Laguerre planes generated by Moebius planes

Laguerre planes generated by Moebius planes

HANSJOACHIM GROH

1. In t roduc t ion . A Moebius plane 1) is a t r iple P , K , _< o f a se t P (points) , a se t K (circles) a n d a re la t ion _< C__ P • K (incidence) such t h a t , i f "p<_ k" is exp re s sed b y "p a n d k are inc iden t" , t he fol lowing holds :

M1 A n y th r ee d i f ferent po in t s are inc ident wi th e x a c t l y one circle.

M2 G i v e n a circle k, a po in t Pl inc iden t wi th k, a n d a po in t P2 no t inc iden t w i t h k, t he re ex is t s exac t l y one circle inc ident w i th p l a n d P2 b u t w i th no o t h e r po in t o f It.

M3 T h e r e are 4 po in t s which are no t all incident wi th one circle, a n d each circle is inc iden t w i th th ree di f ferent points .

A Laguerre plane 2) is a t r ip le S, Z, _< of a se t S (points), a se t Z (cycles) a n d a r e l a t ion _ C__ S • Z (incidence) such t ha t , i f "s <_ z" is e x p r e s s e d b y "s and z a re i nc iden t " , t he fol lowing holds:

L1 A n y th r ee pai rwise jo inable a) po in t s are inc ident w i th e x a c t l y one cycle.

L2 G i v e n a cycle z, a po in t s~ inc ident wi th z, and a po in t s 2 no t inc ident wi th z b u t jo inable to s~, t he re exis ts exac t ly one cycle y inc iden t wi th s~ a n d s2 b u t w i th no o the r po in t o f z.

L3 Given a po in t s~ a n d a cycle z, the re exists e x a c t l y one po in t s 2 inc iden t w i t h z b u t no t ]o inable to s 1.

L4 T h e r e ex i s t s a cycle z inc iden t w i th a t least 3 points , a n d a po in t no t inc iden t wi th z.

T h e classical model of a Moebius p l ane is P = $2 (2-sphere), K = p l a n a r cuts of S~ w i t h m o r e t h a n one po in t , incidence -- c o n t a i n m e n t . T h e classical m o d e l o f a L a g u e r r e p l ane is S = S~ x R = s t a n d a r d cy l inder in euc l idean 3-space, Z = set o f cuts o f S~ x R wi th non -ve r t i c a l p lanes ,

= e. We sugges t to the r e a d e r to c o n s t a n t l y keep these e x a m p l e s in mind .

2) Reports on Moebius (= inversive) planes may be found in [B 60] and [De 68]. 2) For a report on Laguerre planes, see [BM 64]. 3) Two points are joinable iff they are different, and there exists a cycle incident

with both of them.

44 Hansjoachim Groh

For both Moebius and Laguerrc planes, we may identify a circle (cycle) with the set of points incident with it, incidence becoming the membership relation, and we will do so, if convenient. We will often denote such planes simply by the underlying point sets P (resp. S). We say two circles (cycles) touch 4) (resp. intersect properly) iff they have exactly one (resp. 2) points in common. For a Laguerre plane, it follows tha t injoinability is an equivalence relation, denoted by I , whose classes meet every cycle in exactly one point (in the classical model: generators of the cyclinder). We call the point {s~} = I(sl) c~ z in L3 the projection of Sl into z.

Inspired by the classical models, and by a far analogy to topological groups, we define:

By a topological Moebius plane we mean here 5) a Moebius plane whose point and circle set carry topologies such tha t the functions of joining, intersecting, and extended touching4), as induced by M1-M2, are continuous, the domain of proper intersecting is open in K 2, and touching is the limit case of proper intersecting.

By a topological Laguerre plane we mean here ~) a Laguerre plane whose point and cycle set carry topologies such tha t the functions of joining, intersecting, extended touching 4) and projecting, as induced by L1-L3, are continuous.

A fiat Moebius (resp. Laguerre) plane is a topological Moebius (resp. Laguerre) plane whose point space is a 2-manifold ( = surface, i.e. locally homeomorphic to R~).

By [W5 66, Satz 7.1 and Satz 8.1], the flat Moebius planes correspond precisely to the "topological Moebius planes" of EWALD [E 60], i.e. those Moebius planes whose point set can be identified with S 2 such tha t the circles become Jordan curves (i.e. ,~ $1). The flat Laguerre planes correspond precisely to those Laguerre planes whose point set can be identified with the cylinder S 1 x R such tha t the cycles are Jordan curves, and the injoinability classes are closed and homeomorphie to R, the real line ([G 69, 3.10]). In the following, these facts will be used constantly without reference.

4) In the literature, often in addition each circle (resp. cycle) is said to touch itself (in each of its points). By "extended touching" we mean the function derived from ]~[2 (resp. L2), extended correspondingly, but restricted to PlY=P2 (resp. sl ~: s~).

5) See [W6 66], [St 70]. In [H 69], HEISE has investigated several alternatives to this definition.

6) See [G 69]. Alternatives to this definition have been investigated recently in [I 70].

Laguerre planes generated by Moebius planes 45

B y the Jordan curve theorem in a flat Moebius plane P each circle k separates P into connected components. This makes meaningful the following

Definition: In a flat Moebius plane, an oriented circle (abbreviation: o-circle) is a pair (k, C), where k is a circle, and C is a connected component of P \ k . Two oriented circles (k i, Ci) touch iff kl and 1~2 touch each other, and C1 and C2 arc comparable (i.e. C1 C_ C 2 or C 2 _C C1). -

F la t Moebius planes abound: A convex surface P in Euclidean 3-space together with its planar cuts constitutes a (flat) Moebius plane iff it is differentiable ([E 60, Satz 2]). Two such Moebius planes are isomorphic only if the corresponding surfaces are projectively equivalent ([lV[ 67, 4.3.117)). In [B124, p. 227], REIDEMEISTER attr ibutes to STUDY the incorrect s ta tement tha t every flat Moebius plane is obtainable as above (i.e. "ovoidal"). In [B1 30, p. 229] (attr ibuted to HSELMSLEV) and [E 60], examples of non-ovoidal flat Moebius planes were constructed. I n [E 67], to show the existence of as many as possible HERII~TG types [He 65], EWALD constructed a number of large classes of Moebius planes, which all turn out to be flat ones in a natural topologization. Thereaf ter the classification of all flat Moebius planes whose automorphism group is at least 3-dimensional b y STl~AMBACH ([St 72a], [St 72b]) necessi tated the explicit construction of fur ther classes.

Comparat ively few Laguerre planes are known: Those generated by cutt ing an ovoidal cone with planes, as constructed b y B ~ z [B 64], and a generalization of M;4uR~R [M 72] by cutting with a certain system of broken planes. In [Me 70], a construction principle for finite Laguerre planes of even order is given.

The occasion of this paper was to improve this si tuation b y constructing further Laguerre planes. Actually, we divert the wealth of known Moebius planes to Laguerre planes by constructing a (flat) Laguerre plane out of every flat Moebius plane P and any of its points p ~ P. In a later paper [G 72b] we will intrinsically characterize those flat Laguerre planes thus obtainable. Precisely:

Theorem 1. I ] P is a f iat M o e b i u s plane, p e P , we def ine:

S = set o / a l l oriented circles (k, C) con ta in ing p (i.e. p e k), Z = (P \ {p}) w set o / a l l or iented circles not con ta in ing p, s <_ z i~ s = ( k, C) e S a n d either ( a ) z e P \ {p } , z e k, or (b ) z = ( k ' , C' ) e Z ,

and s and z touch.

T h e n S , Z , <_ is a f iat s) Laguer re plane. -

v) The result of ~u~E~ is not restricted to fiat Moebius planes. s) The topologies will be defined in section 6.

46 Hansjoaehim Groh

In view of this, tile original definition of the classical Laguerre plane (S = oriented lines = "spears", Z = oriented circles) may be interpreted as applying the construction of Theorem 1 to the classical miquelian (flat) Moebius plane with one assigned point p = oo. An isomorphism between the classical (Laguerre) model and the planar cuts of the cylinder embedded in 3-space, based on algebraic and order properties of the real field R, has been established by BLASCHKE ([B1 11]). A generalization of this isomorphism to pythagorean (ordered) fields is included in [M 66]. - Our result establishes the construction of a cylinder (as a topological space) by purely topological tools. We remark tha t no assumptions about configuration or algebraization properties (cfi [B 60 b], [B 61]) or existence of automorphisms (cf. [St. 70], [St. 72a], [St. 72b]) will be made.

The proof of Theorem 1 depends heavily on the fact that a big portion of the incidence properties (beyond M1-M3) of the classical Moebius plane is forced, by topology, to carry over to arbitrary flat Moebius planes. This portion concerns the existence and precise number of common touching circles. I ts validity is proved in [G 72 a].

The question appears reasonable whether the generating procedure of Theorem 1 yields flat cycle planes from flat circle planes (i.e. incidence structures satisfying all axioms of a Laguerre plane (resp. Moebius) plane except L2 (resp. M2)).

We conjecture that Theorem 1 can be extended to generate Lie planes in the sense of C~EN [C 72]: Let L be the set of points and oriented circles of any flat Moebius plane. Conjecture : L, with touching of o-circles as relation (naturally extended to points) is a Lie plane.

Notations: Besides the usual set theoretical and topological notat ions (see e.g. [Du 66]) we use: For a topological space X and a subspaee A of X, bd z A (or shortly bd A) stands for ~4 n X \ A , the boundary of A in X. A ~ X separates B 1 and B~ if B 1 and B~ are contained in different components of X \A.

We base the following notation on the Jordan curve theorem (JCT; see e.g. [Du 66, X V I I 5.4])9): I f C is ~ component of S~\] for ] z $1, the other one will be denoted by C-. Similarly, if b is a component of ] \ {P~, P2}, P~ 4 P2 e ], then the other component will be denoted by b-.

In convenient abuse of language, we will often at t r ibute to oriented circles (k, C) incidence properties of k. For example : "(lc, C) goes through p" means that k goes through p.

2. Oriented circles. From now on, with the exception of Section 5, suppose we are given a flat Moebius plane (P, K).

~) If S~ ,~jC $2, then S21j has precisely two components C1, C2, and bd Ci = j .


We first note the following straight-forward implications of the JCT on our basic definitions:

2.1 Let (]ci, Di) be two touching oriented circles. Then

a. (]Cl, D1) and (k2, D[) do not touch b. (kl, D~) and (k2, 1)~) touch

2.2 Corollary. Let (k 1, DI) be an oriented circle, k2 a circle touching k 1. Then there exists precisely one component D 2 o/ P \ k ~ such that (k 1, D1) and (k2, D2) touch.

2.3 Let (kl, D~) be three o-circles such that the k s touch mutually in p. Then at least two o / the (k i, D~) touch.

P r o o f : The Jo rdan Curve Theorem implies, simply by the fact tha t the k i ~ $1 have mutual ly just p in common, tha t there exist two of t hem (say kl, k2) such tha t ka\{p} ~ C , where C is the component of P \ ( k 1 u k2) with kl u k2 C_ bd C. Therefore by [G 72a, 2.4] ka separates k~\ {p} from k~\ {p}, and hence Da is comparable to either DI or Ds.

2.4 (Transi t iv i ty) Let (ki, Di) be three o-circles such that (kl, D1) touches (k~., Ds) in p, and (k~, D~) touches (k a, Ds) in p. Then (kl, D1) touches (k3, D3) in p, provided (k 1, D~) ~ (ks, Da).

P r o o f : kl = k3 would imply D3 = D~, contradicting 2.1a. Thus kl n k s = = {p} = ks n/ca, kl # k3, and M2 together imply tha t kl touches k3 in p. Because of 2.1b we m a y assume D~ C D2. In case D2 C_ D3 this implies D~ ~ D3, proving the s tatement . In case D3 ~ D~, [G 72a, Corollary 2.5] implies tha t D1 and Da are comparable.

2.5 Let (k~, Di) be two o-circles, k a common touching circle o / k l and k~. Then there exists a component D of P \ k such that (k, D) touches both (k i, Di) i / a n d only i /:

(1) in case I k 1 (~ k21 = 2: k.C D1 n 1) 2 or k C_ D~ n D-~,

(2a) in case k I n IQ = {p} C It: (kl, ~)1) touches (/c~, De),

(Zb) in case ]c 1 n]c a = (p} ~ k: k ~ 1) 1 n D2 or ]c C__ D-~ n D~,

(3a) in case k~ n k2= 0 and k does not separate k~\]c and k~\k: k C__

CC_ D1 n D~ or ]c C D 1 c~ D~,

(3b) in case kl c~ k~ = r and k separates: k C_ .D 1 (~ D~ or ]c C__ D 1 ~ D~. -

P r o o f : (1) If : By 2.1b we may assume k c__D~. Then by 2.2 there exists a component D of P \ k with D _C D 1. This implies D c~ k s = 0: This follows from D - D__ D~ and since by [G 72a, 2.1] a l ready ]c~ n D~4= 0.

Therefore D C_ D~ or D C_ D~. The latter would imply /c C D C D~ n D~,

contradicting tha t we are given k c__D~ c~ D~. Hence (k, D) touches

48 Hansjoachim Groh

(kl, D~) and (ks, Dz). Only i/: Since (k, D) touches (lcl, D1), we may assume, by 2.1b, t ha t D _C D1. Since (k, D) touches also (k~, D2), we have D C De or / ) ~ C D . The latter would contradict [G72a , 2.1]. Hence k_C/ )C

C D1 n D2. (2a) If : By 2.1b we may assume D1 c D2. Then D~ is the component

of P\kl disjoint to k2. I f k _C/)1, let D be the component of P\k contained

in D1. I f k C_ DT~, then the above proper ty of 1)1 implies the existence of a component D of P\k with D~C_D, and b y [G 71a, Corollary 2.5], D and De are comparable. Hence in both cases (/c, D) is a common touching o-circle. Only i/: By 2.1b we may assume D C_ D 1. Since (k, D) touches also (k S, D2), we have D C D2 or De C D. In the first case [G 72a, Corollary 2.5] implies tha t D1 and D e are comparable. In the second trivially De C__ D1.

(2b) I/: B y 2.1b we may assume k C__ D1. Thus there exists a component

D of P\ /c with D C_ D1. Hence k 1 C_D-, and since (/c I u ]cz)\k is connected

we also have /c~ ~ D-. Therefore ks n D = O, implying that D _C D~. or

D C_ D~. The lat ter would imply /c C / ) C D~ n D~, contradicting our assumption. Only i/: By 2.1b we ma y assume D c D~. Since (/c, D) touches also (k2, De) we have D C_ D~ or De C D. The latter would imply k 2C_2) and hence k 2 ( ~ k ~ C D n k ~ = k n k l , a contradiction to p ~ k .

Fig. 1

(3a) I/: B y 2.1b we may assume/c C_/)1. Thus there exists a component D of P \ k with D C_ D1. Hence k I n D= 0, and since/c does not separate k I and k2, we also have k S n D = 0. Therefore D C_ D2 or D.C D~. The

lat ter would imply k C_ D C_ D1 n DT, contradicting our assumption. Only i/: We may assume (2.1b) D C__ D1. Since (k, D) touches also (k2, D~), we have D C_ D2 or D2 C_ D. The lat ter would imply k2 C_ D. Since on

the other hand k 1 C_ D- , k would separate.


(3b) I /: We may assume (2.1b) Is C_ D1. Thus there exists a component D of P \/c with D ~ D 1. Hence ]~1 f~ D = 0 , and since k separates,/c~ _C D. Therefore /ce n D - = r implying D - C_ D 2 or D - C_ D~. The first would

imply Is _C D~ n D~, contradicting our assumption. Only i/: We may

assume (2.1b) D C_, D~. We then have D _C De or D2 C C_ D. The first would

imply D ~ D - and hence /c 2 C D - . Since DC_D 1 implies /c 1 C D - , k

would not separate.

kl k

D1

Fig. 2

2.6 Corollary. Let (]ci, Di) be two o-circles touching in p. Then any common touching o-circle goes through p.

P r o o f : Since (Is 1, D1) and (k~, D2) touch, D1 • D2 is a component of P \ k I or o f P \ k 2 , and is disjoint to both /c i. Hence no common touching circle k avoiding p can be contained in the closure of D 1 C~ D2 (or, analogously, of D~ n D~). Exac t ly this however would be required b y (2b) of 2.5 from such a circle k.

3. Characterization of injoinability and touching. From now on, with the exception of Section 5, suppose S, Z, _< are defined as in Theorem 1. " In jo inable" can and will be defined as in footnote 3. Though no incidence properties have ye t been proved, we already call the elements of S "L-points" (to avoid confusion with "Moebius" points) and the elements of Z "cycles".

3.1 Two di~erent L-points si= (ki, Di) are in]oinable i/ and only if (kl, D1) touches (k~, D2).

P r o o f : I / : Let (kl, D1) touch (k~, De). Since both are L-points, we have kl n ks = {p}. By 2.5, no common touching circle k of/Cl and k2 avoiding p can be oriented so as to touch both (kl, Di). Hence no cycle is incident ( >_ ) with both si. Only i/: Since s~, se are L-points, kl r~/c 2 _~ {p}.

4 Hbg. Math. Abh., Bd. X L

50 Hansj oachim Groh

Case 1: k I -- k 2. Then every point q e ]c1\ {p} is a cycle inc ident wi th bo th s i. Case 2:]~1 (-~ k2 = {P, q}, P ~ q" Pick r e D~ (~ D~ 4 0 (by [G 71 a, 2.2]). :By [G 71 a, Theorem 1] there exists a common touch ing circle ]c of k~, and k2 th rough r. Hence/c C_ D1 (~ D~. B y 2.5, k can be given an orienta- t ion D such tha t (k, D) touches bo th (k~, Di). B y M2 we have p ~ k. Hence (k, D) is a cycle incident wi th bo th s i. Case 3: k 1 (~ k~= {p}, Dt and D2 incomparable. This implies t h a t the b o u n d a r y of D~ (~ Dz or D~ n D~, whichever is non-empty , contains k 1 u ks. Picking r ou t of this intersection, we ob ta in f rom [G 72a, Theorem 1 (2b)] and 2.5 a cycle incident with bo th s~.

3,2 Corollary. In jo inabi l i ty is an equivalence relation on S .

P r o o f : The only th ing to show is t rans i t iv i ty , which b y 3.1 is reduced to 2.4.

3.3 T w o cycles z x, z2 have precisely one incident point in common (i.e. " touch") i / a n d only i f

(1) zl = q e P \ {p}, z~ = (]c, D) and q ~ k; or vice versa;

or (2) z~ = (lc~, Di), and (kl, D1) touches (k~, Ds).

P r o o f : I f : Le t s -- (l, E) be a L-po in t incident w i th zl a n d z2.

(1) : I n th is case 1 ~ p mus t touch k and go t h r o u g h q. Hence by M2 1 is the un ique ly de termined circle th rough p touch ing ]c ~ p in q. B y 2.2, the or ienta t ion E is un ique ly de te rmined b y D.

(2) : I n this case by 2.6 1 mus t go th rough {q} = kl (~ k s. B y M2 therefore 1 is the uniquely de te rmined circle th rough p touch ing k~ (and k2) in q. F inal ly , I can be given on (2.5) and only one (2.2) or ien ta t ion E as to touch bo th (ki, Di).

Only i / : Case a. One o] the two cycles is a point: Say z 1 = q e P \ (p}. Then z canno t be also a point : For zs = r e P \ (p } we would have by M 1 the existence of circle 1 th rough p conta ining q a n d r. B u t t h e n z~, z2 would be incident wi th two different L-points (l, E) and (1, E- ) . There- fore z s = (k, C). Assume q ~ It. Then p and q are e i ther in the same or in different components of P\/c. I n the first case, [G 72a, 4.1] and 2.2 would imply the existence of two different L-poin ts inc ident wi th bo th z~. I n the second case, for a n y L-poin t (1, E) inc ident wi th q the circle l ~ S~ would intersect k properly. Hence Zl and z 2 would have no common incident L-poin t a t all.

Case b. Both cycles are oriented circles: z i = (It i, JDi). Case b l : Ikl n k2]=2 . Then by [G 72a, Theorem l] there exist a t

least two circles l~, 12 t h rough p touching/c~ and ]c 2. I f p is no t in ei ther


D1 n 1)3 or D~ n D~, then b y 2.5 there can exist no common incident L-point at all. Otherwise, again by 2.5, l~ and l~ can be given orientations as to touch both (k~, D~). Hence we would have two common incident L-points of z~ and z~. Altogether, Case b l cannot occur.

li

Fig. 3

Case b2: kl r k2 = {q}. Then by [G 72a, Theorem 1 (2b)] there exist at least three circles through p touching kl and k~: One (10) containing q, and two (/1, l~.) not containing q. I f p is neither in D 1 n D , nor in D~- n D~, then l0 is by 2.5 the only candidate for receiving an orientat ion as to touch both (kl, Di). Again b y 2.5, it is a successful candidate iff (kl, D1) touches (k2,/).~). I f p is in D 1 (~ D2 or D~ ~ D~, then 11 and 12 can b y 2.5 be oriented to common incident L-points of z~ and z,. Altogether, the only escape in Case b2 is the s ta tement of our proposition.

Case b3: kl n k2= r Then by [G 72a, 5.4] there exist a t least 4 circles through p touching k 1 and k~: Two (ll, 12) not separating k I and k2, and two (13, 14) separating. I f p e D~ n Dz or D~ r~ D~, then b y 2.5 (3 a) l~ and l~ can be oriented to common incident L-points. Otherwise p e D ~ n / ) 2 or / )1 n D-~, and then b y 2.5 (3b) la and 14 can be so oriented. Altogether, Case b3 cannot occur.

4. Proof of the incidence axioms.

4.1 S, Z, < satisfies L1.

P r o o f : Let s~ = (k~, Di) be three pairwise joinable L-points.

4*

52 Hansjoachim Groh

I f n o t all It, are different, t h en since e v e r y circle can be o r i en ted in a t mos t two ways we mus t have (say) kl = k~ # ka. B y joinabi l i ty , D2 = D 1. The re fo re lca cannot t ouch kl, since o therwise b y 3.1 (It3, Da) would be in jo inable to ei ther s 1 or s2. Hence /c a c~ kl = {p, q}, and z = q is a cycle inc iden t w i th all th ree s,. No o the r cycle is inc iden t wi th all s,, since b y 2.5 a c o m m o n touching circle /c of k 1 an d k 3 canno t be o r i en ted as to t o u c h all (k,, D~).

I f all ]c~ are different, we dis t inguish accord ing to [G 72a, 5.3] the fol lowing cases for t hem:

(1) A n y two o/the ]c i intersect properly. ( la) r k~= {p, q}, q # p . Then z = q is a cycle inc ident wi th all s t. B y

[ G 7 2 a , 5.3 ( l a ) ] there exists no c o m m o n touch ing circle of the /c i. H e n c e q is t he only such cycle.

(lb) n k~ = {p}. B y 2.5 any c o m m o n touch ing circle k or ien tab le as to t o u c h all (ki, Di) mus t be con ta ined in the closure of D1 r~ D2 n Da or D ; r~ D~ r D~. B y [G 72a, 5.3] prec ise ly one of these ~~ is a c i rcular t r i angle a n d hence contains, b y [G 72 a, T h e o r e m 2] precisely one c o m m o n touch ing circle k in its closure. B y t he suff iciency p a r t of 2.5, I~ can be g iven an or ien ta t ion so as to t o u c h all (k~, De).

Fig. 4

(2) There exist two circles (say kl, k2) touching each other. I f ka would t o u c h k 1 (and hence ]c2, name ly in p), then , since all k~ are different , b y 2.3 a t leas t two of the (k~, Di) would touch . H e n c e b y 3.1 these would be in joinable , con t r a ry to our assumpt ion . H e n c e k3 intersects k 1 an d k~ proper ly . B y 2.5 any common touch in g circle k m u s t be con ta ined in the closure o f n D~ or n D T. B y [G 72a, 5.3] an d since D 1 an d D~ are b y

10) In the terminology of [G 72a, 5.3]: I f D ~ C i for all t h r e e i ' s : D lc~D 2c~D 3. I f D~----C i for 2 i ' s : D l nD2c~D3 . I f D i - - C i for one i :D~-~D~-c~D 3. I f D i = C i for no i: D~- 6~ D~- (~ D~-.


3.1 incomparable , precisely one of these is a circular triangle. Hence i t contains by [G 72a, Theorem 2] precisely one common touching circle k in its closure. B y the sufficiency par t of 2.5, lc can be given a unique (2.2) orientat ion so as to touch all (l% D~).

4.2 S, Z, <_ satisfies L2.

P r o o f : L e t z be a cycle, S l - z a L-point , and s 2 ~ z a L-poin t joinable to s : s / = (k~, Di).

Case 1: z = q e P \ {p}. Then sl <_z implies q e kl, and s2:~z implies q ~ k 2 and in pa r t i cu la r ]c 1 :t = k2.

Case la . l~ and Ic~ intersect properly. Since q ~ I%, q is e i ther in D~ or in D~. Correspondingly using [G 72a, 2.2] and [G 72a, 3.1], precisely one of D1 n D2 a n d D~ (a D [ contains a common touching circle 1 of k 1 and k~. th rough q in i ts closure, and there is only one such I. B y 2.5 and 2.2, I can be given a un ique or ientat ion E so as to t ouch bo th (1% Di). The other direction of 2.5 implies t h a t any common touching circle conta ined in the closure of D1 c~ D [ or D~ c~ D2 cannot be so oriented. Al together , b y 3.3 there exists precisely one cycle y incident wi th s~ and s 2 bu t wi th no other L-po in t of z, name ly y = (1, E).

P

~ D2

Fig. 5 Case lb . kl n k2= {p}. Since s~ and s2 are joinable, 3.1 implies t h a t

D 1 and D2 are incomparable . Hence precisely one of D 1 n D~ and D~ • D~ conta ins kl w kz in its boundary . B y [G 72 a, Theorem 1 (2 a)], this componen t conta ins a common touching circle 1 th rough q in its closure, a n d there is on ly one such 1. The rest of the p roof is now ( though a different pa r t of 2.5 is used) verbal ly the same as for Case 1 a.

Case 2: z = (k, D). Then s 1 < z implies k ~ kl = {q} for some q e P \ {p}. Case 2a. k~ = k~. Then b y 3.3 y = q is a cycle inc ident wi th Sl a n d s2

bu t no o ther L -po in t of z. Conversely, i f y' is a n y o ther candidate , i t mus t be an o-circle (l, E) touching bo th (ki, D~). However , since s I :~= 8 2 implies D~--D-~, 2.1a excludes the existence of such an o-circle.

Case 2b. k I n k2 = {P, q}. Then by 3.3 y = q is a cycle sat isfying L2. Conversely, i f y' is a n y other candidate, i t mus t b y 3.3 be an o-circle

54 Hansjoachim Groh

(l, E) touching (k, D) and both (k i, Di). Since (k 1, 1)1) and (k, D) are themselves touching (in q), 2.6 implies tha t l goes through q. But then by M2 it could not touch ]Q. Hence no other candidate y' exists.

Case 2c. I k l n k 2 1 = 2 , q~k2. Then q is either in D s or in D~. Correspondingly, using [G 72a, 2.2] and [G 72a, 3.1], one of D 1 n D2 and /)~ n D~- contains a common touching circle l of kl, k2 through q in its closure. By 2.5 and 2.2, 1 can be given an orientation E so as to touch both (ki, D~). Because of Ikl (~ ksl = 2 b y M2 we have p ~ l and therefore y = (1, E) is a cycle with s~, ss_< y. Fur thermore s s < y, ss:~z imply y 4 z. Hence by 2.4 the o-circle y touches also z. By 3.3 this implies tha t y satisfies L2.

Conversely, if y" is any candidate, by 3.3 and since k n (kl n ks)= ~, it cannot be a point. Hence y ' = (l', E ') is an o-circle touching (k, D) (by 3.3) and both (k~, D~). Since (kl, Dx) and (k, D) are themselves touching (in q), 2.6 implies tha t l goes through q. B y 2.5 l lies in the closure of D1 n Ds or D 1 n D~. [G 72a, 2.2] and the uniqueness part of [G 72a, 3.1] imply then l '= I. Finally 2.2 implies E ' = E .

Case 2d. kl n k s = {p}. Since Sl and s2 are joinable, 3.1 implies tha t D1 and D~ are incomparable. Hence precisely one of D 1 n Ds and /)~ n D~ is not empty and contains ]c~ w k s in its boundary. B y [G 72 a, Theorem (2a)] this one contains a common touching circle l of k~ and k s through q in its closure. B y 2.5 and 2.2 I can be given an orientation E so as to touch both (k~, D~). Because of q =~ p we have p e I. The rest of the existence proof for y can be verbally carried over from case 2 c.

Conversely, if y' is any candidate, it must be an oriented circle (1, E) and touch (k, D) (by 3.3) and both (k~, D~). The rest of the proof for y ' = y can be verbally carried over from case 2 c, except tha t a different par t of 2.5 is used, [G 72a, 2.2] is not used, and [G 72a, 3.1] is replaced b y [G 72a, Theorem 1 (2a)].

4.3 S, Z, <_ satisfies L3.

P r o o f : Le t z be a cycle, Sl =(k l , D~) a L-point. We may assume 81~Z.

Case 1: z = q e P \ {79}. - Then q ~ k~. B y M2 and 2.2 there exists precisely one oriented circle (ks, D2)= s2 through q touching s~ in 79. By 3.1 (both directions) this establishes L3 for this case.

Case 2: z = (k , / ) ) . - In subcase Ikx c~ kl = 2, using [G 72a, 2.2], by [G 72a, 3.1] precisely one o f / ) c~ D~ and D - n D~ contains a common touching circle k2 of k~ and k through 79 in its closure. Hence by 2.5 (both directions) and 2.2 there exists one and only one oriented circle s s = (k s, D2) through 79 touching both sx and z. B y 3.1 (both directions) this establishes L3 for this subcase. - To establish it for the subcases


[k~ n / r = 1 resp. 0, we have 2 choices: We m a y e i the r p r o c e e d similarly, rep lac ing [G 72a, 3.1] b y [G 72a, 3.2] resp. [G 72a, 3.3]. Or we m a y use t h a t in jo inab i l i ty is a l r eady k n o w n to be an equiva lence r e l a t i on (3.2): W e m a y a lways replace sl b y an injoinable L - p o i n t s~ = (k~, D~) inducing t he subcase [k~ r~ k[ = 2 a l r eady considered. Because o f 3.1 such an L - p o i n t can be cons t ruc ted .

]Q ]r

Fig. 6

5. Topology of "oriented lines" in R2-planes. In this sect ion we expose facts on the space o f o r ien ted lines (2 di f ferent ways) o f a f lat affine plane. These will be needed in Sect ion 6 to show t h a t the Lag u e r r e p lane ju s t cons t ruc t ed can be topologized to a flat one. Since these fac t s can be p r o v e d w i t h o u t a n y more effor t for a r b i t r a r y R2-planes, we will do so.

An incidence space 11) is a pai r o f a se t L 0 (points) a n d a se t L 1 (lines) o f subsets o f L 0 o f a t least 2 e lements , such t h a t a n y two d i f fe ren t po in t s Pl, P~ are c o n t a i n e d in precisely one line (denoted b y P l v P2)-

A topological plane 12) is an incidence space where L o a n d L 1 c a r r y topologies such t h a t the funct ions o f joining a n d in te r sec t ing are cont inuous , a n d t he doma in o f in tersec t ing is open in L~.

F o r a topologica l space X, a X-plane is a topologica l p lane w i th L 0 ~ X. I n par t i cu la r , a fiat 12) plane is a X-p lane where X is a 2-manifold .

I n a R2-plane, each line 1 is ~ R and closed in L 0 ([Sa 67, 2.4]). H e n c e the J o r d a n curve t h e o r e m implies t h a t Lo\l has precise ly tw o components . Th is gives rise to two defini t ions o f " o r i e n t e d l ines" : (a) An oriented line is a pa i r (1, C) of a line 1 an d one o f the c o m p o n e n t s C o f Lo\l. T h e set o f all those will be d e n o t e d b y L~. - (b) A functionally oriented line is a pa i r (l, F , ) of a line 1 an d an equiva lence class F l o f t h e set o f all h o m e o m o r p h i s m s / : R g l, w h e r e / 1 a n d / 2 are cal led e q u i v a l e n t (deno ted b y fl=]~) iff /1/~ 1 is an o r i en ta t ion ( = order ) p re se rv ing h o m e o m o r p h i s m of R. The set o f all those will be d e n o t e d b y L~ . - LI+ is g iven the fol lowing topo logy : L e t F = L~0 = set o f all f unc t ions

11) I t would not be justified to call (L 0, L1) a plane. We therefore adopt the name "incidence space" from [KP 70].

12) We adopt these names from [Sa 67].

56 Hansjoachim Groh

f rom R in to L 0 be given the compac t open topology . T h e n L + is a quo t i en t space of the subspace of all funct ions which are h o m e o m o r p h i s m s and whose image is a line. - L + is re la ted to L 1 as follows [Sa 67, 2.9 and 2.11]: The topology of L1 is the quo t i en t t o p o lo g y ob ta ined f rom L + b y ident i fy ing func t ion classes whose e lements h a v e the same lille as image. W e denote this na tu ra l map by n. F o r the l a t e r s t a t emen t s , we list t he following facts :

5.1 I n each R2-1)lane:

a n : L + --* L1 is cont inuous .

b ([Sa 67, p. 10]) 1, --* 1 in L 1 i~ for each ] : R ~ 1 there ex is t / , , : R x l, such t h a t / , --* / i n L f (i.e. u n i f o r m l y on compact subsets). - (1,, f ,) ---, (1, /) in L + i~ for each f' e ] there e x i s t / , E s w i t h / , ~ [' i n LRo .

e ([Sa 67, p. l l ] ) l, --* l i n L 1 i / the sequence 1, is set convergent to l (i.e. each p e 1 is l imi t of a sequence p , w i th p , e 1,, a n d no such sequence accumulates at a po in t p r l). -

An or ien ta t ion 0 of R 2 m a y be defined (cf. [ST 34, p. 100 - 101]) as an equiva lence class of coheren t ly or ien ted simplieial decomposi t ions . Here a coheren t ly or iented simplicial decompos i t ion is a simplicial decomposi- t i on where each s implex (here: t r iangle) has an o r i en t a t i on ([ST 34, p. 40]) such t h a t each two ad j acen t simpliees induce ([ST 34, p. 59]) oppos i te or ien ta t ions ([ST 34, p. 88~) (here also: l inear orders) on the i r in te rsec t ion (here: c o m m o n edge, ~ [ 0 , 1]). T w o co h e ren t l y o r ien ted simplicial decomposi t ions are equ iva l en t iff t h e y have a c o m m o n cohe ren t ly or ien ted ref inement .

E v e r y o r ien ta t ion (9 o f L 0 z R ~ induces a b i jee t ion (also d en o t ed b y (9) f rom L~ on to L + ( thus car ry ing over a t o p o l o g y to L ~ ) L e t (l, C) e L~ be given. P i ck a (geometric) t r iangle la) T mee t ing b o t h co m p o n en t s of L o \ l (We will abbrev ia te this b y " b a l a n c e d a r o u n d l") . P ick a simplicial decompos i t ion of L 0 conta in ing bd T ~ S~. This simplicial decompos i t ion can be coheren t ly o r ien ted (in a un ique way) so as to be long to (9. Thus bd T ~ S~ inheri ts an o r i en ta t ion (9r, which in t u r n induces a l inear order -< (9r on each p roper subset o f bd T. - Define b = C c~ bd T. B y assump-

t ion on T, bd T has precisely 2 poin ts in c o m m o n wi th l, an d b is a c o m p o n e n t o f bd T a f t e r r emova l of these two points . We now deno te these 2 poin ts in such a way t h a t on b we h av e q~ < ~r q~. P ick [ : R ~ 1 such t h a t / - ~ (ql) -< f-l(q2) on R. Define f inally (9 (l, C) = (l, ]).

13) I f Pl are three noncollinear points (of L0), the (geometric) triangle associated with them is defined as the intersection of the components of Lo\pi v p j containing Pl ({i,J, /}----- {1, 2, 3}). I t follows that b d T ~ S , , and that P i V P i n b d T is the closure of the relatively compact component of P l v p j \ {p l , pj} .

Laguerre planes generated by Moebius planes 5 7

We sketch a p roof t h a t this funct ion 0 : L~ -~ L + is well defined: We have to show t h a t ] does not depend on the three "p ick ings" in the above construct ion: (1) o f ] : R ~ l , (2) of the simplicial decomposi t ion containing b d T , (3) o f T (must be shown in this order). - (i) Suppose t h a t ] , / ' : R ~ l such t h a t ql/-1 <_ q2/-: and ql/'-1 <_ q2/'-1 on R. Then there exists an order preserving homeomorphism / : R ~ R' wi th qi / -1 /= q~/,-1 for bo th i. I n par t icular / -1/ / , has two different fixed points and hence canno t be order reserving. Since / is order preserving, along with / -1 / / , also /-1/ , mus t be order preserving. Hence [ ' = ]. - (2) follows f rom the equivalence class definition of the or ien ta t ion 0. - (3) Suppose T, T' are two tr iangles balanced (see above) a round I. We have to show t h a t for a n y homeomorphism ] r e s p . / ' : R m I 0-associated to T resp. T' we have ] ' -- [. Since for any two tr iangles ba lanced a round 1 there exists a th i rd one dis joint to both of them, we need on ly consider the case T n T' = 0. Fo r th is i t suffices to settle the case t h a t bd T and bd T' are ad jacen t a t an edge c with endpoints x ~ C and x - G C- (see Fig. 7). Hence C meets l in a point q. Denote l(~ b d T = {q,r}, l ~ b d T ' = {q, r '}. Since b d T and b d T ' arc adjacent , q separates r f rom r' on lm R. - Fo r the same reason, there exists a simplicial decomposit ion of L 0 conta in ing bd T and bd T'. This, or iented coherent ly so as to belong to 0, induces orientat ions 0 r resp. Or on b d T resp. bd T' such t h a t the induced l inear orders _< O r resp. _< Or are opposite to each other on C. Hence wi th the nota t ions of the cons t ruc t ion of 0 : L ~ - ~ L ~ +, using {x,q} C b, { x , q } C b ' , we have (say) (r, q ) = (q:, q~) and (q, r') = (q~, qs Therefore i f ] : l ~ R satisfies /-X(ql) <_/-i(q~), then (since q separates) also /--l(ql) ~_~/-i(q~). Thus ] = ['. -

x

l

X-- Fig. 7

F o r the r ema inde r o f Section 5, suppose we are given a Re-plane (Lo, 51)'

5.2 Let 1 be a line, Pl and p~ points separated by 1. Then I, --, l implies that Pi and P2 are also separated by l~ finally.

58 Hansjoachim Groh

P r o o f : The assumptions imply t h a t m = p l v p ~ R meets 1 in a point q separat ing Pl and P2 on it. Hence b y the openness of the domain of in tersect ing q. = l. ^ m is a point f inally. The con t inu i ty of in tersect ing then implies q~ - . q. Hence qn f inal ly separa tes p~ and p~ on m. B y [Sk 54], [Sk 57] (restated in [Sa 67, 2.8]), the two components of m \ { q ~ } are separa ted by l~, implying our s t a t emen t .

5.3 ;Let l, m be lines intersecting in q. Le t 1, --* 1 be a sequence. Then f inally q, = 1. n m is a point. Let / : R ~ l, / , : R ,~ I. be such that / . --* f pointwise . Then /~1 (q.) ~ / - l (q ) .

P r o o f : I t suffices to show t h a t the re exists a compact set conta in ing f inal ly all/-~l(q,). Pick Pl, P2 e l \ {q} such t h a t t h e y are separa ted b y q on 1. Hence by [Sk 54], [Sk 57] (see p roo f of 5.2), Pl and P2 lie in different components of L o \ m . Denote r i = ]-1 (Pi)- Since In (ri) = Pi. -~ P~, the same is f inal ly t rue for Pin and P2.. Hence q . - - (Pl. v p~,) ^ m finally separa tes Pin and P~n on l,. Therefore/~1 (q,) is f inal ly be tween r 1 and r e a n d hence in the compac t set [r~, r~].

5.4 T h e / o l l o w i n g proper t ies /or (l,, C,) , (l, C) e L~ are equivalent:

(1) (1., c . ) ~ (l, c), (2) l . ---) l, a n d / o r all x ~ C, f inally x ~ C. ,

(3) l , --* l, and ]or some x e C, f inal ly x e C..

P r o o f : (1) ~ (2). First , by 5.1a we have l, - , I. - Secondly, pick a t r iangle T balanced a round 1 hav ing x as a ver tex. Define b = C r~ bd T. Deno te the two points of l r~ bd T in compliance wi th the defini t ion of (P(I, C) by q~, qe; i.e. q l < 0r q2 on b. P ick / : R ~ l such t h a t / - l (q~)< < ]-1(q2). Then r C ) = (l, ]).

B y 5.1b there exists a sequence / , : R ~ 1, wi th / , --. / un i fo rmly on compac t subsets, such t h a t r C , ) = (l,, In). B y 5.2, l~ r~ b d T f inal ly consists o f two points, denoted b y ql, , q2n such t h a t q~, < 0 r q~. on b. = C. r~ bd T. On the other hand , b y the con t inu i ty of intersect ing, the qi, can be " rea r ranged" to sequences q~, -* qi, i.e. {q~, q~} = {ql., q~}.

B y 5.3 ~ ( q ~ . ) - , / - X ( q i ) . Hence f inal ly /-~X(q~.)</~l(q~.). On the o ther

hand , ~ (l., C,) = (1. , / ' ) implies /ix (qi.) "~ /~-I (q2n) for all n. Hence q[. = ql, finally.

Now b is the component bd T \ (q~, q2 } wi th q~ < 0 r q~ on b. and likewise b. the componen t of bd T \ (q~., q~,} wi th q~. < r q2. on b,. Hence x e b and qi. ~ qi together imply x e b, f inally. Therefore x e C. finally.

(3) ~ (1). Pick a tr iangle T ba lanced a r o u n d 1 having x as a ver tex . Define b, ql, q~, and pick / as in the first pa r ag raph of (1) ~ (2).


Since l a -~ l, b y 5.1b there exists a sequence / , : R ~ l a with / a - * / uni formly on compac t subsets. This implies (la, ],) -* (l, f). Therefore i t remains to show (la, fa )= ~P (la, C.).

b,,= Ca n b d T is b y 5.2 a sequence of finally proper in tervals on bd T ~ $1. Deno te the endpoints of ba b y qxa, q2. such t h a t q~. _< Or q2a on ba. Then b y the con t inu i ty of intersect ing the sequence {qla, q2a} of endpoints o f the b a converges to {q~, q~}. Now, since b y assumpt ion also finally all ba con ta in x, t h e y are conta ined in some proper in te rva l I C bd T. Hence ql. -< Or q~. on I implies q~a -~ qi separa te ly for i = 1, 2. Therefore b y 5.3 ~ (q la )</~a(q~ , ) , yielding ~(1, Ca)= (1, ],), as desired.

O O 5.5 Corollary. The /unct ion -: L~ - , L~, defined by (1, C ) - = (l, C-) , is a fixed po in t / ree homeomorphism.

P r o o f : L e t (1,, Ca) ~ (1, C). We have to show (1 a, C[) -~ (1, C-). P ick q e C-. B y 5.1c, q cannot lie in infini tely m a n y 1,. I f q e C a for inf ini tely m a n y n, t h e n b y 5.4 (l,, C,) would have a subsequence convergent to (1, C-), con t rad ic t ing the fac t t h a t L 1 is Hausdor f f (by [Sa 67, p. 11], the topo logy of L 1 is induced by a Hausdor f f metric). Hence q e C~ finally, which implies via 5.4 our s ta tement . -

Definition: I n an incidence space (L o, L1), a subset 0 ~ L 0 is called an oval i f (1) I1 • e l _< 2 for each 1 e L 1, a n d (2) 0 has in each of i ts poin ts x precisely one " t a n g e n t " t (x), i.e. O n t (x)-- (x}.

5.6 Let 0 C Lo be an oval, 0 ~ S 1. Let t : 0 ~ 51 be the/unct ion assigning to each x ~ 0 the " tangent" to 0 in x. Denote by D the relatively compact

0 component o / L o \ O . Let t o (resp. t o-) : 0 ~ L 1 be the /unc t ion assigning to each x ~ 0 the oriented line (t(x), Dx) (resp. (t(x), D-~)), where Dx denotes the component el L o \ t ( x ) wi th D C_ D x. - Then t(x) r~ D = 0 /or all x ~ 0 , and t, t o and t o- are in]ective and continuous.

P r o o f : Since D is re la t ively compact , whereas each of the componen t s of t ( x ) \ (x} is not , none of t hem can be conta ined in D.

The inject iveness o f t, t ~ and t ~ is an immedia te consequence o f ax iom (2) for an oval.

Fo r the con t inu i t y o f t, we first no te t h a t by [Sa 67, 2.13] the set o f lines mee t ing the compac t set 0 ~ S 1 is compact . Hence i t suffices to show t h a t for x,, x e O wi th x a -~ x and t (xa) - . I we mus t have l -- t (x). - B y 5.1c, x e l . Hence l r would imply l r ~ O = ( x , q } wi th x :~q . De no t e b y b the re la t ive ly compac t componen t of l \ (x, q}. Then b C D: suppose b C D - . T h e n by the O-curve Theorem [Wh 58, I I I 1.4 a n d 1.5] D - \ b has two components E~ wi th b d E ~ = b u 0 i, 0 i a componen t o f O \ { x , q } , 01:~0~. One of the Ei, say El , is relat ively compact . P ick

60 Hansjoachim Groh

r cO1. T h e n each of the componen t s ( ~ R) o f t ( r ) \ {r} C D - meets E 1

and, since E~ is re la t ive ly compact , also E2. H en ce I t ( r ) c ~ b I>_ 2, a

contradic t ion . On the o ther hand, b C D is equal ly impossible: B y 5 . 1 c each p o i n t

of i t would have to be accumula t ion po in t o f a sequence r , e t ( x , ) .

However , b y our first s t a t emen t , t (x,) n D = 0. - Al toge ther , the only escape is l = t (x).

The con t inu i ty of t ~ now follows i m m e d i a t e l y f ro m t h a t of t, and 5 .4:

F o r x,, x c O with x , ~ x we have t ( x . ) --* t ( x ) . Pick ing q e D, we have b y defini t ion q e D~, q e D~,. Hence by 5 .4 (3) =~ (1) we have tO(x.) ---,

t o (x) . - The con t inu i ty of t ~ follows f rom t h a t o f t ~ the observa t ion t o - = (to) -, and 5.5. -

Definition: In an incidence space (Lo, L~), a l i n e p a r t i t i o n is a subset B ~ L l s u c h t h a t tJ l = L 0 , a n d l l r ~ 1 2 = o f o r l l ~ e l 2 , 1 i c B . -

l e B

o Definition: An o r i e n t e d l i ne p a r t i t i o n of aR2-plane is a subset B o ~ L 1

such t h a t B = {ll(1, C) e B ~ is a line par t i t ion , an d for (l~, C~) e B ~ we have t h a t C~ and C2 are comparab le (i.e. C~ ~ C2 or C~ C C1). -

Define u~ = ulso : B ~ --, B . - ST~A~BACH p r o v e d B ~ R and B closed in L~ for a n y line pa r t i t i on B of a R~-plane [St 67, Hi l fssatz 3.2]. We will need :

5.7 L e t B ~ be a n y o r i e n t e d l i n e p a r t i t i o n o / a R 2 - p l a n e . T h e n : a. 7cso i s a o

h o m e o m o r p h i s m / r o m B ~ on to B . b . B ~ i s ~ R a n d c losed i n L 1. -

P r o o f : a. Since uso is by defini t ion b i jec t ive a n d b y 5 . 1 a cont inuous , i t remains to show t h a t u~} is cont inuous . L e t l,, 1 e B wi th I, -~ I. Deno te u~( l , , )= (1,, C,), u~]( l )= (1, C). P ick x e C. B y 5 .4 ( 3 ) = ~ (1) it suffices to show x e C, finally. According to the def in i t ion of an or ien ted line pa r t i t i on C ~ C, or C , ~ C for each n. In the first case x ~ C n. Hence we m a y assmne C , ~ C for all n.

P ick q e 1 and deno te m = x v q. B y [Sk 54], [Sk 57] (see p ro o f of 5.2)

b = m (~ C is the c o m p o n e n t of m \ {q} con ta in ing x. B y the openness o f the dom a in of in tersect ing, m A 1, = q , is a po in t finally. C~ g C implies q, e b. The con t inu i ty o f in tersec t ing implies q, -~ q. Hence finally q,, separa tes x f rom q on b. B y [Sk 54], [Sk 57] (see p r o o f of 5.2) this implies t h a t x and q are in di f ferent componen t s o f L o \ l . f inally. Since 1 n Cn = 0, we have x e C , finally.

b. [St 67, Hilfssatz 3.2] and a. imp ly B o ~ R . - L e t ( l , , C , ) ~ B o be a o

sequence convergen t to (l, D ) e L 1. T h e n b y 5 . 1 a , I , - ~ 1. B y [St 67, Hilfssatz 3.2], 1 e B. Hence a. implies ( l , , C , ) -~ ~-~1o (1) = (l, C) . Since L 1

is Ha usdo r f f ([Sa 67, p. 11], see p r o o f of 5.5) , (l, D ) = (1, C) ~ B o. -

Laguerre planes generated by lVIoebius planes 61

6. Topologizing of the constructed Lagucrre plane. - We now return to the structure (S, Z, _< ) before Section 5.

6.1 = Theorem 1.

P r o o f : By 4.1, 4 .2 and 4.3, (S, Z , <_) is a Laguerre plane. The axioms of a Moebius plane imply tha t the pair L o = P \ (p},

L I = (/c\{p}lp c/c e K} is an incidence space (in fact, an affine plane). The axioms of a topological Moebius plane imply tha t L 0, L 1 becomes a topological plane if we endow L 0 with the subspace topology and identify L 1 by k\ (p } ~ k with the subspace of K of all circles through p. Since p ~ $2, we have L 0 ~ R e, i.e. (L0, L1) is a Re-plane.

The set S of L-points may be identified by (k, C) - , (k\ {p}, C) with O the set L~ of oriented lines of the above R~-plane. By Section 5, L1 again

may be identified, via a fixed orientation r of L ~ with LI+, the space of functionally oriented lines. We endow S with the topology carried over from L~ + via these identifications.

By [Sa 67, 2.14] L +, and hence S , is homeomorphic to the cylinder S 1 • R. - We first show t h a t the set z ~ of L-points incident with a cycle z is, as subspaee of S, homeomorphic to $1: (a) I f z = q ~ P \ (p}, the set .q+ of functionally oriented lines through q is ~$1 ([Sa 67, 2.14(b)]). (b) I f z = (m, E) is an o-circle, the axioms for a Moebius plane imply tha t m is an oval in (L 0, L1). I n case E is relatively compact, z ~ is identified with the set to(m) consisting of oriented tangents to m. This set is as continuous injective image (5.6) of m homeomorphic to S~. In ease E is not relatively compact, we have identification with t ~ (m) which again by 5.6 is ~ $1.

Secondly, we observe t ha t each injoinability class J is ~ R and closed in S : J is identified with the set B o = {(lc\ {p} , C)](/~, C ) e J}. This set is by 3.1 an oriented line part i t ion of (L 0, L1), and hence by 5.7 home-

O omorphic to R and closed in L~. By the preceding two facts, the assumptions of [G 69, 3.10] are

satisfied. Hence there exists a (unique) topology on Z such tha t (S, Z, < ) is a topological (hence flat) Laguerre plane. -

For [G 72b] we will need 6.3. For this we need

6.2 F o r each topological Laguer re p lane , convergence i n Z i s equ iva len t

to set convergence.

P r o o f : Let z , --* z in Z. Then (1) I f s _< z, the continui ty of projecting implies s n = I ( s ) • Zn ~ S. (2) I f t e S is limit of a (generalized) sequence (= net), the cont inui ty of projecting implies s , = I (s ,) (~ z , ~ I (s) r~ z.

Hence, since S is Hausdorff ([G 68, 1.7]), t e Z.

62 Hansjoaehim Groh

L e t z, be set convergen t to z. P i ck 3 d i f ferent po in ts s~ < z. T h e n each s i is l imi t of a sequence s~,<_z,. B y [G 6 8 , 2.1] S~n, S2n, S3~ are f inal ly jo inable ( to z,,). The con t inu i ty o f jo ining t h e n implies z, -* z in Z. -

6.3 The natural injection Z : P \ ( p ) -* Z is continuous.

P r o o f : Le t q,, q e P \ (p ) wi th q, --* q in the t opo logy o f P \ (p}. To show t h a t i (q , )= q, -* q = i (q) in Z14), b y 6.2 i t suffices to show:

(1) L e t (k, C ) e S be inc ident w i th q, i.e. q e k. Th en t h e r e exis ts a sequence (It n, C,) e S convergen t t o (k, C) wi th q, e It,. - P r o o f : Def ine sn = (k~, C,) to be the p ro jec t ion (L3) o f s = (]c, C) on to the cycle q,, B y 3.1, B o = ( ( l \ ( p } , D)t(1, D ) e I ( s ) } is an o r i en ted line p a r t i t i o n o f (Lo, Lx). B y [St 67, Hiffssatz 3.2 (Proof ) ] t h e n a t u r a l map, assigning to each po in t x e L0 the line l~ e B = uB (Bo) conta in ing it, is cont inuous . H e n c e q. -* q in L 0 implies k , \ ( p ) -* k \ (p} in L 1. T h e n b y 5.7a also ( k , \ (p} , C ~ ) = ~ ( k ~ \ (p}) --. ~ ( k \ ( p } ) = ( k \ (p ) , C) in L~. B y the iden t i f ica t ion of L o and S, (/c~, C,) -* (k, C).

(2) L e t (kn, C,) e S be a sequence c o n v e r g e n t (in S) to (k, C) e S such t h a t q, e k,. Then q e k. - P r o o f : B y o u r ident i f ica t ion (see p r o o f o f 6.1) t h e sequence (k, \ (p ) , C,) of o r i en ted lines converges in L~ to (k\ ( p ) , C).

O H e n c e b y the ident i f icat ion of L~ w i th L + an d b y 5.1a, t he sequence k , \ ( p ) o f lines converges to k \ {p). B y 5.1c th is implies q e k. -

[B 60a] [B 60b]

[B 61]

[B 64]

[BM 64]

[B1 11]

[B1 24]

[B1 30]

References

W. BENZ, (fber Moebius-Ebenen. J. Ber. DMV 1960 (68, 1--20). W. BENZ, (fiber Winkel- u. Transitivit/~tseigenschaften in Kreis-Ebenen I. J. Reine Angew. Math. 1960 (205, 48--74). W. BE•Z, (fber Winkel- u. Transitivit~tseigenschaften in Kreis-Ebenen II. J. Reine Angew. Math. 1961 (207, 1--15). W. BENZ, Pseudo-Ovale und Laguerre-Ebenen, Abh. Math. Sem. Ham- burg 1964 (27, 80--84). W. BE~z and H. M~URER, Grundlagen der Laguerre-Geometrie. J. Ber. DMV 1964 (67, 14 42). W. BZASCHKE, (fiber die Laguerresche Geometrie orientierter Geraden in der Ebene I. Arch. Math. Phys. 1911 (18, 132--140). W. BLASCHKE, Vorlestmgen fiber Oifferentialgeometrie I. 2. Auflage, Springer Verlag, Berlin 1924. W. BI~scnv~,, Vorlesungen fiber Differentialgeometrie I. 3. Auflage, Springer Verlag, Berlin 1930.

~4) Note that, besides its existence, we have no information about the topology of Z.

[c 72]

[De 68] [I)u 66] [E 60]

[E 67]

[G 68]

[G 69]

[G 72a] [G 72b]

[H 69]

[He 65]

[i 70]

[KP 70]

[M 66]

[l~I 67]

[M 72]

[Me 70]

[Sa 67] [Sk 54]

[Sk 57]

[St 67]

[St 7o] [St 72a]

[St 72b]

[ST 34]

[Wh 58] [W5 66]


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Eingegangen am 7. 3. 1972

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Laguerre planes generated by Moebius planes