for topo- logical covering morphisms.

Tbeorem 13. Let G be a locally trivial, transitive topological groupoid with IIAUSDORFF object space, and such that all its closed subgroups are good. Then there i x a bijectwn between conjugmy classes of closed subgroups of 0 and equivalence tlusses of locally trivial, transitive, topological coverings of G with HAUSDORZT o?!ject space.

This theorem follows from previous results and the following proposition, \rtIiose proof is given in $ 4.

Proposition 14. Let G be a locally trivial topological groupoid with HAUSDORFF ihjwt space. Let D be a subgroup of Q{x,)}. Then the following conditions are equi- tarleni:

Ob (9 ) : Ob (H) -0b (0) i s an open map.

of G{zn}.

I~AUSDORFF. Let D be a good subgroup of G. Then GZG, is ~ocaldy trivial.

( i ) D is Closed in G{x,}, (ii) G D is TI, (iii) V D is HAUSDORFF.

Vinally, in Topological Groupoids I ([5]) we proved the existence of quotient 1q)ological groupoids. We now show that the quotient topological groupoid G/N

152 Brown/Dsnnsh-Neruia/Hardy, Topological groupoids

of a locally trivial topological groupoid Q by a wide, totally disconnected, normal subgroupoid N is locally trivial. In fact we prove more, namely

Theorem 16. Let Q be a M l y trivial, topological groupoid, and N a wide mmucl subgrmpoid of Q such that Ob (p) : Ob (Q) -0b (QlN) ia a aubmer&n. Then QlN is a locally trivial topological groupoid.

The proof is given in 8 4.

Proof of Proposition 1. We have a commutative diagram

- H P H'

Let q : H -Q and r : H -. H be topological covering morphisms. Then clearly, for each h'E Ob (W), p 1 St&' is a homeomorphism onto StQp(h') so that p is an abstract covering morphism and the unique lifting morphism aP : Q % Ob (H') +H' exists. Continuity follows easily from the diagram,

QROb (H') - 8P-+ H

where 322: Q z O b (61')-Ob (61') is the obvious projection. Conversely, let q : H 4 and p : 61' -4 be topological covering morphisms.

Then again it is easy to see that for any h ' ~ Ob (IT), r ISt&' is a homeomorphism onto St,r(h') so that the unique lifting morphism 8,: HROb (H')-H' exists. Continuity follows from the diagram

- H' H 2 Ob (H') ' r

Brown/D~nesli-Naruie/H~rdy, Topological groupoids 153

Proof of Theorem 4. We are considering the Q-H-bispace (q, y; 8; p , 9). We have to prove that the action q~’ : Q x SIH -SIH induced by Q is continuous.

Let 7 : S-SIH be the quotient mapping. Then r is an open mapping, hence 1 Xr: B X S -Q XSIH is open, and so a quotient mapping. But (328 is a (1 xr)- saturated subset of QXS and is also, since Ob (G) is HAUSDORFF, closed in QxS. So 1 2 r : Q % S -Q 2 SIN is a quotient mapping. Since p‘ (1 Z r) =q, it follows that ‘p‘ is continuous.

Proof of Theorem 6. By Corollary 6, QD is a left Q-space and we let q : H - 4 be the corresponding topological covering morphism. Let y be the coset D in QD.

Then q(H{y}) =D the group of stability of D in Q,. Now suppose r : K - Q and zEOb (K) given as above. Then Q operates on

Ob ( K ) and so we can define h: Sl# -0b (K), a H a - z. Since z is stable under D, h fwtorises through A’: Gr,-Ob (K). Clearly A’ is a Q-map and so defines by Theorem 2 a morphism 8: H-K such that rs=q and Ob (s)=h’, whence s(y)=z. By Proposition 1, s is a topological covering morphism. This proves existence of 8

and uniqueness is simple to verify.

Proof of Proposition 7. Let {A,: U , - F } be a trivialising cover of 3’. Let St,U, ={a€ F: 8’aE U,} ; then {Stpu,} is an open cover of F and so it is sufficient to prove!* continuous on StFTJ,.

Let aESt,U,. Then

f*(a)=f*(a .1,(8’a)) .f*(1,(a’a)-J) , = a,(!@ *~, (a’a) ) , f* (z , ) ) - sp(!J.(8’a),f*(z,))-’

which is clearly a continuous function of a.

Proof of Proposition 10. (i)=+(ii) Let aEStGp(s). Then by hypothesis there exists an open neighbourhood U of a - s in S and a continuous function 1 : U -Q 2 8 such that A(a - s) = lPtCr4) = (la,, a - e) and 1(l) EB 2 S(a - s, t ) whenever tE U. Let A ( t ) = ( A i ( t ) , a - s), so that I , ( t )EQ and Ai(t) * a - 8=t , for tE U , while &(a - 8) = 1,. Then if p(t)=i l t ( t ) -a, lEU, we have p ( a - s ) = a andQ,p(t)=iZt(t).cc.s=t. (ii)*(i) Let sES, and suppose qS: St,p(s)-S is a submersion, and p : U-StGp(e) is a continuous function from an open neighbourhood U of s such that q8p = 1 and p ( s ) = ld8). Let1: U 4 2 8 begiven byt*(p(t), 8 ) . Then 8’1(l)=s, aA(t )=p( t ) . s=t . So Qj?S is locally trivial. (ii)*(iii) Let sES, and let p : U-StGp(s) be a local w x w i & i m d m , a u c h $hat p(s) = lp(,). Then U is contained in the orbit Qs of s; thus Qs is o p n in 8. Now StGp(s) is clmrly a principal Q{p(s)}-bundle in the sense m. Renee i t is also a principal Q,-bundle. But a principal bundle under a group action is locally trivial if and only if the projection to the orbit space is a submersion. This proves (iii). (iii)*(ii) Let sES. If StGp(s) -Ga is alocally trivial principal Q,-bundle then i t is a submersion. Since as is open in S, i t follows that 8tGp(8) -8 is a submersion.

154 Brown/Dsnesh-Nmuie/Herdy , Topologicel groupoids

Proof of Proposition ll(ii). Let sES. The following diagram is commutative. Since G is transitive and locally trivial, Proposition 10 implies a is an open map. Since S is a transitive G-space, qs i s surjective. Hence p is an open mapping.

Proof of Theorem 12. We need the following lemma which is probably well- known.

L e m m a . Let H be a topological group and X a locally trivial principal H-bundle over XIH. Suppose that D is a subgroup of H such that H -c HID is a locally trivial principal D-bundle. Then X --c XID is a locally trivial principal D-bundle.

Proof. Let p : X - X l H , q : H-HID, r : X-CXlD be the projections. Let xE X ; we wish to find a neighbourhood W of r ( x ) such that r - j ( W) is D-isomorphic to WXD.

Let U be a neighbourhood of p ( x ) such that there is an H-isomorphism a: ?-I( U ) -c U XH. Let a@) = ( p ( x ) , z‘). Let V be a neighbourhood of &’) such that there is a D-isomorphism p : 8- *( V) V X D. Then W =a7*( U X q - j ( V ) ) is an open neighbourhood of z which is D-isomorphic to U x V x D. Let W =r( W). Then W is homeomorphic to U x V , and r-1( W) is D-isomorphic to W X D . This proves the lemma.

We now obtain Theorem 12, which is equivalent to St&o+GD being a locally trivial principal D-bundle, by taking X = S t g o and H =G{xo}.

Proof of Proposition 14. Clearly (iii) =-(ii) =-(i). So we need only prove (i) *(iii). Now a : St+o+Ob (G) is a locally trivial, principal G{xo}-bundle. By [12], page 70, Theorem 1.1, a,: G,-Ob (G) is canonically isomorphic to the associated fibre bundle, with fibre G{zn}/D. Therefore a,: G, -0b (G) is a locally trivial fibre bun- dle. But Ob (G) is HAUSDORFF, and G{zo}/D is HAUSDORFF, since D is closed in G{zo}. Hence G, is HAUSDORFF.

Proof 01 Theorem 16. Let p : G-GIN be the canonical %@I-morphism. Let CEOb (GIN), and let xEOb (a) satisfy p ( x ) = C . Then, since Ob ( p ) : Ob (a)- -0b @IN) is a submersion,,.there is an open neighbourhood V of C and a con- tinuous function p : V-Ob (G) such that p(C)=z and Ob ( p ) p= 1,.

Now a is locally trivial, so there is an open neighourhbood U of z and a con- tinuous function 1: U -G such that A(%) = 1, and a1= lo. Let 0 =p-1( U). Then 0 is an open neighbourhood of C, and 1 =PAP : l? -.GIN satisfies a l = 10 a8 required.

Brown/Danesh-Naruie/Hardy, Topological groupoids 155

References

11) R. &ENS, Topologies for homeomorphiem groups, Amer. Jour. Math. (2) 88, 693-010

[2] R. BROWN, Elements of Modern Topology, McGraw-Hill (1968). r31 -, Fibratione of Groupoide, Jour. of Alg. 16 (l), 103-132 (1970).

(1946).

1 R. BROWN and G. DANEBH-NUUIE, The fundamental groupoid a8 a topological groupoid. Proc. Edinburgh Math. SOC. 19. 237-244 (19761.

. I

R. BROW and 5. P. L. HARDY, Topologioal groupoids: I. Univereal Conatructione. Math. Nachr. 71, 273-286 (1976). G..DANEBH-NARuIE, Topological groupoide, Ph. D. Theeie, Southampton (1970). C. EEREBMANN, Categories topologiques et catbgories diff6&ntiables, Coll. Geom. diff. gl,obalee, Bmxefles, 137 - 160 (1968). A. F~OLIOH, Groupids and groupoid spacee, unpublished lecture notes given a t London (1 986). P. GABRIEL and M. ZISMAN, Categories of fractions and homotopy theory, Berlin, New Yorli, London (1986). J. P. L. HARDY, Topological groupoids, M. A. thesis, Wales (1972). P. J. HIQQXNB, Notee on categories and groupoide, Van Noetrand (1971). D. HUBERIOLLER, Fibre bundles, McGraw-Hill (1906). D. MONTQOMERY and L. ZIPPIN, Topological transformation ~ O U P E , New York (1955). W. NEWNB, Functional Analyeis, unpublished lecture notes given in Liverpool. N. STEENROD, Topology of fibre bundles, Princeton (1951). F. WARNEB, Foundations of differentiable manifolds and lie groups, Scott (1971).

Added in Proof:

Additional Bibliography

3 . Topological groupoids and topological categorios

[A621 M. K. DAKIN, Topological groupoids, U-spaces end Hurewicz lifting functions, M. Sc. Dis- aertation, Univereity of Walee (1974)

[A631 M. K. DUIN and A. K. SEDA, U-epacee and topological groupoide, Glaenik Mat. [A641 T. TOMDIECK, On the homotopy type of classifying epacee, Manuecripta Math. 11, 41-49

[A051 J. P. L. HARDY and S. A. MOBRIS, A note on free topological groupoide, Math. Nachr. (to

[A661 J. P. L. HARDY and D. PUPPE, Classifying epaces and universal categories (to appear) [A67) A. HELLER, Adjoint functors and the bar construction, Advances in Math. 12,8-31(1974) [A681 P. NICHOLAS, Free topological groups and free products of topological grOUpS, Ph. D. thesis,

I A691 A. K. SEDA, Un concept de mesuree invariantes pour lea groupoides topologiques c. R.

(A701 J. STA~HEFF, construction of BC; Lecture Notes in Mathematics No. 279, 81-86 (1972)

(1974)

appear 1977)

Univereity of New South Wales (1978)

Acad. Sci., Paris, 280, 8-1003-6 (1975)

t. LIE groupoids and the theory of connectlons

[ A711 B. FERRIF, Introduction H. 1’6tude dee catbgories differentieblee et de leure prolongements,

[A721 I. K o ~ R , On the reducibility of connections on the prolongations of vector bundles, Colloqu.

[A731 I. KOLLR, Some higher order operatione with connectione, Czech. Math. Jour. 44,311-330

[A741 I. KOLRL, The contact of spaces with connection, Jour. Diff. Geom. 7, 683-670 (1972)

Eequieaes Math. e2, Paris (1974)

Math., 80, 246-267 (1974)

(1974)

156 Brown/Danesh-Naruie/Hardy, Topological groupoids

[A7a] I. KO&, On the abaolute differentiation of geometric object fielde, Ann. Poloniai Math.

[A761 N. VAN QW& Non-abelian Spencer cohomology end deformation theory, J. Diff. &om. 8,

[A773 A. SZYBIAK, Canonicel forme on a groupoid related to rngular prolongatione of Lie groups,

27, 293404 (1973)

166-211 (1969)

C O ~ ~ O ~ U . Msth. 26, 229-239 (1972)

8. r-etructurea and foliations [A781 J. STABEEBF, C l d i c s t i o n via Br, , Lecture N o h in Math. NO. 279,86-94 (1972) [A791 P. STREDDEB, The classifying apace of a topologiod groupoid, preprint, Warwick (1974) [A801 T. VEsoar, On the relationehip between the groupoid and the pwudogroup of tranefor-

metione, Bull. Kyushu Inet. Tech. le, 4146 (1985).

Correction: On p. 273 of [a] the first four references ehould be to A13, A62, A64 and A29, while on p. 280 the referenoe to A16 ahould be to 86.

R. Brown and J . P . L. Hardy 8 h l of Mat,hema.tica and Computer Science University College of North Walea Bangor, C a e w . U.K.

Q. Dawh-Naruie Daneahamye-Ali Roozvelt Ave. Tehran Iran