Bull. Kyushu Inst. Tech.
(M.&N. S.) )iTo. 12, 1965 .
ON THE RELATIONS BETWEEN IHE GROUPOID AND THE
PSEUDOGROUP OF TRtMNrSFORMATIONS
By
Toshitane UEsuGi
(Rcccived Nov. 16, 1964År
If a pseudogroup To of transformations of a topological space is given, then we can eonstruet the topological groupoid 17r, which is called the tepological groupoid associated with the pseudogroup ro of transformations [1], [6]. In the present paper we shall show that the pseudogroup r of transformations is constructed from a topologieal groupoid C with seme properties. These pToperties hold good for the topologieal groupoid 17r, assoeiated with a given pseudogroup ro of transformations. When we construct the topologieal groupoid G' from the pseudogroup T of transformations, we shall see that this topological groupoid C' is homomorphic with the initial topologieal groupoid G and that the homomorphism theorem in the group theory holds good.
Let B be a topological space and an open and continuous map p of B onto the subspace E of all units of C be given. If G is a topologieal groupoid of operators of (B, p, E), then we can eonstruet the new topological groupoid GB with the same properties as above [1]. Accordingly we can also construct the Pseudogroup r of transformations from CB. We shall investigate the relations between fi and r.
In section 1, we eonstruet the pselldogroup r of transformations from a
tepologieal groupoid G with some properties, This pseudogroup r of trans-
fOrmations operates on the space E of all units of a given topological groupoid
C (Theorem 1). In section 2, we shall prove that the topologieal groupoid G'
assoeiated with this pseudogroup r of transformations is homomorphic with the
initial topo]ogical groupoid G (Theorem 2). In section 3, we define the kernel
Of the homomorphism and also define the topological factor groupoid by this
kernel. Then we shall see that the topological factor groupoid is isomorphic
With the topological groupoid C' (Theorem 3). Further we have a necessary
and suthcient condition that the homomorphism in Theorem 2 be an isomer-
Phigm (Theorem 4), In section 4 and seetion 5 we assume that G is a topological
grOUpoid of operators of (B, p, E). In seetion 4, we constTuet a new topological
grOUpoid ea and show that GD has the same properties as C. Then the pseudo-
grOUP r of transformations whieh is constructed from GB operates on B• We
Shall prove a general relation between i and r (Theorem 5). Lastly, in section
41
42 Toshitanc UEsvGr
5, we shall find a neeessary and suMeient condition that r be an isomorphie prolongation of r (Theorem 6).
g1. The pseudogroup of transfermations of the space of units oÅía certain topologicaJ groupoid. Let G be a topologieal groupoid in the sense of A. Haefliger [1]. The definition of the topological groupoid is as follows.
DEFirsTiTioN 1. A set e of elements {e, g, lt, ---} is ealled a multiplicative
system if, for some pair g,hEG, aproduct g.hGG is defined. An elementeof G is cal!ed aunit if e•g=g and h•e=lt whenever e•g and h•e are defined. A rnultiplicative system is ealled a gToupoid jf the following conditions are
satisfied :
a) The triple product f•(gth) is defined if and only if (f•g)•h is defined and, when one of them is defined, the associative law f•(g•h)=:(f•g)•h holds.
b) The triple produet f•(g•h) is defined whenever both products f•g and g•h are defined.
c) For each gEG there exist right and left unit$ e', e", Tespeetively, such that g•e' and e".g are defined.
d) For each gEC there exists an inverse g-i, i.e., an element g-i of e sueh that g-'•g is a right unit, g•g-i a left unit, of g.
rt follows from the axioms that each element g of a groupoid G has a uni- que right unit, a unique left unit and a unique inverse.
A groupoid with a topology is called a topaolegical groupoid if the following conditions aTe satisfied :
e) The rnap (g, h)-,g•h of subspace of CÅ~G formed by the multiplicatible pairs to G defined by the produet of e is eontinuous.
f) The map g-,g-i of G to G is continuous.
We shall denote a: C-ÅrE and B: C-) E, where E is the subspace of all units of G, the projeetions whieh map eaeh element gGG respeetively into lts right unit a(g) and left unit B(g).. Then ct and B are both continuous maps of G onto E,
From the axioms of the greupoid we see that the product g•h is defined if and only if ct(g)= B(h) for g, h E G. In Åíact, if g•h is defined, then g•h=(g•ct(g))•h•
From a) of Definition 1 we see that g•((cr(g))•h.) is defined. Therefore we have . a(g)=B(h) by means of the uniqueness of the left unit of h. Conversely, sm9e
g•a(g) and B(h)•h are defi-ned, if a(g)=B(h), then we see that the product g•h iS
defined from b) of Definition l. '
Sinee g"-i•g and g•g-i are defined, it is trivial that a(g-')=B(g) and B(g")
=a(g) for any gE G.
Hereafter we always assume that the topological groupoid G has the
On the Rclations Between thc Groupoid and thc Pscudogroup of Transformations 43
following two properties: (i) the subset E is an open set of G and (ii) ct is a loeally homeomorphic map. (If C is the topological groupoid eonsisting of the germs of elements of a pseudogroup of transformations of a topological spaee, then G has these properties.) Since B(g)==cr(g-r) for any geC, we see that the projeetion B is also a locally homeomorphic map, Therefore we have the following
LEtsf"fA 1. 1. Ro7' an?J element g of the topological groupoid C there exists an open ?zeighborhooa ti of g such that the Testrictions of a and 3 to ti are Tespec- tively hemeomorphis7ns onto open subsets ev(ti) and B(ti) of E.
By Lemma 1.1-we take an op-.en neighborhood ti of g for an element g of G. We put U= af(U) and V=B(U). Let q be the map of U onto J7 definedas
follows.
For any x E U there exists one and only one element g E t7 such that tz(g)=
.x•. Then we put q(x)==B(g), i.e. q=(Blt7)o(crlt7)'i, where aldr and B]t7 are respectively the restrictions of ev and B to ti. Aceordingly the map op: U--ÅrV is a homeomorphism.
Let T' be the set of all sueh maps ca which we eonstruct for every gEG and every neighborhood of g in Lemma 1.1 by the way mentioned above. For any element q of r', we shall denote by U(ip) and V(q) respectively its domain and range. The composition map ipoÅë of anyelements g, ip of r' means the map of op-i(V(Åë)All(ap)) onto op(V(Åë)AU(ip)) defined by qoÅë(x)=ip(Åë(x)) for .r E yt -i (7(Åë)A U(q)),
PRoposTT'ioN 1. 1. The set T' has thefolto2ving p7'operties;
a' ) lf q, gi E 1"' and qoÅë is dejined (i.e. V(ip)AU(ip)eE di), then there exists an open neighboo'hood U. of .x (where x is any point of the domain of qoÅë) such that the 7'estriction of caoÅë to U. belongs to r'.
b) lf op G r', then the inverse map op-i ofq belongs to 1".
c) The identity map e ofEbelongs to r'.
d) The 7'estrictio7z ofany element of r' to an open subset ofE belongs to 1"'.
PRooF. Let q == (tS? l t7)o(ct l ti)-' and gb = (B [ J7)o (cr l l7)-i where t7 and J-i are
OPen neighborhoods of two elements of G respectively. Let x be a point of U(geogb)=:st!)"'(V'(gb)AU(op)). If we put h=(cv11'X)'i(rc) and g=(culti)-i(B(h)) then bY definition we have q,Q{b(x)=B(g). Since a(.cr)=:iEP(h) the product g•h is defined.
We take an open neighborhood pt of g•h in Lemma 1.1. Then a(M-i) is an epen neighborhood of .r because x=a(h)==a(g•h), Here we may assume that a'(pt)(
U(caoÅë)• On the other hand using the continuity of the multiplication of the
tOPologieal groupoid G, we see that there exist the open neighborhoods C(t7
and D(7 of g and h respeetively such that g'•h' is in M' for any g' EC and any
4r4 Toshitane UEsuGr
h' ED whenever their produet is defined. We put O=B(D)Act(C).
We shall define U. by U.==a(JI7)Aa((Bl7)-'(O)). If we put tf.=(a1pt)-'(U.), then t7. is an open neighborhood of g•h and a: ti.-, U. is a homeomorphism To prove that gpogblU. G r' we take a point x' E U.. We have gpoop(x')=ip(B(h'))=
B(g') where h' = (a i P')-i (x') and g' --- (a l li7)d' (B (h')). Since h' G (l? l l7-')-i(O), we
have h' ED and g' G C, Accor dingly g' .h' E pt. Moreover g'.h' E U. beeause cx(g'•bl)=cu(h')=x'. Thus we have qoÅë(x')==B(g')=B(g'•h'). Therefore a') holds good.
Proof of b). Let ca==(BIV)e(cxltT)-i as in a'). Since at(g"i)==B(g) and i9(g")=a(g), if we put t7-i={g-'igG ti}, then a: Cr'Z- J7(go) and B; t7-'. U(ep)
are both homeomorphisms. For given yE V(ip) we have q-i(y)==ev(g) where
g E CiL sueh that B(g)= F. I.e. gp-'(sy-)=i9(,g•-i) where g-i e tT-i such that cv(g`-')= pt.
Hence q-iET'.
The property c) is proved immediately from the faet that E is an open set of G.
The property d) is trivial. Q.E.D.
The local homeomorphism q: U-, V of E is called a unton of the elements of T' if U==vUi where eaeh Ui is an open set of E and each q71Ui belongs to i
r'. Then we shall denote ip==vopi, where qi=qiUi.
i
Let T be the set of all local homeomorphisms of E which are the unions of elements of r'.
THEoREM 1. r is a pseudogroup of tran$foTmations ofE
PRooE It is suMeient to prove that the following conditions are satisfied:
.
a) lf q, ÅëEr and caoÅë is defined, then opoÅë E r, and b), c) and d) of Proposi- tion 1.1 in which r' is replaced by r.
Let op,ÅëEr and gpovt be defined. We put op=y. epi: y. U(qi)--Åry. V(epi) and op=VÅëj;VU(Åëj)--ÅrVV(opi) where {q7i;iEI}(r' and {Åëj;iE,J}(jr'. If we put
tt,=,.e.elU.('O,'•'.A,,U(,op,o,'Åë,',.9".d•%',-'-,g6IK•.2op,y';Y(,g(g,09i'l6S,h8,2,.W.e,h,iV.e.3ag.i',Y,q,O:,Ce"-f.
ii'a union of the elements of r' by a') of Proposition 1.1, we have qeO E r.
If op==vopi E r, then ip-i= vca;• i. From b) of Proposition 1.1 op7• L e r', there-
is fore wehave g"E,r. The set r' isasubset of r. Hence the identity map e of E belongs to r.
Let JJ7 be an open set of E such that B7(vU(opi). Then each opiI n7 belongS
's to r'. Therefore we have opIJI7=v(opilM), Q.E.D.
i
On the Relations Between the Greupoid amd the Pseudogroup ofTransformations 45
REMAiuc. I"rom the definition of the set r, it is clear that a loeal homeo- morphism Åë defined on Ubelongs to r if and only if there exists a family Ui of open sets of E sueh that U=vUi ,and ql Ui Er for each i.
s
S2. The topological groupoid G' associated with r. Let C' be the topologieal groupoid consisting of the germs of elements of the pseudogroup r of transformations of E, Then we may identify the subspace of units of C' with the spaee E. We denote by q, a germ of an element ip of r at a point
.x E V'(op).
For every g E G we take an open neighborhood U of g in Lemma 1.1. Then a loeal homeomorphism op == (Blti)o(alti)-i of E is an element of T. We put e(g)==Åë.c.), It is evident that the value e(g) is independent of the choiee of a neighborhood Cr of g. Thus we have the map e:G-ÅrG'.
DEFmiTi6N 2. Let Gi and C2 two topological groupoids. A map p of Gi into C2 is called a homomorphism of Ci into Cz if the following conditions are
satisfied :
a) If g,hEGi and g•h is defined, then p(g)•p(7a) is defined and equals to
p(g•h)•
b) If gE Gi is a unit, then p( g) is a unit of C2.
c) P is a continuous map.
THEoREM 2. The map e is a homomorphism of G onto G'.
PRooF. Let g, hEG andg•h be defined, We put e(g)=ip.c.) and e(h)==ea(h)•
Since g•h is defined, we have a(g)=B(h). Hence we see that the product q.(g)•
ep.(h) is defined, i.e, e(g)•e(h) is defined. On the other hand we have q.(g)•Åë.(h)==
(epeÅë)a(h) and e(g•h)=(qoop).(..h)=(ipeip).(h) which is easily seen from the proof of
a' ) in Proposition 1.1, Therefore we have e(g•h)=e(g)•e(h)•
If gEG is a unit and e(g)=op.(.), then q is an identity map. Hence e(g) is a unit of C. (In general, b) follows from a) because G' is a groupoid, [3, Satz
1]).
For any gEG, let e(g) =gp.(.) and ep=(Bit7)o((xlt7)-i. We take an open neighborhood V'={ep,.syGV} of q.(.) where 7is an open set of E sueh that V(U(op) and a(g) E V. If we put j7=(alt7)-i(V), then J7 is an open neighbor-
hoodofgsuchthat e(P7)(p". Thereforeeiscontinuous. .
For any e!ement op. of G' we may assume that qis in r'. If ca=(BIU)o (cr fiU)-' where ti is an open neighborhood of an element of G, then there exists an element h of t7 such that cr(h)=.x. Then e(lt)=ip.(h) and hence e is onto-
Moreover we have the Åíollowing
46 Toshitanc UEsuGr
LE"frttx 2.1. The homomorphism e: G-ÅrG' has tJte pToperties 1) if e(g)•e(h) is defcned, then g•h is also aefcnea.
2) O is an open map.
PRooF. Let e(g)=op.(,), e(h)=Åë.{h) and g.(,År•op.(h) be defined. Then a(g)=
Åë(ct(h)). By definition of vt,Åë(ct(h)) equals B(h). IIence cr(g)=B(h) and there- fore g•h is defined.
For the open set ti in Lemma 1.1 e(J)={q.;xE ev(ti)} is an open set of C'.
2) follows easily from this fact. Q.E.D.
This lemma will be used in the next section.
g 3• The factor groupoid oÅí G by the kernel of e. The kernel Ker(e) of e is the set of the elements of G whieh are mapped on units of G' by e. For tvvo elements g, h E G we shall say that g is equivalent to h, denoted by g-vlt, if g•h-i is defined and g•h-i G Ker(e).
If g is equivalent to h, then e(g•h'i)==e(g)•e(h)-i is a unit of G'. Hence we have e(g) =e(h). Conversely if e(g)=e(h), then a(g)=cu(h) and B(g)==B(h). Since B(h-')=af(h), g•h-i is defined and henee O(g.h-r)=e(g)•e(h)ti is a unit of G'. Thus we have the following
LElitx(A 3. 1. IiTor two elements g,hEC,g is equivalent to h if and only if e(g)=e(h).
From this lemma it is trivial that the relation "-v" is an equivalence relation. And rnoreover the relation "--" is compatible with the groupoid structure of C. The following lemma holds good.
LEi rMA 3.2.
1) ifg•--h, g'•ybl ancl g•g' is definea, then h•h' is atso defcnd aml g•g'•'-h•h'•
2) lf g'vlt, then g-'-vh'i.
3) lfg•-h, then a(g)==ev(h) and B(g)=B(h). (Hence cr(g)--cu(h) and B(g)-- B(h)).
PRooF. The property 3) was proved in the proof of Lemma 3.1. If g--lt and g'--h', then e(g) =e(h) and e(g')= O(h'). On the orther hand, since g•g` iS defined, we have e(g•g')==e(g)•e(g'). Henee e(h)•e(lt') is defined. By Lemma 2•1 we see that h•h' is defined. Therefore we have e(g•g')=e(g)•e(g')=O(h)•e(h')=
e(h•h`), from which we see that g•g'--h•h'.
If e(g)=e(h), then e(g)-'=e(h)-i. Therefore e(g-')=e(g)"=:e(h)'i=e(h'"')- Q.E.D.
We shall denote the set of the equivalenee elasses by G!Ker(e). LetPbe -i the natural map ef G onto G/Ker(e) which maps g E G onto the class contaming
g. We denote p(g) by g-. Thengisarepresentativeelementof g". Now we
On thc Relations Bctwccn the Groupoid and thc Pscudogroup ofTransformations 47
can define a groupoid structure in C• /Ker(e). For two elements g-, 7, E G/Ker(e), we define the product g•ii is equal to g•lt if and o'nly if g•h is defined in G. It is easily seen by Lemma 3.2 that this definition is independent of the choice of the representative elements and that G/Ker(O) is a groupoid with respeet to the above product. (see [2, Satz 1.]).
We shall say that a subset O of G/Ker(e) is an open set of G/Ker(e) if p-i(O) is an open set of G. In this manner we introduee the topology in C/IÅqer(e), We want to show that the groupoid G/Ker(0) is a topological groupoid with respect to this topology.
Firstly we remark that the map p is open. In fact, for any given open set U of C we take any element g of p-'(p(U)). Then there exists an element he U such that p(g)=p(lt) and hence O(g)=e(h). By the definition of e we have open neighborhoods V and V' of g and lt respectively such that av(V)=cr(V') and ca= ep where q= (i9 1 V)o(ct [ J7 )-' and Åë == (B I J7')o (cu l J7')-'. Here we can assu me th at V'(U. For an element g' E V we ean take h' E V' such that cr(g')==cv(h'). Sinee q=Åë we see that e(g')=e(h'). Hence p(g')=:p(h'). Since h'E J7'(U we have g'cp-i(p(U)), i.e. Ii(p"i(p(U)). Therefore p-i(p(U)) isan open set ofC.
Let g-, i,EC/Ke7'(e) and g•IHt be defined. We take an open neighborheod lr7 of g-•Zl. Since p-i(t-t7) is open and since g•hEp-i(IP) which follows from g•it==
g•lt, there exist open neighborhoods U and V of g and h respeetively such that if g' E U, h' EV and g'•lt,' is defined, then always g'•h' cp-i(l-S7). Put U=p(U) and Ii==P(V). Then U and Ji are open neighborhoods of g- and h respectively be- Cause Pis an open map. For any g'E(Z and i,.'Efi sueh that g"•ZF,' is defined, there exist g{EU and h.{EJi such that p(g')=p(g{) and p(h')=P(hf). From
Lemma 3.2, we see that gl•hl is defind and g'•Z7'= gi'•Z-tl. Since g{nyh{ E p-i(i-V) we have g'•it'==g{•z-,{EIT7. Therefore the product operation of G/Ker(e) is continu- ous.
Take any g-' e G/lÅqer(e) and let JTi be any open neighborhood of g--i. Since g-i==17" we have g-' ep-i(!-L7). For p-i(JTi) there exists an open neighborhood g of g such that U'i(p"(JP). If we put V=p(U), then we ean easily see that U-:(l-Y. Thereforethemap g'.g"' iscontinuous.
Thus we have the topological groupoid GIIÅqer(e). Then it is immediate]y fOIIows that the natural map p is a homomorphism of G onto GIKe7'(e) as the FOPolegical groupoid. Moreover if p(g)•p(h) is defined, then it fol!ows that g•h iS defined because of the definition of the product in C/Ker(e),
We shall summarize the above discussion into the following
PRoposiTioN 3.1. The set C/Ker(e) of the equivalence classes is a topological grOeepoid. The naturat map p of G onto GIKer(e) is an open homomorphism as
the topological g7•oupoid such that if p(g)•p(h) is defi?ted, then g•h is atso defined
48 Toshitanc UEsuei
for g, heC.
Next we define the map b of C/Ker(e) onto C' by the natural way, i.e.
O(g-)=e(g) for any gEG!Ker(e). Then wehave e'ep=e,
Ti•iEoREAf 3. G/KeT(e) is isomo?'phic with e' under the map b.
PReoF. Obviously O isaone-to-one map. [Vake any g-,iiEC!Ker(e) such that g•it is defined. Then, sinee g•h is defined, we have e(g)•e(h)=e(g•h). Sinee eig-)=e(g) and O(l-t)==e(h) we see that Oig-)•b(Z-L) is defined. On the other hand, O(g-•ri)=O(g•h)==e(g•h). Therefore we have O(g-)•O"(is)=e(g-•Z-,).
Let geG/Ker(e) be a unit. Since g----g•a(g)=g-•a(g) we have g----a(g).
Hence O(g-)=e(a(g))=e(a(g)) is a unit of G' because a(g) e Ker(O). Thus O is a hemomorphism. (Or, see [3, Satz 1]).
If O(g-)•O(Z-L) is defined for g-, E e G/Ker(e), then e(g)•e(h) is defined. By Lemma 2.1 we see that g•h is defined. Therefore g-•Z-i is also defined.
Next we prove that e is a homeomorphism of e/KeT(e) onto G'. Let Ube any open set of C/Ker(e). Sinee eH(U)=Oep(p-i(U))=e(p-i(U)) and sinee e is an open map by Lemma 2.1, we see that O(U) is an open set of C'. For any g- eG /Ker(e) let V be an open neighborhood of e(g-). Then there exists an open neighborhood Mofg such that e(M)(V beeause O(g-)=e(g) and the continuity of e. Sinee p is an open map, p(B7) is an open neighborhood of g- sueh that
O(p(M))=e(B7)(V. Q.ED.
Further we shall find a necessary and suMcient condition for G to be isomor- phie with G' under the map e. By Lemma 2.1 it is necessary and sufrleient for eto be an isomorphism that ebe one-to-one. Evidently, e is one-to-one if and only if p is ene-to-one.
LEMAfA 3.3. p is one-to-one if aoz(l enty if Ker(e)==E.
PRooF. We assume that Ker(e)=E. If p(g)=p(lt) for g,.hEG, then g•h-' is defined and g•h-i E Ker(e), Since g•hm' is equal to a unit of e we have g=h.
Conversely, let p be ene-to=one. For any gGKer(e), g.a(g)=g•a(g)'i is defined and g==g.a(g)-ZEKer(e). Hence p(g)==p(cr(g)). Therefore we have g=a(g).
Thus we have proved the following
THEoRE"r 4, A tnecessary and sufiicient comlition foT G te be isomorphie
tvith G' is that of e(g) is a unit of G', tlten g be also a unit of C.
S1. The topological groupoid oÅí operators. Let B be a topological
space and let a map p of B onto E be open and eontinuous. We denote by GB
the subspaee of the produet space GxB consisting of all the elements (g, N) Of
On the Rclations Betwcen thc Groupoid and the Pseudegroup of Transformatiens 49
GÅ~B such that ce(g)=p(bl).
DEFiNiTioN 3. ([1]) A topologieal groupoid G is called a groacpoi(l ofopera- tors of (B, p, E) if there exists a continuous map (g, X)-)-g•di of CB onto B such that:
a) p(g•di)=B(g)•
b) (g•h)•te=g•(h•di).
c) e•N=N, if e is a unit of G.
In this section and the next section we assume that our topological groupoid G is a groupoid of operators of (B, p, E).
For two elements (g, i), (h, Y)ECB, we define the product (g, i)•(h, Y) is equal to (g•h, N) if and only if I==h•S. Then, since a(g)=p(di)==p(h.Y)==B(h), we see that g•h is defined and cv(g•h)==ec(h) =p(jy). Therefore this definition of the preduet is wel! defined.
PRoposmoN 4.1. ([1]) GB is a toipologicae gro2tpoiel with respect to the above procluct.
PRooF. It is trivial to verify the eonditions a) and b) of Definition 1. We see that (g, bl)EGB isaunit of(B if and only if g isaunit of G. In fact, if (g, i) is a unit of CB and g•h is defined for some hE C, then vve have p(r)=a(g) =B (h,). Since B(h) = a(h-i), We have (h", i) E( B. We put jF = h-i • .-r, Then (h, Y) E GB and (g, rr)•(h, 57) is defined. Therefore we have (gtlt, bl)=(h, Y), Hence
g.h=h. If ing is defined, then we have h•g=h as above. Thereforeg is a unit of C. Conversely, ifg is a unit of G, then (g, .'v) is obviously a unit of GB.
We easily see that (a(g), iii) and (B(g),g•it) are Tespectively the right and the left unit of (g, I). Also (g-i, g•.-v) is the inverse of (g, v).
We ean prove the continuity of the product and the inverse operations.
Q.E.D.
As mentioned in the proof of Proposition 4.1, the set of all units of GB is the set of the elements (e, .-r) of GÅ~B. Since e= a(e)=p(N), the set of all units Of CB is the set of elements (p(di), I) where .-T EB. If we define !Zr(.r-) =(p(al), fi)
fOr eaeh N E B, then we see that the map V is a homeomorphism of B onto the SUbsPaee of all units of GB. Therefore we identify the subspace of all units of CB with B by !P'. Let crB be the restriction of the natural projection GxB--.B tO CB. Then aB{(g, di)} is the right unit of (g, hi).
PRoposmoN 4.2. ([1]) B is an open set of GB and ctH is a localev homeomor- phic map.
PRooit, LetNbe an open set of B, Obviously, !r(N)(p(N)xArAGB. If
(P(;), )-') Cp(iV)xNAGB, then p(:)=p(Y), Hence (p(:), Y) =lr(Y). Therefere we
50 Teshitane UEsuGr
have Vr(N) =p(N)xNAGB. Sinee p(N)Å~NACB is an open set of CB we see that B is an open set of CB.
Let (g, X) cCB and U be an open neighborhood of g in C such that culU is a homeomorphie map. We take an open neighborhood Nof I such that p(N)(
cr(U). Then we see that ecB isaone to one map on UxNACB and crB(UxIVA CB)==N. Let J-Å~OACB be any set which is contained in UxIVAeB. If we take an element (h, Y) of VÅ~ OAGB, then aB{(h, Y)} =Y E O(N and p(Y)=a(h) e a(V)A p(O). Since p is an open map, a(fi)Ap(O) is an open set of cr(tT) and hence there exists a neighborhood P' of Y such that p(P')(ec(rt)Ap(O), We put P=OAP', For any :- e P, there exists an element ls E 17' sueh that ct(k)==p(Z) because p(:-) E a(l7). Then (le, ;) e 17Å~OACB and crB{(k, :-)}=::. Therefore vve have P(aB(l-7Å~
OAGB). Thus we see that ctB is an open map on UxNAGB, Since cuB is a eontinuous map of GB onto B, aB is a laeally homeomorphie map. Q.ED.
Novv we ean construet the pseudogroup r of transformations of B by the consequence in the section 1. In the following and the next section we shall investigate the relations between r and L
Let di and q be local homeomorphisms of B and E respectively. If p(U(di)) ==
U(ip) and podi=Åëop on U(ip), then we call di a prolongation of ip.
Let Åë=(B l tT)o(a l ti)-i be an element of 4 where t7 is an open neighborhood in Lemma 1.1. If we put D= tixp'i(U(g))AGB, then aBID and BBID are homeo- morphisms, where BB{(g, I)} is the left unit of (g, fi) for eaeh (g, di)eGB.
Therefore the local homeomorphism di(q)=(BBID)e(aBlD)"i belongs to r. More- over we have the following
LEMMA 4.1. di(ip) is a prolongation of ca.
PRooF. It is evident that ctB(D)(p"(U(ca)). For any NEp"(U(q)), there existsgEU such that cv(g)=p(di). Then (g, :.) GD and aB{(g, di)}=X. Therefore crB (D) =p'i (U(q)) . Si nce U(di (ip )) = ctB (D), we have p (U(to (ip))) == U(q).
Let N be an element of U(pt(ca)). Then there exists an element (g, .-x')ED and we have di (ca) (di)=BB {(g,- re)} ==g•X, Hence pe di(q) (di)=p(g•di)=B(g). On the other hand caop(bl)==ip(cr(g))=B(g). Therefore we have poto(op)=ipop on V(tu(op))•
Q.E.D.
Let e be an element of i such that ip=(BBIC)o(cvBIC)-i, where ctBIC and
BBIC are homeomorphisms. If we take C small enough, then we may assume that C=j7xNAGB. Let i be any element of U(di)=aB(C) and (g, N) be an element of C (such a (g,Ti) exists). Let ti' be an open neighborhood of g in Lemma 1.1 sueh that ti'(P7. Since p(re)E cr(ti'), there exists an open neighbor' hood O of di such that p(O) ( cr(ti') and O ( N. If we put q= (B l ti')o (a I ti')-' 1 p(0), then ip E r and op == (B 1 t7)o (a 1 t7)"' where ti = (t t 1 CT')"' (p (o)).
Put C== t7 xOAeB. Since we can easily see that aB(C') ==O, we have poaB(C')
On the Relations Bctwccn thc Groupoid and the Pscuclogreup of Transformations 51
.= p(O) == U(ge) and di 1 O= (BB I C')o(ctB I C')-i. It is easily proved that po di = q7 op on
o. Therefore di10 isa prolongation of op.
From the above consideration and Lemma 4.1 we can conelude the following Ti-iEoREif 5. Let q- be any eleme?tt of 1'`. Then, for any poiott hi of U(di) there
extsts an open ozeighbo7'heoa O of di such that ip1O is a prolongation of an element of r. Also, let Åqp be any element of r. Then, for aozy point xG U(ip) there exist an open neighborleoed U of .v an(l an element di of i such that di is a prolongation of opIU•
S 5. A nesessary and suflicient condition that r be an isornorphic prolongat ion of T'. Let Gb be the topologieal groupoid associated with i". Then
we have an open homomorphism O of GB onto Cb by the eonsequence in the
seetion 2,
We shall construct a homomorphism n of Ck onto C'. The methed of the construetion of r, is the same as the case that r isa prolongation of L (See [6], g2). For each g" G Cb, if g'"=ei, where di E r' and di E l7(di), then by Theorem 5, there exist an open neighborhood Oand an element ip of r such that di1O is a prolongation of q. We define n(g)=qp(i).
Let l-t'=gZiil (T Ch and gN•jii be defined. If n(1"L')=Åëp(il), since p(fii)=p(di(5}))=cb(p(iP)), then we see that r,(g)-n(/-i.) is defined and equals to (qoop)p(v). Since di1O and g71N (where N is an open neighborhood of )'"f) are respectively prolongations of ip and Åë, we see that there exists an open neighborhood V of p(y) such that (di 1 O)o(di lN) is a prolongation of qoÅë1 V. (In faet, we find ip"'(p(di(N)AO)) as V.) Therefore we have r, (gN•h)==ff((eodi)il)=(qDÅë)p(il) and hence z(g-)•r,(rt)=n(g"•7,).
If g-=dif is a unit of Ck, then we ean assume that di is an identity map.
Therefore q is also identity map and henee n(g)=qp(i) is a unit of C'.
Prom the latter part of Theorem 5 we see that n is onto.
For any element op of r, S(U)={q.;.xGU(q)} is an element of the base of open sets of C'. Since n-i(S(U)) is the union of all the sets {{Pi; fv- G U(e)} such that p(U(e))(U(ip) and rr(e.-)=q. for iG l7(di), where eaeh di is an element of i,
rt'-i
(S(U)) is an epen set of Gh. Therefore Tt is continuous, 11hus we have a homomorphism n of Gk onto G'.
Next, let c be the restriction of the natural projeetion CxB-}G to GB.
Thenc isahomomorphism of Ca onto C. In faet, let (g, .') and (h, bl) be two elements of GB such that (g, hi)•(lt, y) is defined. Then, since (g, .r'L)-(h, )'f)=
(g'Jt, jif), we have o{(g, N)}•a{(lt, 5F)} =g•h=o{(g•h, 5)}. Since if (g, x') is a unit Of CB, then g is also, we see that a{(g, fi)} is unit of G.
We shall prove the following
PRopostTioN 5.1. The following diagram is comen2etative:
52 Toshitane UEsver
o
G. .Gk ifi e .n
C- G'
PRooF. Let (g, di) be any element of GB, We cap find open neighborhoods
lgl,:•g.{.a"S,ilr6,f,i,X".Ch.Bh,.at.g,E[,",,Å~,i\61,}gB.,a}d,.Bli.V.Å~.i.Vtl.9B,.a,',e,h,o.m,e3-el:rt;
2.:n.d,,O.b,,e,,g-R?p,?n,,yeS3bs.ELhso.d,o.fZ.s,":,h,khCet,pgO,2S,a.(,U- la)=a",,d.9,g.N,t,.IS,W.P,P,":
pro!ongation of op= (B1ti)e(ait7)-i. (See the discussion in the proef of Theorem
5.)
Then, since O{(g, di)} == ip.-, we have noO{(g, X)} = opp(.b. While we have eoa{(g, N)} =e(g)= ga.(.). Sinee cz(g)= p(rx), we can conclude that 7rog=eoo.
QE.D.
CoRoLLARy. C(Ker(e))(Ker(e).
PRooF. Let (g, di) be any element of Ker(O). Then, noO{(g, N)} isaunit of G'. Since r,oO{(g X)}=eea{(g, N)}, we see that c{(g, hi)} EKer(e). Q.E,D.
Now we reeall the definition of the isomorphic prolengation as follows:
DEFzNmoN 4. ([4], [5]) l" is called an isomorpJtic pTolongation of r, if the following conditions are satisfied:
1) Every element of fi isaprolongation of an elgment of r. . 2) There exists thecorrespondenee to from r to 7 whieh hasthe following
properties :
a) For every opcT, tu(q) isaprolongation of q. .
b) Any prolongation oÅí op is a restriction of to(ip). (In this sense tu(q) ]s ealled the maxdanal 1""otongatton q).
c) Let q and ep be two elements of r such that caoip is defined. Then
o(q)odi(Åë) is also defined an'd it is a prolongation of ipoip.
d) If Åë is a restriction of op, then to(Åë) is also a restrietion of tu(q).
e) if q is an identity map, then o(ip) is also an identity map.
Then we can prove the fol!owing
.
THEoREM 6. A necessary and sufiicient cndition for r to be an isomorphw rolongatdan of r is that-
1) Every element of fi be a pTolongation of an element of r, 2) Ker(e)=ad'(Ker(e)).
PRooF. Firstly, we assume that 1" is an isomorphie prelongation and We
On the Rclations Betwccn thc Greupoid and the Pseudogroup of Transforrnations 53
shall prove 2). Let gN= ei E Ch and r, (gN) be a unit of C'. If we put z(g") == q., then there exists an open neighborhood N of di sueh that eIN is a prolongation of q. Moreover we may assume that q is an identity map. Aecordingly the maximal prolongation al(q) of g is an identity map. (By e) of Definition 4.) By the maximality of tu(q), di]N is a restriction of di(op). Hence we see that g- is aunit of Cb. Now, let (g, i) be any element of GB such that o(g, e)GKer(e), Sinee eoc{(g, X)}=noa{(g, X)} by Proposition 5.Z, neO{(g, bl)} is a unit of C'.
Therefore O{(g, di)} is a unit of Gh and hence (g, bl) E Ker(e). Thus we see that
if'i (Ker(O))(Ker(op. On the other hand, a-'(Ker(e)))Ker(oj in general from the Corollary of Proposition 5.1. Therefore the necessity of the conditions is proved.
Conversely, we assume that the conditions of TheoTem 6 hold. We eonstruct the eoTrespondence to in 2) of Definition 4. Let ca be an element of r' such that q==(B]ti)o(aiti)H'. Then we take di(ca) in Lemma 4.1. While let di be any prolongation of q. Since p(U(e))=U(ip), we have U(di)(p-i(U(ip))=U(to(q)). For any .i E U(di), there exists an element (h, .r"") of GB such that di(x-)=BB{(1- .-r)}=h•ft.
lf we take a suiliciently small open neighborhooa N of i, then we can find an open neighborhood D' of (h, di) such that ip1N=(BB1D')o(aBID')-' and D'=PÅ~OACB, where l7 and O are respectively open neighborhood of h and ft. Further let 17 be so small that exlfi and Blrt are homeomorphisms. On the other hand, since tu(ep)=(BBID)o(cxB1D)-i, there exists an element (g, ft) of D such that to(op) (di)=
BB{(g, N)}=g.I. If g.h'ibelongs to Ker(e), then a{(g•h-i, .-x)} EKer(e). By the condition 2), we see that (g•h-i, hi) belongs to .Ker(O). Therefore we have O{(g, di)} ==O{(h, i)} and hence g•N=h•ft. This means that e=to(q)IU(di), i.e, e is a restriction of to(q),
We shall prove g•h'L EKer(e). Let x' be any point of p(N)Acu(Jt). We take g' EU and h' E 17 sueh that a(g')=x/ and cu(h')= x' respectively, Next, we take ft'GIV sueh that p(/)=.x'. Then we see that (h', bl')eP7Å~OAGB beeause N=
crB(D')(O and cp (i')=h'•hi'. Sinee {Rf is a prolongation of gP, we have po4i(r.') = ipop(i'), While poip(di') =p(h'•x""')=B(bl) and q•p(di')=q(.x')=B(g'). Therefore we haVe B(h')= B(g'). Thus we see that gp(.x') = B( g') == B(h') = (B 1 )7)o (af I tz)-i(x'). I. e.
9----(BIP)o(crlrt)'i on p(N)Acr(fi). Sinee p(N)Acr(rt) is an open neighborhood Of a(g)=p(di)=ct(h), therefore we have e(g)=e(h), i.e. g•h-iEKer(e).
Now let ep be any element of r. Then we may vvriteqin the form of op = Y. ipi, where qi=(Bltii)o(aItii)'i. For a point ."x'-ep-i(U(opi))Ap'-i(U(epj)), let
to-(g'i) (bl)=g•lt and a)(g,i) (il)==h•il, where (g, lii) e [7iÅ~p-'(U(gPi))AGB and (h, sc') E Ui X p-: (U(opi)) AGB. Since p (di) E U(ua i)A U(ipi) and qi 1 U(qi)A U(ipi) == opj l U(cai)A U(ipi), if we put ti(+ = : (a l tii)-'(U(ipi)A U(opj)) and tiS = (a l tii)-i(U(qi)AU(ipi)), then we haVe (Bl tri)o(ct 1 t7;)'i=(ig L t7;)e(ct 1 ti;)-i. Therefore, O(g)=e(h), i.e. g•h-i E Ker(e), frOM which we see that g•.'.'=h•I. We put (D(op)=vto(ipi). It is easily seen that
-l
54 Toshitane UnsvGT
to(ip) isalocal homeomorphism. Since the domain U(to(op)) of to(q) is equal to vU(tu(opi))==vp'-'(U(q,))==p-'(U(ca)), we see moreever that to(q) is a prolonga- tion of q.
Let ip be any prolongation of q. Then vve have U(di)(U(al(q)). If we put A'i=U(di)AU(to(cai)), then diINi isaprolongation of gilp(Ni), Henee diiNi isa
restricti on of to (goi l p(Ni )). While, since gt,f lp(Ari) = (B I Ii7 i•)o (a l t7 i• )- i, where l! i• =
(alOi)-i(p(Ni)), we see that tu(qi1p(Ni))=(BBlDi)e(aB1DI•)-', where Dl• =Ui+Å~
p-'(p(Ni))AGB. Sinee Df•(tiixp-i(U(opi))ACB, tu(qilp(tVi)) is a restriction of to(opi). Therefore cliINi isa restriction of tu(qi) and hence we see that di is a restriction of to(q). Thus we have a) and b) in the condition 2) of Definition 4.
Next, let Åë=vÅëi be an element of i sueh that qoop is defined. Sinee i V(tu(Åë)) ==p-i(V(ep)), we see that U(to(q))A 7(o(Åë)) ==p-i(U(ip))Ap-i(V(ip)) =-p-i(U(op)A
V(Åë)). Henee, to(q)eto(Åë) is defined. Further we see easily that p(U(to(ip)eto(di)))=
U(geesb). Since poa)(op)otu(Åë) (it)=qopotD({b) (.'x")=gpogbop(iii) for any di E U(ct)(gp)oa)(gb)), tu(ca)otu(gi) is a prolongation of opeip.
Let Åë=vipj be a restriction of ip =vqi. Since U(ip)(U(q), we have
p-i(u(Åë))(p-i(u(ip)), i.e., U(o(Åë))(U(tu(op)). If di is a point of U(di(ip)), then p(i) is a point of U(Oi)AU(qi) for some i and some i. Sinee Åëjl CJ(ipi)AU(qi) =opil U(dii)
A U(qi), if we take g E t7i and h E l7'i, where epi --- (B l t7 i)e (af 1 tT i) '-i and Åëj --- (B l 17i)e
(ccll--i)-i, sueh that ct(g)=a(h)=p(x'), then we see that g•hd'K{zr(e), Therefore we have g•.-x==h•S. On the other hand, to(Åë)(.-v)==tu(Åëi) (N)= BB{(h, .i)} =h•I and to(ca)(.'k"-)=tu(qi)(rr)=g•a-. Henee tu(ip)(hi)=tu(op)(x).
Lastly we assume that g=vcai is an identity map. Then, eaeh ipi is an i
identity map. Since to(q)=vtu(ipi), it is surncient to prove that eaeh to(cai) is i
an identity map. Aceordingly, we shall prove that to(q) is an identity map when op=(Blti)o(cr1ti)-' is an identity map. Then e(g) is a unit of G' for any
gEU- L I.e. gGKer(e) for anygEtT. Let now di be any point of U(tt)(q)). Then
di (op) (m) == g•N, wh ere (g, e) E t7 x p" (U(op)) AGB. Since (g, di) E Ker (O) by meGnS of gEKer(e), there exists ah`open neighborhood D' of (g, N) such that D'(UÅ~
p-i(LT(ip))AGB and to(q)1crB(D') is an identity map. Therefore we see that to(q) (di)==l foT any i. E U(to(ip)). Thus we complete the proof.
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[3] M. HAssE uncl L MicHLER, tlhber die Einbettbarkeit van Kategericn in Gruppoide, Math. Nachr. 25 (1963),
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On the Relations Betwecn the Groupoid and :he Pscudogroup ef Transforrnations 55
[41 Y. MATsvsmMA, Sur le Prelongement d'un PseudegrouPe d'isemorPhismes tocaux d'une variiti diVirentiabtc, IA"agoya Math•J., 7 (1954), 103-110.
[s] T. UEsuaT, On the isomerPhism and the hememorPhism ef the bases of the pseudegroups of transformations, Mem. Fac. Sci. Kyushu Univ., Scr. A, 14 (1960), 34--44.
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Department of Mathematics Kyttsh2e Institute of Technology
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