A NOTE ON THE TOPOLOGICLt- GROUPOM ASSOCIATED M7Tl H A PSEUDOGROUP OF TRAJNrSFORMATIONS

Bull. Kyushu Inst. Tech.

(M. & DLg. S,) No, 12, 1965

A NOTE ON THE TOPOLOGICLt- GROUPOM ASSOCIATED M7Tl H A PSEUDOGROUP OF TRAJNrSFORMATIONS

By

Toshitane UEsvGi

(Receivcd Nov. 16, 1964)

A, M. Rodrigues proved in [4] the following theorem: lf r isaregular and connected Lie pseudogroup of order s, then the set iS(b of all s-jets of elements of r is generated, as a differentiable groupoid, by any neighborhood of the set I of all units of iS(r). This is a generalization of the well known theorem of O. Schreier for a connected topological group. In the present paper we generalize this Rodorigizes' theorem in the case of that r is a pseudogroup of transformations (not neeessarily Lie pseudogroup). The theorem will be proved when T is a simply transitive pseudogroup of transformations of a conneeted topologieal space.

In section 1, we explain the definitions and notations and give a faetor groupoid. In section 2, we firstly prove a theorem for this factor groupoid (Theorem 1). Then, our intended theorem is obtained as a speeial case of Theorem 1. In seetion 3, we g,ive a proof of the theorem of O. Sehreier from our standpoint.

g 1. Defuitions andnotations. Let E bea topological spaee. A loeal

homeomorphism e of E is a homeomorphism from an open set of E onto an open set oÅí E, in this ease we shall denote by U(q) and V(q) respectively the domain and range of q.

DEFiNmoN !. 1. A pseuclogroup of transfor7nations of E is a set r of local homeomorphisms of E satisfying the following conditions:

a) If ip, ip e r, then the composition opoip belongs to r.

b) If qGr, then the inverse map op-i of q belongs to r, e) The identity map ofEbelongs to r.

d) The restriction of any map of r to an open set of E belongs to l'.

Let G be a set of elements {e, g, h, ••-}, C is called a muttiplicative sJstem If, fOr some pair g, hEG, a product g, hEG is defined. Then an element e of e

rS called a unit if e•g==g and h•e==h whenever e.g and h.e are defined.

DEFrNiTioN 1. 2. A multiplicative system C is ealled a groupoid if the

35

36 Toshitanc UEsvG:

following conditions are satisfied:

a) The triple product f•(g•lt) is defined if and only if (f•g)•h is defined and, when one of them is defined, the assoeiative law f.(g•h)==(f•g)•h holds.

b) The triple produet f•(g•h) is defined wheneveT both product f•g and g•h are defined.

c) For each geG there exist right and left uatits e', e", respectively, sueh that g•e' and e".g are defined.

d) For eaeh g E G there exists an inverse g'i, i.e., an e}ement g-i of C such that g-i.g is a right unit, g•g-i a left unit, of g.

It follows from the axioms that eaeh element g of a groupoid G has a unique right unit, a unique left unit and a unique inverse.

Hereafter the right unit and the Ieft unit of g e G are respective]y denoted by cu(g) and B(g)•

Further a groupoid G with a topology is called a topological groupoia if the following conditions are satisfied:

e) The map (g, h)-g•h of subspace of CÅ~G formed by the multiplicatib!e pairs to G defined by the product of G is eontinuous.

f) The map g-Årg'i of G to G is eontinuous.

It is easily seen that a and B are both continuous maps of G onto the sub- spaee ef all units of G.

Now, let llr be the set of the germs of elements of apseudogroup r ef transformations of E. We denote the germ at the point xeE of qEr by op..

For two elements q., Åë, G rrr we define the product Åë,•q. equals to (Åëoip)x if and only ify==q(x). Then, units of i7r are the germs of identity map of'Eand (ip.)-i=qe(i.). It is easily seen that ITr is a groupoid with this produet. On the

other hand 17r possesses a natural topology: a base of open sets with respeet to this topology consists of the subsets {op:; xE U(g)}, where each ca is an element of L Then we may easily prove that ITr is a topologieal groupoid with this topology. Since a unit oÅí lrr is written by c., where c is the identity map of E, we identify the space E wi,th the subspaee of units of ZTr by the map x-Årex•

Then a(4,.)==x and B(q.)=op(x) for any ep.EZTr (see [1]).

rrr is called the topological groupoid associated with r,

Let U=={Ui;iEI} be any open eovering of Eand ru be the subset of ip of r such that U(op)( Ui and V(q)( Ui for some ieI. We put then ti={op.;qGrll}•

LE"i"iA 1. 1. The s2ebset t7 of llr is an open set of Lrr which contains E.

PRooF. For any element g=op.Gt7 there exists an element Ui ef 11 such that Lr(op)( Ui and V(go)( th Then {ep,; vE U(gp)} is an open set of 17r and {q,;

ve U(Åë)} (ti. Henee ti is an open set of nr.

.

Letg be an element of E. Theng is written by the forrn g=c.. If Ui is an

A Note en the Topological Greupoid Asseciated with a Pseudogroup of Transformations 37

element of U such that xE Ui, then clUbelongs to ru. Therefore g==c.EtT.

Q.ED.

Next, we introduce an equivalence relation "•v" into ]7r as follows: For two elements g and h of ZTr, we eall that g is equivalent to h if and only if a(g)=ct(h) and B(g)=B(h), This relation is evidently an equivalence relation.

]i'urther the following lemma holds.

LElifiiA 1, 2. Letg be eq2eivalent to h, where g, h E ZTr. Then, (i) g-' is equivalent to h-i.

(ii) a(g) is equivalent to cx(h) and B(g) is equivalent to B(h).

(iii) If g' is equivalent to h' and g•g' is definedi, then h•h' is atso defined ana g•g' is eguivalent to h•h'.

PRooF. The property (ii) is trivial frorn the definition of the relation "--".

In general, we may prove that g•h is defined if and only if a(g)=B(h) for tvvo elements g and h of a groupoid G. In fact, if gth is defined, then (g•a(g))•h is defined. By condition a) of Definition 1.2, theng•(a(g)•h) is also defined. Sinee ev(g) is a unit, a(g)==B(h) from the uniqueness of the left unit of h. Conversely, if cx(g)=B(h), then g•t9(h) is defined. Therefore g•(B(h)•h) is defined by condition b) of Definition 1.2, i.e. g•h is defined.

Now let g and h be two elements of nr and g be equivalent to h. Sinee g•g-i and g-'•g are both defined, we have ev(g)=fi(g"') and a(g-')=B(g). Hence, a(g-')=i9(g)=B(h)=cv(h-") and R(g-i)=ev(g)= ct(h)=B(h-i). I.e., g-iNis'.

]i'urther, let g' be equivalent to h' and g•g' be defined.' Since cM(g)=B(g'), a(g)=a(h) and B(g') =B(h'), we have a(h)=:B(h'). [I]herefore h•bl is defined. It is trivial that g•g' is equivalent to h•h' using the faet that a(g•g')==a(g') and B(g.g')=B(g) whenever g•g' is defined for two elements g and g' of a groupoid.

Q,E.D.

Let ttr be a set of equivalence classes of nr under the equivalenee relation and p: ZIr.rrr be the natural map. We denote p(g) by g- for gEITr.

]i'or two elements g', rt G jEir we shall define the product g•it equals to g'•h if and only if g•h is defined in Lrr. By property (iii) of Lemma 1,2 this definition iS well defined. Using Lemma 1.2, we easily see that irr is a groupoid with thiS produet. Further we have a(g)=ev(g), B(g)=B(g) and g-i =Ei:"i' (see, [2, Satz 1]).

DEFiNiTroN 1, 3, Let Gi and G2 be twe groupoids and P be a map of Gi into C2• Then p is called a ho7no7norphisni of Ci into G2 if the following eonditions are satisfied:

a) If g, hEet and g.h is defined, then p(g)•p(h) is also defined and equals

te p(g•h).

38 Toshitane UEsvai

b) If g is a unit of Gi, then p(g) is also a unit of G2.

The natural map p.: ZTr.ltr is then an onto homomorphism. In faet, the eondition a) of Definition 1.3 is tTivial from the definition of the product in li'r.

Let e be aunit of 17r and g-•e be defined for some gEnr. Since g-•e :ge•Ee=g, we see that e' is also a unit of i7r. Hence the condition b) of Definition 1.3 is satisfied. Moreover it is evident from the definition of the product in llr that if p(g)•p(lt) is defined, then g•h is also defined.

Consequently we have

PRoposrTroN 1. 1. ttr is a groupuoidi anda natuTal 7nap p is a homomorplvism of llr Onto rrr. Moreover, fo7' two eleme7tts g and lt of nr, if p(g)•p(h) is defined

in rrr, then g•h is also alefined in 17r.

S 2. The groupoid ITr.

DEFrNiTioN 2. 1. ([3]) A pseudogroup r of transfoxmations of E js called, transitive if, for ahy point (x, s,) EExE, there exists an element ca c r such that q(.x)=y•

Then the following theorem will be proved.

THEoREM 1. Let r be a tTa7zsitive pse2tdogroup of transformations of a connected topological space E. Then, for any open covering n={Ui;ieI}, rrr is generated, as a groupoul, by p(U).

PRooF. Let g- be any element of rrr. If we put cx(g)==.e and B(g)==x then we can find the two element Ui and Ui of U such that .xE Ui and yG Ui. Since E is connected, there exists a finite subfamily {Ui,, Ui,, •--, Uf,} of U such that

Ui, --- Ui, Ui,== Lri and UitAUi,.,\ e5, t = O, 1,--•,s-1. Taking the s points xi, x•2,-••, pt.

such that xt e Ui,.-,AUi, for each t==1, --•, s, then, since r is transitive, there exist the elements opO, Åë',••-, caS'i and q" of r sueh that opO(x)=xi, cai(xi)==.r2,••+, qS'i(pt,-i)= x. and q'(.x,)==y.-'For each 9' (r=O, 1, ---, s), since x,+iE Ui.AV(of), there exists an open neighborhood U' of x, such that ip'(U')( Ui,AV(op'). When we put U== U'AUi., we have U( lli. and op'(LT)( Ui,. Henee, q'lU belongs to ru- Accordingly, we may assume that each q' is an element of ru. Therefore each element q;. of nr belongs to ti, where we put pto=fr. On the other hand, evidently the produet q;,••+••ca1,•qe is defined in rrr and a(gi,•+--•q9) =a(q9)==•v and B(opl,•••-•opg)=a(Åë;,)=opS(x.)=:7. Thus we see that g----ipi,••+••ip2=of/,••-••qg,

where ip;'.Gp(tT) (r==O, !,•••, s). Q.E.D.

DEFiNrrroN 2.2. ([3]) A transitive pseudogroup r of transformations of E

is ealled simply transiti7ve if an element ca of r fixes a point x of E, then there

A Note en thc Topological Greupoid Associaied with a Pseudosi'oup or Transforrnations 39

exists a neighborhood Uof .x such that qlU is an identity map.

PRoposiTioN 2. 1. If i' is a simpey transitive, then llr=Ztr.

PRgoF. Let g=ee. and h=gb, be two elements of ZTr and g be equivalent

8g,ig-,,g,'"=C9.cr.(.g,)=,ex,:.h)a."S,,B,(g,),=,e8'L•.ll•l?,Se:t.h,a,kir,=,\,a,",d,q8x).r];9E,g)li,,"e,fi:2

vi-'eqlU is an identity map. Then iplU==ÅëlUand hence we have g=h. Q.E.D.

From the Theorem 1 and the Proposition 2.1 we have

THEoRE"f 2. Let r be a simply transitive pseudog?'oup of t7'ansformations ofa connected topologicae space E. Tleen, for any open covering 11 of E, ZTr is generatea, as a topological groupoid, by an open neighborhood ti of the set of all units of nr.

Since the map ev of j7r onto E is continuous, if nr is eonneeted as a topological spaee, then E is connected. Therefore we have

Ti{EoRE:f 3. Let 1' be a simpey transitive psezedogroup of to'ansformations of a topotogical space E. If llp is a connected topologicae space, then, fo7' an?J open eovering U of E, 17r is generated, as a topological g?'ozapoid, by an open

neighborhooal ti of the set of alt eenits of ll'r.

li 3. The theorern of O. Schreier for a connected toporogical group.

In this sectiop we give a proof of a well known theorem of O. Schreier for a Åëonnected topologieal group. This theorem states thata connected topotogical gr02Lp zs ge7zerated b2Ll any neighborhooae of the itnit element.

First, let C be a connected topological group and r be the set of loeal homeomorphisms of G such that every ]ocal homeomorphism is a restrietion of a left translation of C to an open set. Then I' is evidently a simply transitive

Pseudogroup of transformations of C. Let l(c) be the Ieft translation of C by ifECand l(a). be the germ of l(o) at .v EG. When we map l(a). to (e, .r) E6-Å~C, }vhere C" is the underlying abstraet group of G with the diserete topology, ITr ]S homeomorphic onto G- Å~C by this map. Then, the induced groupoid structure Of C-Å~C by ITr is given as follows: For two elements (o, A:), (t, J,) GG-xG, the P.roduet (r, y)•(a, .r) is defined if and only if y=a•.v and equals to (r•a, .as). The right unit ev(a, .v) and the left unit B(a, x) of (o, x) are respeetively .r and o•.x'•

Now Iet U be any neighborhood of the unit element of G. Sinee we can

take an open neighborhood li7 such that jJ7(U and ll7==JIF-i, we may assume

without loss of generality that U is open and U=Um'. Let us take an open

eOVering {U•rsrGG} of G. For any element oeG, we seleet an element (a, .x')

Of C--XC. Then there existsa finite subfamily {U•ro,--•, Uer,} of {U•r;teG}

40 Toshitanc UasuGr

sueh that xE Lr.ro, c.x E CT.r, and V.r,AU.r,.i=7!=szS (r=O, 1,•••, s-1) and we have•

(ff, rc)=(o., x.).-+•.(co, xo) from Theorem 2, where .xr e U.rr and Cr.xr=xr+i E V.tr for eaeh r=O, 1,d••, s-1. (See the proof of Theorem 1). If we put k'.=P..rr and Or.xr =P;•rr, P;, P;EU, then we see that c,•p,.T,=p;.r.. Henee we have if,=

p;•p;i. Since (o, x)=(c.•...•oo, x) we have a=o..•••.ao=pg•pyi•••••p6•pii, where p;, p;i c U. Thus we see that the theorem of O. Sehreier holds good.

References

[1] A. HAEFLTGER, StructuTesfeuitleties et cohemelegie d vateur dans un faisceau de greuPoides, Cemm, Math.

Helv. 32 (1958), 248-329.

[2] M. HAss4 Einigt Bemerkungen fiber GraPhcn, Kategorien und GrttPPeidc, Math. Nachr., 22 (1960), 255- 270.

[S] Y. MATsusHn!.A, PseudogrouPes dc Lie transittltls, SEminairc Bourbaki, rnai 1955.

[4] A. M. ReDR:GuEs, 7The dirst and secend fundamental theorcms of Lie fer Lie Pseude grouPs, Amer. J. of Math., e4(1962), 265-282.

Department of Matleematics

Kyushu Institute of Tecltnology