The fundamental groupoid of the quotient of a Hausdorff space by a discontinuous action of a discrete group is the orbit groupoid of the induced action Ronald Brown*, Philip J. Higginsy, Mathematics Division, Department of Mathematical Sciences, School of Informatics, Science Laboratories, University of Wales, Bangor South Rd., Gwynedd LL57 1UT, U.K. Durham, DH1 3LE, U.K. January 17, 2003 University of Wales, Bangor, Maths Preprint 02.25 Abstract The main result is that the fundamental groupoid of the orbit space of * *a discontinuous action of a discrete group on a Hausdorff space which admits a universal cover is the orbit gro* *upoid of the fundamental groupoid of the space. We also describe work of Higgins and of Taylor which makes t* *his result usable for calculations. As an example, we compute the fundamental group of the symmetric square of* * a space. The main result, which is related to work of Armstrong, is due to Brown* * and Higgins in 1985 and was published in sections 9 and 10 of Chapter 9 of the first author's book on * *Topology [3]. This is a somewhat edited, and in one point (on normal closures) corrected, version of those * *sections. Since the book is out of print, and the result seems not well known, we now advertise it here. It is hoped that this account will also allow wider views of these resu* *lts, for example in topos theory and descent theory. Because of its provenance, this should be read as a graduate text rathe* *r than an article. The Exercises should be regarded as further propositions for which we leave the proofs t* *o the reader. It is expected that this material will be part of a new edition of the book. MATH CLASSIFICATION: 20F34, 20L13, 20L15, 57S30 1 Groups acting on spaces In this section we show some of the theory of a group G acting on a topological* * space X, and describe the orbit topological space, which is written X=G. There arises the problem of relating topological invariants of the orbit sp* *ace X=G to those of X and the group action. In particular, it is a complicated and interesting question to f* *ind, if at all possible, relations between the fundamental groups and groupoids of X and X=G. This we shall do for* * a particular family of actions which arise commonly, namely the discontinuous actions. The resulting * *theory generalises that of regular covering spaces, and has a number of important applications. A useful s* *pecial case of a discontinuous *_____________________________________ email: r.brown@bangor.ac.uk yemail: p.j.higgins@durham.ac.uk 1 Orbit groupoids * * 2 action is the action of a finite group on a Hausdorff space (see below); there * *are in the literature many interesting cases of discontinuous actions of infinite groups (see [2]). We now come to formal definitions. Let G be a group, with its group operation written as multiplication, and l* *et X be a set. An action of G on X is a function G x X ! X, written (g, x) 7! g . x, satisfying the following pr* *operties for all x in X and g, h in G: 1.1 (i) 1 . x = x, (ii) g . (h . x) = (gh) . x. Thus the first rule says that the identity of G acts as identity, and the s* *econd rule says that two elements of G, acting successively, act as the product of the two elements. There are some standard notions associated with such an action. First, an e* *quivalence relation is defined on X by x ~ y if and only if there is an element g of G such that y = g . x. Th* *is is an equivalence relation. Reflexivity follows since G has an identity. Symmetry follows from the existenc* *e of inverses in G, using 1.1 (i), (ii). Transitivity follows from the product of two elements in G being in * *G. The equivalence classes under this relation are the orbits of the action. The set of these orbits is written * *X=G. Suppose given an action of the group G on the set X. If x 2 X, then the gro* *up of stability of x is the subgroup of G Gx = {g 2 G : g . x = x} The elements of Gx are said to stabilise x, that is, they leave x fixed by thei* *r action. If Gx is the whole of G, then x is said to be a fixed point of the action. The set of fixed points of th* *e action is often written XG . The action is said to be free if all groups of stability are trivial. Another useful condition is the notion of effective action of a group. This* * requires that if g, h 2 G and for all x in X, g . x = h . x, then g = h. In this case the elements of G are entir* *ely determined by their action on X. We now turn to the topological situation. Let X be a topological space, and* * let G be a group. An action of G on X is again a function G x X ! X with the same properties as given in 1.* *1, but with the additional condition that when G is given the discrete topology then the function (g, x) 7* *! g . x is continuous. This amounts to the same as saying that for all g 2 G, the function g] : x 7! g . x * *is continuous. Note that g]is a bijection with inverse (g-1)], and since these two functions are continuous, ea* *ch is a homeomorphism. Let X=G be the set of orbits of the action and let p : X ! X=G be the quoti* *ent map, which assigns to each x in X its orbit. For convenience we will write the orbit of x under the action* * as ~x. So the defining property is that ~x= ~yif and only if there is a g in G such that y = g . x. Now a topol* *ogy has been given for X. We therefore give the orbit space X=G the identification topology with respect to * *the map p. This topology will always be assumed in what follows. The first result on this topology, and one w* *hich is used a lot, is as follows. Proposition 1.2The quotient map p : X ! X=G is an open map. Proof Let U be an open set of X. For each g 2 G the set g . U, by which is mean* *t the set of g . x for all x in U, is also an open set of X, since g]is a homeomorphism, and g . U = g][U]. But [ p-1p[U] = g . U. g2G Since the union of open sets is open, it follows that p-1p[U] is open, and henc* *e p[U] is open. 2 Orbit groupoids * * 3 Definition 1.3An action of the group G on the space X is called discontinuous i* *f the stabiliser of each point of X is finite, and each point x in X has a neighbourhood Vx such that any elem* *ent g of G not in the stabiliser of x satisfies Vx \ g . Vx = ;. Suppose G acts discontinuously on the space X. For each x in X choose such an o* *pen neighbourhood Vx of x. Since the stabiliser Gx of x is finite, the set " Ux = {g . Vx : g 2 Gx} is open; it contains x since the elements of Gx stabilise x. Also if g 2 Gx the* *n g . Ux = Ux. We say Ux is invariant under the action of the group Gx. On the other hand, if h 62 Gx then (h . Ux) \ Ux (h . Vx) \ Vx = ;. An open neighbourhood U of x which satisfies (h.U)\U = ; for h 62 Gx and is* * invariant under the action of Gx is called a canonical neighbourhood of x. Note that any neighbourhood N o* *f x contains a canonical neighbourhood: the proof is obtained_by replacing Vx in the above by N \Vx. The* * image in X=G of a canonical neighbourhood U of x is written U and called a canonical neighbourhood of ~x. In order to have available our main example of a discontinuous action, we p* *rove: Proposition 1.4An action of a finite group on a Hausdorff space is discontinuou* *s. Proof Let G be a finite group acting on the Hausdorff space X. Then the stabili* *ser of each point of X is a subgroup of G and so is finite. Let x 2 X. Let x0, x1, . .,.xn be the distinct points of the orbit of x, wi* *th x0 = x. Suppose xi= gi.x, gi2 G, i = 1, . .,.n, and set g0 = 1. Since X is Hausdorff, we can find pairwise di* *sjoint open neighbourhoods Ni of xi, i = 0, . .,.n. Let " N = {g-1i. Ni: i = 0, 1, . .n.} Then N is an open neighbourhood of x. Also, if g 2 G does not belong to the sta* *biliser Gx of x, then for some j = 1, . .,.n, g . x = xj, whence g . N Nj. Hence N \ g . N = ;, and the acti* *on is discontinuous. 2 Our main result in general topology on discontinuous actions is the follow* *ing. Proposition 1.5If the group G acts discontinuously on the Hausdorff space X, th* *en the quotient map p : X ! X=G has the path lifting property: that is, if ~a: I ! X=G is a path in X=G and* * x0 is a point of X such that p(x0) = ~a(0), then there is a path a : I ! X such that pa = ~aand a(0) = x0. Proof If there is a lift a of ~athen there is an element g of G such that g . a* *(0) = x0 and so g . a is a lift of ~a starting at x0. So we may ignore x0 in what follows. Since the action is discontinuous, each point ~xof X=G has a canonical neig* *hbourhood. By the Lebesgue covering lemma, there is a subdivision ~a= ~an+ . .+.~a1 of ~asuch that the image of each ~aiis contained in a canonical neighbourhood. * *So if the path lifting property holds for each canonical neighbourhood in X=G, then it holds for X=G. * * __ Now p : X ! X=G is an open map. Hence for all x in X the restriction px : U* *x ! Ux of_p to a canonical neighbourhood Ux is also open, and hence is an identification map. So we can id* *entify Ux with the orbit space (Ux)=Gx. The key point in this case is that the group Gx is finite. Thus it is sufficient to prove the path lifting property for the case of th* *e action of a finite group G, and this we do by induction on the order of G. That is, we assume that path lifting hold* *s for any action of any proper Orbit groupoids * * 4 subgroup of G on a Hausdorff space, and we prove that path lifting holds for th* *e action of G. The case |G| = 1 is trivial. Again let ~abe as in the proposition, and we are assuming G is finite. Let * *F be the set of fixed points of the action. Then F is the intersection for all g 2 G of the sets Xg = {x 2 X : g . * *x = x}. Since X is Hausdorff, the set Xg is closed in X, and hence F is closed in X. So p[F ] is closed, sinc* *e p-1p[F ] = F . Let A be the subspace of I of points t such that ~a(t) belongs to p[F ], that is, A = ~a-1p[* *F ]. Then A is closed. The restriction of the quotient map p to p0: F ! p[F ] is a homeomorphism. * *So ~a|A has a unique lift to a map a |A:A! X. So we have to show how to lift ~a|(I\A)to give a map a |I\Aand t* *hen show that the function a : I ! X defined by these two parts is continuous. In order to construct a |I\A, we first assume A = {1}. Let S be the set of s 2 I such that ~a|[0,s]has a lift to a map as starting* * at x. Then S is non-empty, since 0 2 S. Also S is an interval. Let u = supS. Suppose u < 1. Then there is a y 2 * *X\F such that p(y) = ~a(u). Choose a canonical neighbourhood U of y. If u > 0, there is a ffi > 0 such that* * ~a[u - ffi, u + ffi] ~U. Then u - ffi 2 S and so there is a lift au-ffion [0, u - ffi]. Also the stabiliser o* *f y is a proper subgroup of G, since y =2F , and so by the inductive assumption there is a lift of ~a|[u-ffi,u+ffi]t* *o a path starting at au-ffi(u - ffi). Hence we obtain a lift au+ffi, contradicting the definition of u. We get a similar co* *ntradiction to the case u = 0 by replacing in the above u - ffi by 0. It follows that u = 1. We are not quite fi* *nished because all we have thus ensured is that there is a lift on [0, s] for each 0 6 s < 1, but this is not t* *he same as saying that there is a lift on [0, 1). We now prove that such a lift exists. By the definition of u, and since u = 1, for each integer n > 1 there is a * *lift an of ~a|[0,1-n-1]. Also there is an element gn of G such that gn . an+1(1 - n-1) = an(1 - n-1) Hence an and gn.(an+1 |[1-n-1,1-(n+1)-1]) define a lift of ~a|[0,1-(n+1)-1]. St* *arting with n = 1, and continuing in this way, gives a lift of ~a|[0,1). This completes the construction of the l* *ift on I\A in the case A = {1}. We now construct a lift of ~a|I\Ain the general case. Since A is closed, I\* *A is a union of disjoint open intervals each with end points in {0, 1} [ A. So the construction of the lift i* *s obtained by starting at the mid point of any such interval and working backwards and forwards, using the case A* * = {1}, which we have already proved. The given lift of ~a|A and the choice of lift of ~a|I\Atogether define a li* *ft a : I ! X of ~aand it remains to prove that a is continuous. Let t 2 I. If t =2A, then a is continuous at t by construction. Suppose the* *n t 2 A so that y = a(t) 2 F . Let N be any neighbourhood of y. Then N contains a canonical neighbourhood U of y. * *If g 2 G then g.y = y, and so U is invariant under the action of G. Hence p-1p[U] = U. Since ~ais continuo* *us, there is a neighbourhood M of t such that ~a[M] ~U. Since pa = ~ait follows that a[M] p-1[U~] = U. T* *his proves continuity of A, and the proof of the proposition is complete. * * 2 In our subsequent results we shall use the path lifting property rather tha* *n the condition of the action being discontinuous. Our aim now is to determine the fundamental groupoid of the orbit space X=G* *. In general it is difficult to say much. However we can give reasonable and useful conditions for which the qu* *estion can be completely answered. From our point of view the result is also of interest in that our sta* *tement and proof use groupoids in a crucial way. This use could be overcome, but at the cost of complicating both* * the statement of the theorem and its proof. In order to make the transition from the topology to the algebra, it is nec* *essary to introduce the notion of a group acting on a groupoid. Let G be a group and let be a groupoid. We will write the group structur* *e on G as multiplication, Orbit groupoids * * 5 and the groupoid structure on as addition. An action of G on assigns to eac* *h g 2 G a morphism of groupoids g] : ! with the properties that 1] = 1 : ! , and if g, h 2 G t* *hen (hg)] = h]g]. If g 2 G, x 2 Ob( ), a 2 , then we write g . x for g](x), g . a for g](a). Thus t* *he rules 1.1 apply also to this situation, as well as the laws 1.1 (iii) g.(a + b) = g.a + g.b, and (iv) g.0x =* * 0g.x for all g 2 G, x 2 Ob( ), a, b 2 such that a + b is defined. The action of G on is trivial if g]= 1 for all g in G. Definition 1.6Let G be a group acting on a groupoid . An orbit groupoid of the* * action is a groupoid ==G together with a morphism p : ! ==G such that: (i)If g 2 G, fl 2 , then p(g . fl) = p(fl). (ii)The morphism p is universal for (i), i.e. if OE : ! is a morphism of g* *roupoids such that OE(g.fl) = OE(fl) for all g 2 G, fl 2 , then there is a unique morphism OE* : ==G ! of g* *roupoids such that OE*p = OE. The morphism p : ! ==G is then called an orbit morphism. * * 2 The universal property (ii) implies that ==G, if it exists, is unique up t* *o a canonical isomorphism. At the moment we are not greatly concerned with proving any general statement abou* *t the existence of the orbit groupoid. One can argue that ==G is obtained from by imposing the relations * *g . fl = fl for all g 2 G and all fl 2 ; however we have not yet explained quotients in this generality. We will* * later prove existence by giving a construction of ==G which will be useful in interpreting our main theorem. B* *ut our next result will give conditions which ensure that the induced morphism ßX ! ß(X=G) is an orbit morph* *ism, and our proof will not assume general results on the existence of the orbit groupoid. The reason w* *e can do this is that our proof directly verifies a universal property. First we must point out that if the group G acts on the space X, then G act* *s on the fundamental groupoid ßX, since each g in G acts as a homeomorphism of X and g] : ßX ! ßX may be defi* *ned to be the induced morphism. This is one important advantage of groupoids over groups: by contrast* *, the group G acts on the fundamental group ß(X, x) only if x is a fixed point of the action. Suppose now that G acts on the space X. Our purpose is to give conditions o* *n the action which enable us to prove that p* : ßX ! ß(X=G) determines an isomorphism (ßX)==G ! ß(X=G), by verifying the universal property* * for p*. We require the following conditions: Conditions 1.7(i)The projection p : X ! X=G has the path lifting property: i.e.* * if ~a: I ! X=G is a path, then there is a path a : I ! X such that pa = ~a. (ii)If x 2 X, then x has an open neighbourhood Ux such that (a)if g 2 G does not belong to the stabiliser Gx of X, then Ux \ (g . Ux) * *= ;; (b)if a and b are paths in Ux beginning at x and such that pa and pb are h* *omotopic rel end points in X=G, then there is an element g 2 Gx such that g . a and b are homotopi* *c in X rel end points. a_______o0y0___________________________* *____________________________________________________________________________@ __________________________________________* *_______ ____________________________________________* *___ x o__________g_._a______________________________* *____________________________________________________________________________@ ____________________________________________* *_________________________________________________________________ _________''_____________________________* *_____.._____________________________________________________________________@ b o g . y Orbit groupoids * * 6 For a discontinuous action, 1.7(ii:a) trivially holds, while 1.7(i) holds b* *y virtue of 1.5. However, 1.7(ii:b) is an extra condition. It does hold if X is semi-locally simply-connected, since t* *hen for sufficiently small U and x, g . y 2 U, any two paths in U from x to g . y are homotopic in X rel end poi* *nts; so 1.7(ii:b) is a reasonable condition to use in connection with covering space theory. A neighbourhood Ux of x 2 X satisfying 1.7(i) and 1.7(ii) will be called a * *strong canonical neighbourhood of X. The image p[Ux] of Ux in X=G will be called a strong canonical neighbourh* *ood of px. Proposition 1.8If the action of G on X satisfies 1.7(i) and 1.7(ii) above, then* * the induced morphism p* : ßX ! ß(X=G) makes ß(X=G) the orbit groupoid of ßX by the action of G. Proof Let OE : ßX ! be a morphism to a groupoid such that OE(g . fl) = OE(f* *l) for all fl 2 ßX and g 2 G. We wish to construct a morphism OE* : ß(X=G) ! such that OE*p = OE. Let ~abe a path in X=G. Then ~alifts to a path a in X. Let [b] denote the h* *omotopy class rel end points of a path b. We prove that OE[a] in is independent of the choice of ~ain its homot* *opy class and of the choice of lift a; hence we can define OE*[~a] to be OE[a]. Suppose given two homotopic paths ~aand ~bin X=G, with lifts a and b which * *without loss of generality we may assume start at the same point x in X. (If they do not start at the same po* *int, then one of them may be translated by the action of G to start at the same point as the other.) Let h :* * I x I ! X=G be a homotopy rel end points ~a' ~b. The method now is not to lift the homotopy h itself, but to * *lift pieces of a subdivision of h; it is here that the method differs from that used in the theory of covering spaces* * given in Section 9.1 of [3]. Subdivide I x I, by lines parallel to the axes, into small squares each of * *which is mapped by h into a strong canonical neighbourhood in X=G. This subdivision determines a sequence of homot* *opies hi: ~ai-1' ~ai, i = 1, 2, . .,.n, say, where ~a0= ~a, ~an= ~b. Keep i fixed for the present. Each h* *iis further expressed by the subdivision as a composite of homotopies hij(j = 1, 2, . .,.m) as shown in the * *following picture in which for convenience the boundaries of the hijare labelled: ____~e1//__``` `` ``_____~ej//__```` `` `____~em//__ ~ai| | | | | | | | | | | | | | | | | | ~x= ~c0OO|hi1| O~c1O||~cj-1OOhij|| ~cjOO||~cmOO|him-1| O~cmO|| | | | | | | | | | | | | ~ai-1|___//____```|`` ``|____//____|```` `` `|____//____| ~d1 ~dj ~dm Choose lifts ai-1, aiof ~ai-1, ~airespectively; express ai-1as a sum ai-1= * *dm + . .+.d1 and aias a sum ai= em + . .+.e1 where djlifts ~djand ejlifts ~ej. Choose for each j a lift cjo* *f ~cj(with c0 the constant path at x). For fixed j choose f, g, h 2 G such that g . djhas the same initial poin* *t as cj-1and the sums f . cj+ g . dj, h . ej+ cj-1 are defined. This is possible because of the boundary relations between the pro* *jections in X=G of the various paths. Now our assumption (ii:b) of Conditions 1.7 implies that there is an elemen* *t k 2 G such that the following paths in X k . (f . cj+ g . dj), h . ej+ cj-1 are homotopic rel end points in X. On applying OE to homotopy classes of paths * *in X and using equations such Orbit groupoids * * 7 as OE(g . fl) = OE(fl) we find that OE[ej] + OE[cj-1]= OE[h . ej] + OE[cj-1] = OE[h . ej+ cj-1] = OE[k . (f . cj+ g . dj)] = OE[k . f . cj] + OE[g . dj] = OE[cj] + OE[dj]. This proves that OE[ej] = OE[cj] + OE[dj] - OE[cj-1]. It follows easily that OE[ai-1] = OE[ai]. and hence by induction on i that OE[a] = OE[b]. From this it follows that OE* * *: ß(X=G) ! is a well defined function such that OE*p = OE. The uniqueness of OE* is clear since p* is surje* *ctive on elements, by the path lifting property of Conditions 1.7. The proof that OE* is a morphism is simple. This c* *ompletes the proof of Proposition 1.8. * * 2 In the next section, we introduce some further constructions in the theory* * of groupoids and groups acting on groupoids, in order to interpret Proposition 1.8 in a manner suitable for c* *alculations. Once again, we will find that an apparently abstract result involving a universal property can, wh* *en appropriately interpreted, lead to specific calculations. EXERCISES 1 1.Let ~ 2 R and let the additive group R of real numbers act on the torus T = S* *1x S1 by t . (e2ii`, e2iiffi) = (e2ii(`+t), e2ii(ffi+~t)) for t, `, OE 2 R. Prove that the orbit space has the indiscrete topology if a* *nd only if ~ is irrational. [You may assume that the group generated by 1 and ~ is dense in R if and only if ~ is * *irrational.] 2.Let G be a group and let X be a G-space. Prove that the quotient map p : X ! * *X=G has the following universal property: if Y is a space and f : X ! Y is a map such that f(g . x)* * = f(x) for all x 2 X and g 2 G, then there is a unique map f* : X=G ! Y such that f*p = f. 3.Let X, Y, Z be G-spaces and let f : X ! Z, h : Y ! Z be G-maps (i.e. f(g . x)* * = g . f(x) for all g 2 G and x 2 X, and similarly for h). Let W = X xZ Y be the pullback. Prove that W* * becomes a G-space by the action g . (x, y) = (g . x, g . y). Prove also that if Z = X=G and f is t* *he quotient map, then W=G is homeomorphic to Y . 4.Let G and be groupoids and let w : ! Ob(G) be a morphism where Ob(G) is c* *onsidered as a groupoid with identities only. An action of G on via w is an assignment to each g 2 * *G(x, y) and fl 2 w-1[x] an element g . fl 2 w-1[y] and with the usual rules: h . (g . fl) = (hg) . fl; 1* * . fl = fl; g . (fl + ffi) = g . fl + g . ffi. In this case is called a G-groupoid. Show how to define a category of G-gro* *upoids so that this category is equivalent to the functor category FUN(G, Set). 5.Prove that if is a G-groupoid via w, then ß0 becomes a G-set via ß0(w). 6.If is a G-groupoid via w, then the action is trivial if for all x, y 2 Ob(G* *), g, h 2 G(x, y) and fl 2 w-1[x], we have g . fl = h . fl. Prove that the action is trivial if for all x 2 Ob(G* *), the action of the group G(x) on the groupoid w-1[x] is trivial. Prove also that contains a unique maximal s* *ubgroupoid G on which G acts trivially. Give examples to show that G may be empty. Orbit groupoids * * 8 7.Continuing the previous exercise, define a G-section of w to be a morphism s * *: Ob(G) ! of groupoids such that ws = 1 and s commutes with the action of G, where G acts on Ob(G) v* *ia the source map by g . x = y for g 2 G(x, y). Prove that G is non-empty if w has a G-section, a* *nd that the converse holds if G is connected. Given a G-section s, let G(s) be the set of functions a : Ob(G* *) ! such that wa = 1 and a commutes with the action of G (but we do not assume a is a morphism). Show * *that G(s) forms a group under addition of values, and that if G is connected and x 2 Ob(G), then G(s* *) is isomorphic to the group (sx)G(x)of fixed points of (sx) under the action of G(x). 2 The computation of orbit groupoids: quotients and semidirect products The theory of quotient groupoids is modelled on that of quotient groups, but d* *iffers from it in important respects. In particular, the First Isomorphism Theorem of group theory (that every surje* *ctive morphism of groups is obtained essentially by factoring out its kernel) is no longer true for groupo* *ids, so we need to characterise those groupoid morphisms (called quotient morphisms) for which this isomorphism theo* *rem holds. The next two propositions achieve this; they were first proved in [5]. Let f : K ! H be a morphism of groupoids. Then f is said to be a quotient* * morphism if Ob(f) : Ob (K) ! Ob (H) is surjective and for all x, y in Ob(K), f : K(x, y) ! H(fx, f* *y) is also surjective. Briefly, we say f is object surjective and full. Proposition 2.1Let f : K ! H be a quotient morphism of groupoids. Let N = Kerf* *. The following hold: (i)If k, k02 K, then f(k) = f(k0) if and only if there are elements m, n 2 N* * such that k0= m + k + n. (ii)If x is an object of K, then H(fx) is isomorphic to the quotient group K(* *x)=N(x). Proof (i) If k, k0satisfy k0= m + k + n where m, n 2 N, then clearly f(k) = f(* *k0). Suppose conversely that f(k) = f(k0), where k 2 K(x, y), k02 K(x0, y0). Th* *en f(x) = f(x0), f(y) = f(y0). x_____k_____//yØOOØ Ø Ø Ø Ø nØØ ØmØ Ø Ø Ø fflfflØ x0_____0____//_y0 k Since f : K(y, y0) ! H(fy, fy0) is surjective, there is an element m 2 K(y, y0* *) such that f(m) = 0f(y). Similarly, there is an element n 2 K(x0, x) such that f(n) = 0f(x). It follows* * that if n0= -k0+ m + k + n 2 K(x0), then f(n0) = 0f(x0), and so n02 N. Hence k0= m + k + n - n0, where m, n - n02 * *N. This proves (i). (ii) By definition of quotient morphism, the restriction f0: K(x) ! H(fx) * *is surjective. Also by 2.1(i), if f0(k) = f0(k0) for k, k02 K(x), then there are m, n 2 N(x) such that k0= m + k* * + n. Since N(x) is normal in K(x), there is an m0in N(x) such that m + k = k + m0. Hence k0+ N(x) = k + * *N(x). Conversely, if k0+ N(x) = k + N(x) then f0(k0) = f0(k). So f0determines an isomorphism K(x)=N* *(x) ! H(fx). 2 We recall the definition of normal subgroupoid. Orbit groupoids * * 9 Let G be a groupoid. A subgroupoid N of G is called normal if N is wide in * *G (i.e. Ob (N) = Ob(G)) and, for any objects x, y of G and a 2 G(x, y), aN(x)a-1 N(y), from which it * *easily follows that aN(x)a-1 = N(y). We now prove a converse of the previous result. That is, we suppose given a* * normal subgroupoid N of a groupoid K and use (i) of Proposition 2.1 as a model for constructing a quotien* *t morphism p : K ! K=N. The object set of K=N is to be ß0N, the set of components of N. Recall that* * a normal subgroupoid is, by definition, wide in K, so that ß0N is also a quotient set of X = Ob(K). Define * *a relation on the elements of K by k0~ k if and only if there are elements m, n in N such that m + k + n is d* *efined and equal to k0. It is easily checked, using the fact that N is a subgroupoid of K, that ~ is an equiv* *alence relation on the elements of K. The set of equivalence classes is written K=N. If clsk is such an equival* *ence class, and k 2 K(x, y), then the elements clsx, clsy in ß0N are independent of the choice of k in its e* *quivalence class. So we can write clsk 2 K=N(clsx, clsy). Let p : K ! K=N be the quotient function. So far, we ha* *ve not used normality of N. Not surprisingly, normality is used to give K=N an addition which makes it i* *nto a groupoid. Suppose clsk1 2 (K=N)(clsx, clsy), clsk2 2 (K=N)(clsy, clsz). Then we may assume k1 2 K(x, y), k2 2 K(y0, z), where y ~ y0 in ß0N. So th* *ere is an element l 2 N(y, y0), and we define clsk2+ clsk1 = cls(k2+ l + k1). We have to show that this addition is well defined. Suppose then k01= m1+ k1+ n1, k02= m2+ k2+ n2, where m1, n1, m2, n2 2 N. Choose any l0such that k02+ l0+ k01is defined. Then w* *e have the following diagram, in which a, a0are to be defined: _a_______________________________* *_________a0_________________________________________ _________________________________* *____________________________________________________________________________@ k1 l ______k2_________________________* *____________________________________________________________________________@ o_____//____o_____//____o____//____o | | | | | | | | | | | | n1 OO|| fm1flffl|| On2O|| fm2flffl|| | | | | | | | | o|____//____|o____//___|_o___//___|_o k01 l0 k02 Let a = n2 + l0+ m1 - l. Then a 2 N, and l = -a + n2 + l0+ m1. Since N is norma* *l there is an element a02 N such that a0+ k2 = k2- a. Hence k2+ l + k1= k2- a + n2+ l0+ m1+ k1 = a0+ k2+ n2+ l0+ m1+ k1 = a0- m2+ k02+ l0+ k01- n1. Since a0, m2, n1 2 N, we obtain cls(k2+ l + k1) = cls(k02+ l0+ k01) as was requ* *ired. Now we know that the addition on K=N is well defined, it is easy to prove t* *hat the addition is associative, has identities, and has inverses. We leave the details to the reader. So we kno* *w that K=N becomes a groupoid. Proposition 2.2Let N be a normal subgroupoid of the groupoid K, and let K=N be * *the groupoid just defined. Then Orbit groupoids * * 10 (i)the quotient function p : k 7! clsk is a quotient morphism K ! K=N of grou* *poids; (ii)if f : K ! H is any morphism of groupoids such that Kerf contains N, then * *there is a unique morphism f* : K=N ! H such that f*p = f. Proof The proof of (i)is clear. Suppose f is given as in (ii). If m + k + n is * *defined in K and m, n 2 N, then f(m + k + n) = f(k). Hence f* is well defined on K=N by f*(clsk) = f(k). Clearl* *y f*p = f. Since p is surjective on objects and elements, f* is the only such morphism. * * 2 In order to apply these results, we need generalisations of some facts on n* *ormal closures which were given in Section 3 of Chapter 8 for the case of a family R(x) of subsets of the objec* *t groups K(x), x 2 Ob(K), of a groupoid K. The argument here is based on [6, Exercise 4, p.95]. Suppose that R is any set of elements of the groupoid K. The normal closure* * of R in K is the smallest normal subgroupoid N(R) of K containing R. Clearly N(R) is the intersection of * *all normal subgroupoids of K containing R, but it is also convenient to have an explicit description of N(* *R). Proposition 2.3Let be the wide subgroupoid of K generated by R. Then the no* *rmal closure N(R) of R is the subgroupoid of K generated by and all conjugates khk-1 for k 2 K, h * *2 . Proof Let bRbe the subgroupoid of K generated by and all conjugates khk-1 f* *or k 2 K, h 2 . Clearly any normal subgroupoid of K containing R contains bR, so it is sufficient to pr* *ove that bRis normal. Suppose then that k + a - k is defined where k 2 K and a 2 bRso that a = r1+ c1+ r2+ c2+ . .+.rl+ cl+ rl+1 where each ri2 and each ci= ki+ hi- kiis a conjugate of a loop hiin by * *an element ki2 K. hl___________________________hi-1__________________* *_________h1___________________________ ____________________________________________________* *____________________________________________________________________________@ _ææ____________________________________________ææ___* *_____________________________ææ________________________________ o o o kl|| ki-1|| |k1| fflffl| fflffl| fflffl| o ook_x_o_rl+1//_orl//_.r.i.//_ori-1//_././._or1//_okx//_o Then a is a loop, since k + a - k is defined, and so also is b = r1+ r2+ . .+.rl+1. Let di= k + r1+ . .+.ri+ ci- ri- . .-.r1- k so that diis a conjugate of a loop in for i = 1, . .,.l. Then it is easily * *checked that k + a - k = d1+ . .+.dl+ k + b - k and hence k + a - k 2 bR. * * 2 Notice that the loop b in the proof belongs to rather than to R, and t* *his shows why it is not enough just to take N(R) to be the subgroupoid generated by R and conjugates of loops in R. The elements of N(R) as constructed above may be called the consequences of* * R. We next give the definition of the semidirect product of a group with a gro* *upoid on which it acts. Let G be a group and let be a groupoid with G acting on the left. The semidirect produ* *ct groupoid oG has object set Orbit groupoids * * 11 Ob ( ) and arrows x ! y the set of pairs (fl, g) such that g 2 G and fl 2 (g .* * x, y). The sum of (fl, g) : x ! y and (ffi, h) : y ! z in o G is defined to be (ffi, h) + (fl, g) = (ffi + h . fl, hg). This is easily remembered from the following picture. ozOO | ffi| | oyOO ohO.Oy fl|| h|.|fl | | ox og . x oh . g . x o ____g___//o ________//_o h Proposition 2.4The above addition makes o G into a groupoid and the projection q : o G ! G, (fl, g) 7! g, is a fibration of groupoids. Further: (i)q is a quotient morphism if and only if is connected; (ii)q is a covering morphism if and only if is discrete; (iii)q maps ( o G)(x) isomorphically to G for all x 2 Ob( ) if and only if * *has trivial object groups and G acts trivially on ß0 . Proof The proof of the axioms for a groupoid is easy, the negative of (fl, g) b* *eing (g-1 . (-fl), g-1). We leave the reader to check associativity. To prove that q is a fibration, let g 2 G and x 2 Ob( ). Then (0g.x, g) has* * source x and maps by q to g. We now prove (i). Let x, y be objects of . Suppose q is a quotient morphis* *m. Then q maps ( o G)(x, y) surjectively to G and so there is an element (fl, g) such that q(fl, g) = 1. So* * g = 1 and fl 2 (x, y). This proves is connected. Suppose is connected. Let g 2 G. Then there is a fl 2 (g . x, y), and so* * q(fl, g) = g. Hence q is a quotient morphism. We now prove (ii). Suppose is discrete, so that may be thought of as a * *set on which G acts. Then oG is simply the covering groupoid of the action as constructed in a previous sect* *ion. So q is a covering morphism. Let x be an object of . If fl is an element of with source x then (fl, 1* *) is an element of o G with source x and which lifts 1. So if q is a covering morphism then the star of at any x* * is a singleton, and so is discrete. The proof of (iii)is best handled by considering the exact sequence based a* *t x 2 Ob( ) of the fibration q. This exact sequence is by 7.2.10 of [3] 0 1 ! (x) ! ( o G)(x) -q!G ! ß0 ! ß0( o G) ! 1. It follows that q0is injective if and only if (x) is trivial. Exactness also s* *hows that q0is surjective for all x if and only if the action of G on ß0 is trivial. * * 2 Here is a simple application of the definition of semidirect product which * *will be used later. Orbit groupoids * * 12 Proposition 2.5Let G be a group and let be a G-groupoid. Then the formula (fl, g) . ffi = fl + g . ffi for fl, ffi 2 , g 2 G, defines an action of o G on the set via the target * *map ø : ! Ob( ). Proof This says in the first place that if (fl, g) 2 ( o G)(y, z) and ffi has * *target y, then fl + g . ffi has target z, as is easily verified. The axioms for an action are easily verified. The formul* *a for the action also makes sense if one notes that (fl, g)(ffi, 1) = (fl + g . ffi, g). * * 2 If X is a G-space, and x 2 X, let oe(X, x, G) be the object group of the se* *midirect product groupoid ßXoG at the object x. This group is called by Rhodes in [9] and [10] the fundamental* * group of the transformation group (although he defines it directly in terms of paths). The following result* * from [10] gives one of the reasons for its introduction. Corollary 2.6If X is a G-space, x 2 X, and the universal cover ~Xxexists, then * *the group oe(X, x, G) has a canonical action on ~Xx. Proof By 9.5.8 of [3], we may identify the universal cover ~Xxof X at x with St* *iXx. The function ßX ! ßX, ffi 7! -ffi, transports the action of ßX o G on ßX via the target map ø to * *an action of the same groupoid on ßX via the source map oe. Hence the object group (ßX o G)(x) acts on StiXx by (fl, g) * ffi = -((fl, g) . (-ffi)) = g . fl - fl. The continuity of the action follows easily from the detailed description of th* *e lifted topology (see also the remarks on topological groupoids after 9.5.8 of [3]). * * 2 Now we start using the semidirect product to compute orbit groupoids. The * *next two results may be found in [7], [11] and [12]. Proposition 2.7Let N be the normal closure in o G of the set of elements of t* *he form (0x, g) for all x 2 Ob ( ) and g 2 G . Let p be the composite -i! o G -! ( o G)=N, in which the first morphism is fl 7! (fl, 1) and the second morphism is the quo* *tient morphism. Then (i)p is a surjective fibration; (ii)p is an orbit morphism and so determines an isomorphism ==G ~=( o G)=N; (iii)the function Ob( ) ! Ob( ==G) is an orbit map, so that Ob( ==G) may be id* *entified with the orbit set Ob( )=G. Proof Let = ( o G)=N. We first derive some simple consequences of the defin* *ition of . Let fl 2 (x, y), g, h 2 G. Then (0g.y, g) + (fl, 1) = (g . fl, g), * * (1) (fl, 1) + (0x, h) = (fl, h). * * (2) Orbit groupoids * * 13 It follows that in we have (h . fl, 1) = (fl, g). * * (3) Note also that the set R of elements of o G of the form (0g.x, g) is a subgro* *upoid of G, since (0hg.x, h) + (0g.x, g) = (0hg.x, hg), and -(0g.x, g) = (0x, g-1). It follows that ß0N = ß0R = Ob( )=G, the set of orb* *its of the action of G on Ob ( ). Hence Ob(p) is surjective. This proves (iii), once we have proved (ii). We now prove easily that p : ! is a fibration. Let (fl, g) : x ! y in * * o G be a representative of an element of , and suppose z = x, where z 2 Ob( ). Then z and x belong to the * *same orbit and so there is an element h in G such that h . x = z. Clearly h . fl has source z and by (3), * *p(h . fl) = (fl, g). Suppose now g 2 G and fl : x ! y in . Then by (3)p(g . fl) = p(fl). This v* *erifies (i). To prove the other condition for an orbit morphism, namely Definition 1.6(i* *i), suppose OE : ! is a morphism of groupoids such that has a trivial action of the group G and OE(g * *. fl) = OE(fl) for all fl 2 and g 2 G. Define OE0: o G ! on objects by Ob(OE) and on elements by (fl, g) 7!* * OE(fl). That OE is a morphism follows from the trivial action of G on , since OE0((ffi, h) +=(fl,Og))E(fl + h.ffi) = OE(fl) + OE(h.ffi) = OE(fl) + OE(ffi) = OE0(ffi, h) + OE0(fl, g). Also OE0(0x, g) = OE(0x) = 0ffix, and so N KerOE0. By (ii), there is a unique* * morphism OE* : ( o G)=N ! such that OE* = OE0. It follows that OE*p = OE* i = OE0i = OE. The uniqueness * *of OE* follows from the fact that p is surjective on objects and on elements. Finally, the isomorphism ==G ~= follows from the universal property. * * 2 In order to use the last result we analyse the morphism p : ! ==G in so* *me special cases. The construction of the orbit groupoid given in Proposition 2.7 is what makes this * *possible. Proposition 2.8The orbit morphism p : ! ==G is a fibration whose kernel is g* *enerated as a subgroupoid of by all elements of the form fl - g . fl where g stabilises the source of f* *l. Furthermore, (i)if G acts freely on , by which we mean no non-identity element of G fixes* * an object of , then p is a covering morphism; (ii)if is connected and G is generated by those of its elements which fix so* *me object of , then p is a quotient morphism; in particular, p is a quotient morphism if the action of G on Ob* *( ) has a fixed point; (iii)if is a tree groupoid, then each object group of ==G is isomorphic to * *the factor group of G by the (normal) subgroup of G generated by elements which have fixed points. Proof We use the description of p given in the previous Proposition 2.7, which * *already implies that p is a fibration. Let R be the subgroupoid of o G consisting of elements (0g.x, g), g 2 G. * *Let N be the normal closure of R. By the construction of the normal closure in Proposition 2.3, the element* *s of N are sums of elements of R and conjugates of loops in R by elements of o G. So let (0g.x, g) be a loop* * in R. Then g . x = x. Let (fl, h) : x ! y in o G, so that fl : h . x ! y. Then we check that (fl, h) + (0x, g) - (fl, h) = (fl - hgh-1 . fl, hgh-1). Orbit groupoids * * 14 Writing k = hgh-1, we see that (fl - k . fl, k) 2 N if k stabilises the initial* * point of fl. Now fl 2 Kerp if and only if (fl, 1) 2 N. Further, if (fl, 1) 2 N then (fl,* * 1) is a consequence of R and so (fl, 1) is equal to (fl1- k1. fl1, k1) + (fl2- k2. fl2, k2) + . .+.(flr- kr. flr,* * kr) for some fli, kiwhere kistabilises the initial point of fli, i = 1, . .,.r. Let* * h1 = 1, hi= k1. .k.i-1(i > 2), ffii= hi.fl, gi= hikih-1i, i > 1. Then (fl, 1) = (ffi1- g1. ffi1+ ffi2g2. ffi2+ . .+.ffir- gr. ff* *ir, 1) and so fl is a sum of elements of the form ffi - g . ffi where g stabilises the* * initial point of ffi. This proves our first assertion. The proof of (i)is simple. We know already that p : ! ==G is a fibration* *. If G acts freely, then by the result just proved, p has discrete kernel. It follows that if x 2 Ob( ), then p* * : St x ! St ==Gpx is injective. Hence p is a covering morphism. Now suppose is connected and G is generated by those of its elements whic* *h fix some object of . To prove p a quotient morphism we have to show that for x, y 2 Ob ( ), the rest* *riction p0 : (x, y) ! ( ==G)(px, py) is surjective. Let (fl, g) be an element x ! y in o G, so that fl : g . x ! y in . Usin* *g the notation of 2.7, we have to find ffi 2 (x, y) such that pffi = (fl, g). As shown in 2.7, (fl, g) = (fl,* * 1). By assumption, g = gngn-1. .g.1 where gistabilises an object xi, say. Since is connected, there are elements ffi1 2 (x1, x), ffii2 (xi, (gi-1. .g.1) . x), i > 1. The situation is illustrated below for n = 2. o6x6 o g1.;x; o g2g1. for all h, k 2 * *H and relations [hk] = [h][k], = , [h] = [h] for all h, k 2 H, where we may identify * *[h] = (h, 1), = (1, k), [h] = (h, k). Factoring out by K imposes the additional relations [h* *] = 1, or equivalently [h] = , for all h 2 H. It follows that (H x H)=K is obtained from H by impos* *ing the additional relations hk = kh for all h, k 2 H. * * 2 Definition 2.11 (The symmetric square of aLspace)et G = Z2 be the cyclic group * *of order 2, with non- trivial element g. For a space X, let G act on the product space X x X by inter* *changing the factors, so that g . (x, y) = (y, x). The fixed point set of the action is the diagonal of X x X* *. The orbit space is called the symmetric square of X, and is written Q2X. Proposition 2.12Let X be a connected, Hausdorff, semilocally 1-connected space,* * and let x 2 X. Let denote the class in Q2X of (x, x). Then the fundamental group ß(Q2X, ) is is* *omorphic to ß(X, x)ab, the fundamental group of X at x made abelian. Proof Since G = Z2 is finite, the action is discontinuous. Because of the assum* *ptions on X, we can apply Proposition 1.8, and hence also the results of this section. We deduce that p* * *: ß(X) x ß(X) ! ß(Q2X) is a quotient morphism and that if x 2 X, z = p(x, x), then the kernel of the quotie* *nt morphism p0: ß(X, x) x ß(X, x) ! ß(Q2X, z) is the normal subgroup K generated by elements (a, b) - g . (a, b) = (a - b, b * *- a), a, b 2 ßX(y, x), for some y 2 X. Equivalently, K is the normal closure of the elements (c, -c), c 2 ß(X, * *x). The result follows. 2 Taylor [12] has given extensions of previous results which we give without * *proof. Proposition 2.13Let G be a group and let be a G-groupoid. Let A be a subset o* *f Ob( ) such that A is G-invariant and for each g 2 G, A meets each component of the subgroupoid of G * *left fixed by the action of G. Let be the full subgroupoid of on A. Then the orbit groupoid ==G is * *embedded in ==G as the full subgroupoid on the set of objects A=G and the orbit morphism ! ==G is t* *he restriction of the orbit morphism ! ==G. Orbit groupoids * * 16 Corollary 2.14If the action of G on X satisfies Conditions 1.7, and A is a G-s* *table subset of X meeting each path component of the fixed point set of each element of G, then ß(X=G)(A=G) i* *s canonically isomorphic to (ßXA)==G. The results of this section answer some cases of the following question: Question 2.15Suppose p : K ! H is a connected covering morphism, and x 2 Ob(K)* *. Then p maps K(x) isomorphically to a subgroup of the group H(px). What information in addition * *to the value of K(x) is needed to reconstruct the group H(px)? There is an exact sequence 0 ! K(x) ! H(px) ! H(px)=p[K(x)] ! 0 in which H(px) and K(x) are groups while H(px)=p[K(x)] is a pointed set with b* *ase point the coset p[K(x)]. Suppose now that p is a regular covering morphism. Then the group G of coverin* *g transformations of p is anti-isomorphic to H(px)=p[K(x)], by [3, Cor. 9.6.4]. Also G acts freely on K. Proposition 2.16If p : K ! H is a regular covering morphism, then p is an orbi* *t morphism with respect to the action on K of the group G of covering transformations of p. Hence if x* * 2 Ob (K), then H(px) is isomorphic to the object group (K o G)(x). Proof Let G be the group of covering transformations of p. Then G acts on K. L* *et q : K ! K==G be the orbit morphism. If g 2 G then pg = p, and so G may be considered as acting tri* *vially on H. By Definition 1.6(i), there is a unique morphism OE : K==G ! H of groupoids such that OEq = * *p. By Proposition 2.8(i), q is a covering morphism. Hence OE is a covering morphism [3, 9.2.3]. But OE is bij* *ective on objects, because p is regular. Hence OE is an isomorphism. Since G acts freely on K, the group N(x) of Proposition 2.7 is trivial. So* * the description of H(px) follows from Proposition 2.7(ii). * * 2 The interest of the above results extends beyond the case where G is finit* *e, since general discontinuous ac- tions occur in important applications in complex function theory, concerned wi* *th Fuchsian groups and Kleinian groups.We refer the reader to [2] and [8]. EXERCISES 1.Let f : G ! H be a groupoid morphism with kernel N. Prove that the following * *are equivalent: (a)f is a quotient morphism; (b)f is surjective and any two vertices of G having the same image in H li* *e in the same component of G. 2.Prove that a composite of quotient morphisms is a quotient morphism. 3.Let H be a subgroupoid of the groupoid G with inclusion morphism i : H ! G. L* *et f : G ! H be a morphism with kernel N. Prove that the following are equivalent: (a)f is a deformation retraction; (b)f is piecewise bijective and fi = 1H ; (c)f is a quotient morphism, N is simply connected, and fi = 1H . Orbit groupoids * * 17 4.Suppose the following diagram of groupoid morphisms is a pushout _________// f || |g| fflffl|__fflffl|// and f is a quotient morphism. Prove that g is a quotient morphism. 5.Let f, g : H ! G be two groupoid morphisms. Show how to construct the coequa* *liser c : G ! C of f, g as defined in Exercise 6.6.4 of [3]. Show how this gives a construction* * of the orbit groupoid. [Hint: First construct the coequaliser oe : Ob(G) ! Y of the functions Ob(f), Ob(g)* *, then construct the groupoid Uff(G), and finally construct C as a quotient of Uff(G).] 6.Suppose the groupoid G acts on the groupoid via w : ! Ob(G) as in Exerci* *se 4 of Section 1. Define the semidirect product groupoid o G to have object set Ob( ) and elements * *the pairs (fl, g) : x ! y where g 2 G(wx, wy) and fl 2 (g . x, y). The sum in o G is given by (ffi, h) + * *(fl, g) = (ffi + h . fl, hg). Prove that this does define a groupoid, and that the projection p : o G ! G, (fl, g) * *7! G, is a fibration of groupoids. Prove that the quotient groupoid ( o G)= Kerp is isomorphic to (ß0 G). 7.Let G and be as in Exercise 6, and let the groupoid H act on the groupoid * * via v : ! Ob(H). Let f : G ! H and ` : ! be morphisms of groupoids such that v` = Ob(f)w and * *`(g . fl) = (fg) . (`fl) whenever the left hand side is defined. Prove that a morphism of groupoids (* *`, f) is defined by (fl, g) 7! (`fl, fg). Investigate conditions on f and ` for (`, f) to have the followin* *g properties: (a)injective, (b)connected fibres, (c)quotient morphism, (d)discrete kernel, (e)covering morphism. In the case that (`, f) is a fibration, investigate the exact sequences of t* *he fibration. 8.Generalise the Corollary 2.6 from the case of the universal cover to the cas* *e of a regular covering space of X determined by a subgroup N of ß(X, x). 9.Let 1 ! N ! E ! G ! 1 be an exact sequence of groups. Prove that there is an* * action of G on a connected groupoid and an object x of such that the above exact sequence is isomor* *phic to the exact sequence of the fibration o G ! G at the object x. [4] 10.Let Xn be the n-fold product of X with itself, and let the symmetric group S* *n act on Xn by permuting the factors. The orbit space is called the n-fold symmetric product of X and is * *written QnX. Prove that for n > 2 the fundamental group of QnX at an image of a diagonal point (x, . .,.x) is * *isomorphic to the fundamental group of X at x made abelian. 11.Investigate the fundamental groups of quotients of Xn by the actions of vari* *ous proper subgroups of the symmetric group for various n and various subgroups. [Try out first the simp* *lest cases which have not already been done in order to build up your confidence. Try and decide whether or no* *t it is reasonable to expect a general formula.] Acknowledgements We would like to thank La Monte Yarroll for the major part of the rendition i* *nto Latex of this work. It is hoped to make further parts of his efforts available in due course. We would also l* *ike to thank C.D. Wensley for helpful comments. Orbit groupoids * * 18 References [1] M. A. Armstrong, On the Fundamental Group of an Orbit Space, Proc. Camb. P* *hil. Soc., 61, (1965), 639-646. [2] A. F. 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