Smooth Stable and Projective Planes - MyWWW ZHAW · to 8-dimensional compact projective planes....

Smooth Stable and Projective Planes

HABILITATIONSSCHRIFT

Der Mathematischen Fakultat

der Eberhard-Karls-Universitat Tubingen

vorgelegt von

Richard Bodi

im Wintersemester 1995/96

Contents

Introduction 1

Part One

Local Theory 9

1. Definitions and Basic Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112. Tangent Translation Planes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183. Regular Stable Planes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .264. Collineations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .295. Central Collineations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

Basic Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32The Structure of Elation Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37Reflections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .44

6. The Structure of Collineations Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47Basic Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47The Stabilizer of Three Concurrent Lines . . . . . . . . . . . . . . . . . . . . . . . . . . .51

Part Two

Global Theory 59

7. Solvable Collineation Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .618. Subplanes of Smooth Projective Planes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .699. Flat Projective Planes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

Skew Hyperbolic Planes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .73Skew Parabola Planes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .77Planes over Cartesian Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

10. 4-dimensional Projective Planes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .8611. 8-dimensional Projective Planes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .9412. 16-dimensional Projective Planes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .98

Appendix on Affine Translation Planes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .108

Glossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .112

Introduction 1

Introduction

The Origins of Smooth Incidence Geometry. The urge of man to understand theworld is as old as mankind itself. Trying to find simple principles that determine thegeometry of the space in which we live is one of the consequences of this impulse. Thefirst characterizations of plane geometry date back more than 2300 years, the most famoustext book on that subject being that of Euclid. The axioms of Euclid essentially remainedunchanged until M. Pasch published his lectures on modern geometry (”Vorlesungen uberneuere Geometrie”, Leipzig) in 1882. The axioms he used for describing euclidean geometryintroduced the concept of an order. In this way he was able to incorporate continuityproperties of the euclidean plane. The axioms of Pasch reappeared in David Hilbert’sbook ”Grundlagen der Geometrie”, published in 1899, and they became well-known tomany mathematicians. In order to extend the axioms of Pasch and Hilbert to othergeometries such as the affine plane over the complex numbers, the concept of an orderhas to be replaced by a more general idea, namely by that of topology. In the early30’s Kolmogoroff was the first who combined projective spaces with topological structures.In the mid-fifties, Skornyakov (1954) and Freudenthal (1957) published two papers ontopological planes, and it was Helmut Salzmann who started the systematic investigationof topological plane geometries in 1955. The programme that emanated from his studiesis the classification of homogeneous compact projective planes, where homogeneity is mostoften measured by the size of its collineation group. The progress that Salzmann and hisstudents achieved within the last four decades is documented in the recent book CompactProjective Planes [114] that appeared in 1995. Although containing almost 700 pages, onlypart of the theory developed by now has found its way into this book. This shows theamount of knowledge that has been obtained in this mathematical field, not to mentionvarious other types of topological incidence geometries (see the articles of Steinke andGrundhofer–Lowen in [28]).

Remembering the initial task of Euclid and Hilbert to characterize the euclidean planeE, it is natural to consider not only the topology of E but also its differentiable structure.This leads from topological geometry to smooth geometry, i.e. incidence geometry onsmooth manifolds such that the geometric operations are smooth. It suggests itself toapply Salzmann’s programme to smooth projective planes. In fact, it was a question ofHellmuth Kneser who asked Salzmann whether or not the Moulton planes (cp. Chapter

2 Introduction

8) can be viewed as smooth projective planes. We will return to this question later.Considering smooth geometries, we have to specify the degree of differentiability to berequired. If we regard smooth manifolds as being complex differentiable, then it turns outthat the only smooth projective plane is the projective plane over the complex numbers.This was proved by S. Breitsprecher [20] and Linus Kramer (oral communication), see also[114]75.1. Thus we have to use weaker notions of smoothness. The case of real analyticitywas also tackled by Breitsprecher. For two-dimensional projective planes, he proved in [20]that such a plane P is isomorphic to the real projective plane (which, of course, is analytic)if P can be embedded in the complex projective plane P2C in the same way as the realprojective plane is embedded in P2C. Up to the present day it is unknown whether thisadditional assumption is necessary or not. In fact, it seems that the real analytic case istoo hard to handle by now. Hence we will use the words smooth and C∞-differentiablesynonymously.

The Development of Smooth Projective Geometry. In sharp contrast to the case of(compact) topological planes, only a few papers on smooth geometries have been publishedup to now. We have mentioned already the pioneer of smooth geometry, S. Breitsprecher,who wrote three papers in the late 60’s, [20], [21], [22]. In 1972, Dieter Betten published apaper [3] on differentiable two-dimensional projective planes, in which he answered Kneser’squestion by showing that the proper Moulton planes cannot be turned into smooth pro-jective planes. We give an alternative proof of this statement in Chapter 9. After a periodof 14 years where nothing was published about smooth geometry, it was Hermann Hahlwho revived this field with two papers, [50] and [45], (one of them is a joint paper withTheo Grundhofer) about spherical fibrations which lead to smooth translation planes de-fined over division algebras. The main result of these two papers is that every smoothtranslation plane defined over some division algebra is isomorphic to one of the classicalprojective planes P, i.e. P is the projective plane defined over the real, complex, quater-nion, or octonion numbers, which we abbreviate by R, C, H, and O, respectively. JoachimOtte (then Hahl’s assistant) generalized this result in 1992 to arbitrary translation planes,see [93] and [94]. Because Otte’s result will play an important role in this manuscript, weformulate it as follows.

Otte’s Theorem. Every smooth projective translation plane is isomorphic (as a smooth

projective plane) to one of the classical projective planes PF defined over an alternative

field F ∈ R,C,H,O.

It is important to note that an analoguous result is false for smooth affine transla-tion planes: Otte constructed non-classical examples of smooth affine translation planes.

Introduction 3

Moreover, he showed that there is also a large variety of non-classical examples of smoothprojective planes.

The most recent results on smooth projective planes are due to Linus Kramer [75].He showed that the point space (and the line space) of a smooth projective plane ishomeomorphic to the point space (and the line space) of the classical projective planeof the same dimension.

The manuscript in hand is divided into two parts. Since smoothness is a local property,it is convenient to adapt the underlying incidence geometry to that local situation. Thisleads to the concept of a stable plane which is already a well-known object of topologicalgeometry. Stable planes originate from Salzmann’s paper Topological planes [100] andhave been introduced by Rainer Lowen, see for example [79] — [83]. The first part ofthe manuscript is devoted to the investigation of smooth stable planes and it deals mostlywith the local properties of smooth planes. The second part treats the global theory,i.e. smooth projective planes. Its aim is the application of the Salzmann programme tosmooth projective planes. First we have to outline the main ideas of this programme in thetopological situation. In order to keep things as simple as possible, we restrict our attentionto 8-dimensional compact projective planes. Smooth projective planes of dimension 8 arediscussed in Chapter 11.

The Salzmann Programme. The projective plane P2H defined over the quaternionskew field H is one example of an eight-dimensional compact projective plane, and we referto this plane as the classical 8-dimensional plane. The collineation group of P2H is iso-morphic to the 35-dimensional Lie group PSL3H. Let P be an arbitrary 8-dimensionalcompact projective plane and denote by ∆ the connected component of its group ofcontinuous collineations. Note that every (abstract) collineation of P2H is in fact con-tinuous (even smooth), but it is unknown whether this is true for an arbitrary planeP. What can be proved is that ∆ becomes a locally compact topological group withrespect to the compact-open topology (which coincides with the topology of uniformconvergence). It is well-known that locally compact groups admit a convenient notionof dimension; all important dimension functions of topological spaces coincide for lo-cally compact groups. Maybe the most intuitive description of dimension is given bydim ∆ = sup n ∈ N | [0, 1]n can be continuously embedded into ∆. In case ∆ is a Liegroup, dim ∆ equals the dimension of ∆ as a manifold. Obviously, the plane P2H is veryhomogeneous, because, among other things, it can be defined in the usual way by a three-dimensional vector space over the quaternions; hence its collineation group is transitiveon the set of quadrangles. The idea is now to use the value dim ∆ as a measure for thehomogeneity of the plane P. It turns out that dim ∆ ≤ dim PSL3H = 35 is true in general.This expresses our intuition: that P2H is the most homogeneous compact projective plane

4 Introduction

of dimension 8. Even better, it can be shown that P is isomorphic to P2H if dim ∆ properlyexceeds 18. This bound is a sharp one. Moreover, every plane P with dim ∆ = 18 is atranslation plane as well as a dual translation plane, and Hahl could even classify theseplanes, see [53] and [55]. If dim ∆ = 17, then we have a similar situation: P or its dualplane is either a translation plane (and again P can be classified) or P is a so-called Hughesplane which can also be described explicitely, [108] and [114], §86. The situation changesradically if dim ∆ = 16 where there seems to exist an overwhelming variety of (even trans-lation) planes which are impossible to classify without imposing further restrictions. TheSalzmann programme for some class G of geometries can be stated as follows:

I) find a convenient measure m for the homogeneity of a geometry G of G.II) find the most homogeneous geometry Gc of G with respect to m.

III) find the critical value g such that every geometry G of G whose measure m(G)strictly exceeds g forces G to be isomorphic to Gc.

IV) classify those geometries that have a measure m(G) which is near to g.

For the classes of two-dimensional, four-dimensional and eight-dimensional compact pro-jective planes, the programme as outlined above is more or less completed, if m(G) isinterpreted as dim ∆. For sixteen-dimensional planes, part IV) still needs further work.

Let us return to smooth projective planes. As in the topological case it is reasonableto take the dimensions of the collineation groups as the homogeneity measure. In order tokeep to the smooth category, we have to replace the group ∆ of continuous collineations bythe group ∆s of smooth collineations. But this causes the first serious trouble, because it isunclear whether or not ∆s is a locally compact group. On the other hand we have to insistthat ∆s is locally compact in order to have a rich structure theory available. In many cases,locally compact groups behave just like Lie groups due to the approximation theorem ofMontgomery–Zippin [88], Chap. IV. For a comprehensive overview of the analogy betweenlocally compact groups and Lie groups the reader is referred to the appendix (Lie Struc-ture Theory for non-Lie Groups) of [122]. The possibility of replacing the compact-opentopology (C0-topology) by some Ck-topology makes things even worse. This might havebeen one of the reasons why no systematic theory of smooth projective planes has beendeveloped until now. Things began to change when Linus Kramer and myself were luckyenough to find an easy way out of this problem. We proved that a continuous collineationof a smooth stable plane is in fact a smooth collineation. Moreover, it turns out thatthis is true even for collineations of more general smooth geometries, namely for smoothgeneralized polygons, [15], [16]. This result enables us to use the old group ∆ = ∆s whichis known to be locally compact. Furthermore, it makes no difference taking either theC0-topology or one of the Ck-topologies. Another consequence is that as in the case of Liegroups the smooth structures of smooth stable planes are uniquely determined by the un-derlying (topological) stable planes. Knowing that ∆ = ∆s is locally compact we can use

Introduction 5

standard arguments of the theory of transformation groups, see [88], in order to concludethat ∆ is even a Lie transformation group on the manifold P of points and the manifoldL of lines. We note one more consequence of the fact that every continuous collineation issmooth, namely that we can easily compare the smooth classification with the topologicalclassification, because the homogeneity of a compact projective plane (measured by thedimension of its collineation group) does not decrease if we add a smooth structure to it.Thus the requirement of a compact plane being smooth actually increases its homogeneity.We will come back to this aspect later in this introduction. We are now going to give arough description of the contents of each chapter.

Local Theory. Smooth Stable Planes. Chapter 1 introduces the reader to smoothstable planes and lists elementary properties of these objects. We prove that point rows andline pencils are smoothly embedded submanifolds of the respective manifolds, a fact thatBreitsprecher did assume in his definition of a smooth projective plane. Perspectivitiesturn out to be local diffeomorphisms. This is used in order to show that the smoothstructures of a smooth stable plane are determined by the smooth structures of the linepencils of a triangle and by the geometric operations. Another important result is the localproduct structure of the point and line manifolds, i.e. every point p (or every line) admitsa chart that is the product of charts of two given lines which intersect in p.

In Chapter 2 we have a look at the tangent spaces of points and lines of a smoothstable plane S = (P,L). Given a point p, we consider the set Lp of all lines that passthrough p. The tangent spaces TpL of these lines L constitute a so-called spread Sp on thetangent space TpP . Thus TpP carries the structure of an affine translation plane whichwe will denote by Ap. We prove that Ap is even a locally compact translation plane. Thislinks the theory of smooth stable planes to the rich theory of locally compact translationplanes. These tangent translation planes turn out to be the most effective tool for theinvestigation of smooth projective planes which starts in the second part.

The short Chapter 3 studies p-regular smooth stable planes. For such planes thetangent translation plane Ap is even a smooth affine plane. In order to apply the results totwo- and four-dimensional smooth projective planes, the concept of p-regularity is weakenedto p-semiregularity. Chapter 3 closes with criteria for a smooth stable plane to be p-(semi)regular.

The remaining four chapters of the local part deal with collineations and collineationgroups of smooth stable planes. We have already mentioned the main result of Chapter 4,namely that every continuous collineation of a smooth stable plane is a smooth collineation.Various consequences of this theorem round off this chapter.

Many important results about collineations are discussed in Chapter 5. In order tosketch the main statements, let S = (P,L) be a smooth stable plane with ∆ as its groupof continuous collineations. Fix some point p and let ∆p denote the stabilizer subgroup of

6 Introduction

∆ at p. We consider the derivation map Dp : ∆p → GL(TpP ) : δ 7→ Dpδ. We show thatthe image Dp∆p consists of (continuous) collineations of the tangent translation plane Ap

which, of course, fix the origin. The map Dp is a continuous homomorphism of groupsand its kernel contains the group of semi-elations having center p. (A semi-elation γ withcenter p is a collineation of S that fixes every line through p and acts trivially on someneighborhood of p in a line A through p, see (5.2).) In the case of a smooth projective planethe kernel kerDp consists exactly of the elations with center p. This allows us to transferstabilizers of the collineation group ∆ into the collineation group of a locally compacttranslation plane, and the geometrical characterization of the kernel enables us to controlthe image Dp∆p. We will use this tool throughout the manuscript. Another focal point ofChapter 5 is the investigation of semi-elation groups. For this part we take pattern fromtwo papers of R. Lowen, [81] and [82]. Many results which has been be proved either forcompact projective planes or for low-dimensional stable planes are generalized to smoothstable planes of arbitrary dimension. In fact, we prove that every semi-elation is an elationand we demonstrate that the groups of elations are solvable groups without non-trivialcompact subgroups. For compact projective planes the proof of such a result is not yetknown. The chapter ends with a few results on reflections.

In Chapter 6, the last chapter of the local part, we continue the study of collineationgroups of smooth stable planes. In the first part of this chapter we focus on the grouptheoretical structure of stabilizer subgroups of ∆. Using the derivation map introduced inChapter 5, we show that the stabilizer of an antiflag is a linear Lie group. Moreover, weprove that Levi subgroups of stabilizers of flags are compact. They are even trivial if thedimension of S is two or four. The second part of Chapter 6 is devoted to the detailedinvestigation of stabilizers Θ of three concurrent lines. The main theorem here statesthat either Θ is isomorphic to the stabilizer Θc of three concurrent lines in the classicalprojective plane of the same dimension as S, or the dimension of Θ is much smaller thandim Θc, see Theorems (6.24) and (6.25).

Global Theory. Smooth Projective Planes. The global theory of smooth planes startswith Chapter 7. We give upper bounds for the dimensions of solvable subgroups of thecollineation group ∆ of a smooth projective plane P. A similar examination has alreadybeen done by Martin Luneburg [87], where this turned out to be a fruitful starting pointfor the analysis of 16-dimensional compact projective planes. Compared to Luneburg’sresults we get much better bounds in any of the cases: compare the table on page 36 of[87] with the table on page 68. It should be noted that our bounds on the dimensionscoincide with the dimensions of solvable groups that appear in the analoguous situationsin the respective classical projective plane.

Chapter 8 deals with the question of how subplanes of smooth projective planes P canbe recognized in the tangent translation planes Ap. In particular, we prove that a Baer

Introduction 7

subplane (i.e. a subplane that has half the dimension of P) induces Baer subplanes in everytangent plane Ap for any point p of the given Baer subplane. This result will be used inChapter 11 which deals with 8-dimensional Hughes planes.

Chapters 9 to 12 are devoted to the Salzmann programme for smooth projective planes.There are separate chapters for each of the possible dimensions 2, 4, 8, and 16. As we havealready mentioned, for every smooth projective plane the group of smooth collineationscoincides with the group of continuous collineations. Thus we may refer to the results ofthe Salzmann programme for compact topological projective planes. Let P be a compactprojective plane with ∆ as its group of continuous collineations. In order to determine thecritical values gs for smooth planes as described in part III) above, we use the classificationof compact projective planes having a collineation group whose dimension is close to thevalue g = gt in the topological case (as formulated in part IV)). According to [114], §65or §87, the critical values gt = gt(n) are 4, 8, 18, and 40, if n = 2, 4, 8, 16. Since thisclassification shows that (except for two-dimensional projective planes) the plane P or itsdual plane is a translation plane if dim∆ = gt, Otte’s theorem implies that for the smoothcase the critical value gs is at most gt − 1. Our aim is to prove the following theorem.

Main Theorem. A smooth projective plane P of dimension n is isomorphic to one of the

classical projective planes P2F, where F ∈ R,C,H,O, if P admits a group ∆ of continuous

collineations with dim ∆ ≥ gt(n)−1, where gt(n) is the critical value in the case of compact

topological projective planes. In other words, we have even gs(n) ≤ gt(n)− 2.

Our strategy for proving this theorem will be the following. The compact projectiveplanes P up to dimension 8 which have a collineation group ∆ of dimension gt−1 are knownexplicitly. We show that every plane appearing in this classification cannot be turned intoa smooth projective plane. It is worthwile to note that in most cases we have to dealwith whole families of planes instead of single projective planes and that for every suchfamily of projective planes we have to use different methods. For two-dimensional planesthe Moulton planes are shown to be not smooth by using the whole collineation group,whereas for the planes defined over cartesian fields, which have a collineation group ofdimension 3, we only need to look at special collineations. The families of skew hyperbolicplanes and skew parabola planes are ruled out by using coordinate methods and the detailedinvestigation of their projectivities which determine the smooth structure of the point set.In the case of four-dimensional planes we have to deal with (flexible) shift planes. We showthat these planes are semi-regular and we use the results of Chapter 3 in order to excludethese types of projective planes. The family of Hughes planes appear in the examinationof eight-dimensional planes. Here, the results of Chapter 8 come into play. We prove thatthe tangent planes Ap of inner points p are isomorphic to the affine quaternion plane A2H.Using the derivation map Dp, the action of the stabilizer ∆p on P is played off against

8 Introduction

the action of Dp∆p on Ap∼= A2H. Since up to the present day the classification for 16-

dimensional projective planes has not been completed, we have to use a different methodin this case. The dissertation of Martin Luneburg [87] will be our guiding light here.

Acknowledgements. I hope that this manuscript will encourage other mathematiciansto work in the beautiful area of smooth incidence geometry in order to push the Salzmannclassification programme out of its starting holes. I would like to thank Jozsef Kozmafor providing important results which enabled Linus Kramer and me to prove that everycontinuous homomorphism between local loops is smooth. Of course, I want to thank LinusKramer. It is a pleasure to work with him. I would like to thank Joachim Otte for hiscontributions to smooth incidence geometry and especially for his theorem on translationplanes. I feel sorry that he has decided to abandon a scientific career in mathematics.I want to thank Michael Joswig for endless hours in front of the computer and for hisinterest in smooth structures. Also many thanks to Martin Luneburg for his well writtendissertation. His clear and precise style saved me many hours of work. Most of all I wantto thank my admired teacher Prof. Dr. H. Reiner Salzmann for being my supervisor, andfor encouraging me to write this manuscript. Also, I want to thank Theo Grundhofer andMarkus Stroppel as well as many other former students of Salzmann for useful discussionsand for simply being good friends. Last but not least I want to express my gratitude tomy kind wife and my lovely daughter Linda without whom I surely would not have hadthe energy to complete this work.

The commutative diagrams appearing in this mauscript have all been typeset by using Paul Taylor’s TEX

macro package diagrams.def, version 3.29. Most of the computations of chapter 9 as well as figures 4 and

5 have been done with Mathematica.

9

Part One

Local Theory

Chapter 1: Definitions and Basic Results 11

CHAPTER 1

Definitions and Basic Results

The notion of a linear space which we are going to write down in our first defini-tion is the incidence geometric foundation of the smooth geometries that we will considerthroughout this text.

(1.1) Definition. A linear space is a triple S = (P,L,F) of sets P , L and F, where Pdenotes the set of points, L is the set of lines and F ⊆ P × L is the set of flags, suchthat for every pair of distinct points p, q there is exactly one joining line L ∈ L, i.e.(p, L), (q, L) ∈ F. If (p, L) ∈ F, we will say that p and L are incident, or that p lies on L,or that L passes through p.

An equivalent formulation of this axiom is that the join map ∨ : P × P \ diagP → L

which assigns to each pair of distinct points its joining line is well-defined. Dually, onecan speak of the intersection relation ∧. Note that in a linear space two lines may notintersect. Hence, for a linear space, the operation ∧ need not be defined on the whole setL × L \ diagL. It is convenient to identify each line L of a linear space with the set ofpoints that are incident with L, i.e. we identify L with p ∈ P | (p, L) ∈ F. If we look ata line this way, we will frequently call it a point row. Similarly, a pencil of lines Lp is theset of all lines that are incident with a given point p. A triangle in a linear space S is a setof three (pairwise distinct) points which do not lie on a common line. A quadrangle is aset of four pairwise distinct points such that any three of them form a triangle.

(1.2) Definition. A stable plane S is a linear space (P,L,F) which satisfies the followingthree axioms:

(S1) There are Hausdorff topologies on both P and L that are neither discrete noranti-discrete such that the join map ∨ and the intersection map ∧ are continuous.Moreover, the domain O of the intersection map is an open subset of L× L.

(S2) The topology on P is locally compact and has positive finite covering dimension.(S3) S contains a quadrangle.

The following theorem summarizes those properties of stable planes which we willneed later. Proofs can be found in [79] and [81].

12 Chapter 1: Definitions and Basic Results

(1.3) Theorem. Let S = (P,L,F) be a stable plane and let p ∈ P , L ∈ L.

a) The point row L is a closed subset of the set P of points, the line pencil Lp is

compact, and the flag space F is a closed subset of P × L.

b) The set L of lines and the line pencils Lp are connected.

c) dimL = dim Lp = l = 2k with k = 0, 1, 2, 3, and dimP = dim L = 2l, dim F = 3l,where dim denotes the covering dimension. The stable plane S is said to be 2l-dimensional.

d) dimU = dimX for every open subset U of X ∈ L,Lp, P,L,F.

Turning to the differentiable case, we use the term smooth in the meaning of differentiableof class C∞.

(1.4) Definition. A smooth stable plane S is a stable plane (P,L,F) such that P andL are smooth manifolds and such that the join and intersection mappings are smooth ontheir (respective) domains.

For the rest of this chapter, let S = (P,L,F) denote a smooth stable plane of dimension2l and let O be the open domain of the intersection map ∧. Note that in contrast toBreitsprecher [21] we do not require the point rows and the line pencils of S to be closedsubmanifolds of P and L, respectively. As we will see below, this follows already from thedefinition of a smooth stable plane. When speaking of a submanifold, we always mean asmoothly embedded manifold. A map f : X → Y , where X is a subset of Y , is called aretraction onto A ⊆ Y , iff f(X) = A and f f = f . The following criterion for a subsetof a smooth manifold to be a smoothly embedded submanifold has turned out to be veryuseful.

(1.5) Lemma. Let N be a subset of a smooth manifold M such that for every point x ∈ Nthere exists a neighborhood base U(x) of x in N consisting of open neighborhoods of finite

covering dimension n = dimN . The set N is a submanifold of M (i.e., the inclusion map

N →M is an embedding), if for every x ∈ N and every neighborhood V of x in M there

is an open subset U of x in V and a smooth retraction f : U →M onto U ∩N .

Proof. The proof of [24], 5.13, shows that N is a local submanifold of M , i.e. for everyx ∈ N there is a neighborhood U of x in M such that N ∩ U is a submanifold of M , seealso [61], p.20, Ex.2. Thus it only remains to show that all of these local submanifolds areof uniform dimension. Since, by hypothesis, the covering dimension of N ∩U is n, we inferthat N ∩ U is an n-dimensional submanifold, which proves the lemma.

(1.6) Theorem. Every point row L is a closed submanifold of P and every line pencil Lp

is a compact submanifold of L.


Proof. Fix a line L ∈ L and some point p ∈ P \ L. By axiom (S1) of a stable plane, themap f : U → P : x 7→ (x ∨ p) ∧ L is defined on some open subset U of P which containsthe point row L. Clearly we have f f = f and thus L = f(U) is a submanifold of Pby (1.5) and (1.3), part d). By Theorem (1.3), part a), the submanifold L is closed in P .For the dual statement, let L ∈ Lp and choose some line K ∈ L \ Lp which intersects L.Consider the smooth map g : U→ Lp : M 7→ (M ∧K) ∨ p which is defined on some openneighborhood U of L in L. Again, we have g g = g, and by (1.5) and (1.3), the line pencilLp is a compact submanifold of L.

As an easy consequence, we get the following simple but nevertheless important resulton perspectivities.

(1.7) Corollary. For L ∈ L and p ∈ P \ L the perspectivity ηp,L : L→ Lp : x 7→ x ∨ p is

a diffeomorphism between L and an open subset of Lp, and ηL,p : U→ L : X 7→ X ∧ L is

the inverse map of ηp,L.

Proof. The map ηp,L : L→ ηp,L(L) is a smooth bijection defined on the smooth manifoldL whose inverse map is given by some restriction of ηL,p : U → L : X 7→ X ∧ L whichis smooth as well. By (S1), the domain U is open and contains the image ηp,L(L). Sinceηp,L(L) = Lp ∩ U, we infer that ηp,L(L) is an open subset of Lp.

The smooth structures of the line set L and the point set P of a stable plane S

are uniquely determined by the smooth structures of the line pencils together with thegeometric operations. In order to prove this, we need two preparatory lemmas. For p ∈ Pwe put ∨p : P \ p → Lp : x 7→ x ∨ p and dually we set ∧L : OL → L : K 7→ K ∧ L forsome line L ∈ L, where OL ⊂ L is the (open) domain of ∧L.

(1.8) Lemma. Let p, q ∈ P be distinct points and set L := p ∨ q. Then the map

χp,q : P \L→ Lp \ L×Lq \ L : x 7→ (∨p(x),∨q(x)) is a diffeomorphism onto the open

subset X := (Lp \ L × Lq \ L) ∩ O of Lp \ L × Lq \ L.

Proof. The set Lp \ L × Lq \ L is a submanifold of L × L by Corollary (1.6). Bydefinition of a smooth stable plane, the map χp,q is smooth. Since every pair of distinctpoints determines exactly one line, the map χp,q is an injection. By definition of the setO, the image of χp,q is just X. Moreover, its inverse map is given by the restriction of theintersection map ∧ to X which is smooth, too.

We will also formulate the dual statement of the previous lemma. However, we omitthe proof, since it is analogous to the one above.

(1.9) Lemma. Let K,L ∈ L be distinct lines, let p ∈ K \ L, q ∈ L \ K, and set

M = p ∨ q. Then there is an open neighborhood UM of M in L such that the map


νK,L : UM → K × L : X 7→ (∧K(X),∧L(X)) is a diffeomorphism onto some open subset

of K × L.

(1.10) Corollary. The smooth structures of the set P of points and the set L of lines

are uniquely determined by the smooth structures of the line pencils Lo, Lu, Lv of some

triangle o, u, v and by the geometric operations of S.

Proof. Set K := o ∨ u, L := o ∨ v, and M := u ∨ v. Since the point rows K, L andM are closed subsets of P , the set P \K,P \ L,P \M is an open covering of P , andthe smooth structures of its members is determined by the smooth structures of the linepencils Lo, Lu, and Lv by (1.8). Thus, the same is true for the smooth structure on P .Let M ∈ L. We may assume that the point u does not lie on M . Then the smoothstructure on M is given by the map ∨u : M → Lu : m 7→ u∨m, see Corollary (1.7). Thus,the smooth structure of a line in S is determined by one of the given line pencils and bysome perspectivity ∨u. By Lemma (1.9), this implies that the smooth structure on L onlydepends on the smooth structures of Lo, Lu, and Lv, because the family UM |M ∈ L isan open covering of L.

A more detailed result on the link between the smooth structures of the set of pointsand the set of lines which we will prove next, yields that two point rows with a commonpoint p intersect transversally in p. Moreover, statement (ii) will play an important rolein Chapter 2.

(1.11) Lemma. Let L1, L2 be two lines that intersect in some point p. Then there exist

open neighborhoods U of p in P and U of L1 in L, and there exist points qi ∈ Li such that

(i) the map ϕ : U → U1×U2 : x 7→ ((x∨ q2)∧L1, (x∨ q1)∧L2), where Ui := U ∩Li,is a diffeomorphism of U onto U1 × U2,

(ii) U ∩ L is diffeomorphic to Rl for every line L ∈ U,

(iii) ρL : U1 → U ∩ L : x 7→ (x ∨ q2) ∧ L is a diffeomorphism for every line L ∈ U.

Proof. Choose points qi ∈ Li \ p. By (S1), there are neighborhoods Vi of Li in Lqisuch

that V1×V2 ⊆ O. By Lemma (1.8), the set V1×V2 is mapped diffeomorphically by χ−1q1,q2

onto some open neighborhood U ⊆ P of p. The map ϕ as defined in the Lemma is well-defined (because of V1×V2 ⊆ O) and smooth on U . In order to show that ϕ maps U ontoU1 × U2, we only have to check that Ui = (U ∨ q3−i) ∧ Li holds for i = 1, 2. The inclusion”⊆” is trivial, since x = (x∨q3−i)∧Li for x ∈ Ui. Conversely, if we have x = (y∨q3−i)∧Lifor some y ∈ U , then y ∨ q3−i ∈ V3−i. Thus, we infer that x ∈ χ−1

q1,q2(V1 × V2) = U , andhence x ∈ U ∩ Li = Ui. Since the smooth map

U1 × U2 → U : (u1, u2) 7→ (u1 ∨ q2) ∧ (u2 ∨ q1)


is the inverse mapping of ϕ, this shows that ϕ is a diffeomorphism.

U

U q1

L1

U1U2

p

L2

q2

In order to prove statements (ii) and (iii), we choose V2 to be diffeomorphic to Rl, which ispossible because Lp is an l-dimensional manifold by (1.6). By (i) and Corollary (1.7), theneighborhood V2 is mapped diffeomorphically onto U1 by ∧L1 : K 7→ K ∧ L1. Thus, theintersection U1 = L1 ∩ U is diffeomorphic to Rl. By (S1), there is a neighborhood U′ ⊆ L

of L1 such that U′ × V2 ⊆ O. Then, for a line L ∈ U′, the projectivity ρL : U1 → L :x 7→ (x∨ q2)∧L is well-defined and maps U1 diffeomorphically onto some open subset ULof L. Recall that ∨−1

q2 (V2) is the set of all those points that lie on some line of V2. Since∨−1q2 (V2) ∩ L1 ⊆ U and

∨−1q2 (V2) ∩ L = ρL(U1) = (x ∨ q2) ∧ L | x ∈ U1 ,

the continuity of the intersection map ∧ implies that there is a neighborhhod U′′ ⊆ L ofL1 such that ∨−1

q2 (V2) ∩ L ⊆ U holds for every line L ∈ U′′. Setting U := U′ ∩ U′′, weconclude that

Rl ∼= ρL(U1) = ∨−1q2 (V2) ∩ L ⊆ U ∩ L. (∗)

By definition of U we have

U = χ−1q1,q2(V1 × V2) = ∨−1

q1 (V1) ∩ ∨−1q2 (V2) ⊆ ∨−1

q2 (V2),

and hence, relation (∗) is in fact an equality. This proves (ii) and (iii).


In Chapter 4 we will also need the dual of Lemma (1.11). For convenience we formulatethe dual statement and sketch a proof for it.

(1.12) Lemma. Let p1, p2 ∈ P be two points and set L = p1 ∨ p2. Then there exist open

neighborhoods U of L in L and U of p1 in P , and there exist lines Ki ∈ Lpisuch that

(i) the map ψ : U→ U1×U2 : X 7→ ((X∧K2)∨p1, (X∧K1)∨p2), where Ui := U∩Lpi,

is a diffeomorphism of U onto U1 × U2,

(ii) U ∩ Lp is diffeomorphic to Rl for every point p ∈ U ,

(iii) ρp : U1 → U ∩Lp : X 7→ (X ∧K2) ∨ p is a diffeomorphism for every point p ∈ P .

U

p1 p2

L

U

K1 K2

Sketch of proof. Choose lines Ki ∈ Lpi\ L. By Lemma (1.9), there are disjoint open

neighborhoods Vi of pi in Ki such that the product V1 × V2 is mapped diffeomorphicallyby ν−1

K1,K2onto some open neighborhood U ⊆ L of L. The map ψ defined in the lemma

is well-defined by the very definition of U and is smooth on U. The proof for ψ being adiffeomorphism now runs as in the proof of Lemma (1.11). This proves (i). Now chooseV2 to be diffeomorphic to Rl by using Lemma (1.6). By (i) and Corollary (1.7), theprojectivity ∨p1 : x 7→ x ∨ p1 : V2 → U1 is a diffeomorphism, whence U1

∼= Rl. Choosean open neighborhood U ⊆ P of p1 which is disjoint to V2. Then, for each point p ∈ Uthe projectivity ρp : U1 → Lp : X 7→ (X ∧ K2) ∨ p is defined. Since the join map ∨ iscontinuous, we may select U in such a way that ∧−1

K2(V2) ∩ Lp ⊆ U holds for every p ∈ U .

Now we proceed as in Lemma (1.11) in order to prove (ii) and (iii).

(1.13) Corollary. If two lines of S intersect, they intersect transversally, i.e. if K,L ∈ L

with p = K ∧L, then there is a coordinate neighborhood (U, h) at p such that h(U ∩K) =Rl×0 and h(U ∩L) = 0×Rl holds. Dually, two line pencils intersect transversally in

L .

(1.14) Theorem. The flag space F of a smooth stable plane S is a closed submanifold of

the product manifold P × L.


Proof. Let (p, L) ∈ F and select some point o ∈ P \ L. By (S1), there are open neighbor-hoods U of p and V of L such that x ∨ o | x ∈ U × V ⊆ O. Then the map

f : U × V→ F : (q,K) 7→ ((q ∨ o) ∧K,K)

is well-defined and smooth. Moreover, we have f f = f , since (q ∨ o) ∧ K = q if(q,K) ∈ F ∩ (U ×V). Thus, by Lemma (1.5) and Theorem (1.3), part d), the flag space F

is a submanifold of P × L which is closed by Theorem (1.3), part a).

The restrictions πP and πL of the natural projections P × L → P and P × L → L

are submersions; for a given flag (p, L) choose some point q 6= p on L and let U bean open neighborhood of p in P that does not contain q. Putting ι : U → F : x 7→(x, x ∨ q), we have πP ι = idU which shows that πP has maximal rank 2l at (p, L). Inparticular, the fibres π−1

L (L) are submanifolds of F and the restriction πL := πP π−1L (L)

is a submersion. Because πL is a smooth injection between two l-dimensional manifolds,invariance of domain yields (see [19], Chap. IV, (19.9)) that πL : π−1

L (L) → L ⊂ P is ahomeomorphism and consequently we have the following result.

(1.15) Corollary. For every line L ∈ L the restriction πP π−1L (L) is a diffeomorphism of

the fibre π−1L (L) onto the point row L of P .

Analogously, we can prove the dual statement:

(1.16) Corollary. For every point q ∈ P the restriction πL π−1P (q) is a diffeomorphism

of the fibre π−1P (q) onto the line pencil Lq of L.

18 Chapter 2: Tangent Translation Planes

CHAPTER 2

Tangent Translation Planes

Let S = (P,L,F) be a smooth stable plane of dimension 2l. In this chapter weintroduce the fundamental concept of a tangent translation plane of the smooth stable planeS. As the name suggests, a tangent translation plane lives on some tangent space TpP . Asa main result we prove that the collection Sp = TpL | L ∈ Lp of vector subspaces of TpPconstitutes a locally compact translation plane Ap on TpP . These translation planes willprove to be one of our most important tool in the investigation of smooth stable planes.

Let us first consider the tangent bundles (TP , P, τP ), (TL,L, τL), and (TF,F, τF).Since the canonical projection πL : F → L is a submersion (see the end of the lastchapter), the pullback

π∗L(TL) := (f, v) ∈ F × TL | πL(f) = τL(v)

with respect to the diagram

TL

FπL - L

τL

?

defines a vector bundle (π∗L(TL),F, τ ′L) with τ ′L(f, v) = f , see [64], Chap.3, Prop.3.1.Moreover, we have the commutative diagram

TP DπP TFDπL - TL

I@@

@@

@

Dπp K

⊆

@@

@@

@

π′′L

R

π′L

K π∗L(TL)

@@

@@

@

τK

R

τ ′L

P

τP

? πP

F

τF

? πL - L

τL

?

Chapter 2: Tangent Translation Planes 19

where π′L(f, v) = v and π′′L(v) = (τF(v),DπL(v)). The triangles on the left hand side ofthe diagram will be explained below. For a fixed flag f = (p, L) ∈ F we have

τ ′−1L (f) = (f, v) ∈ F × TL | πL(f) = τL(v) = f × τ−1

L (πL(f)) = f × TLL. (∗)

Since πL is a submersion, the tangential map DπL maps every fibre TfF of τF epimor-phically onto some fibre TLL of τL. Hence, by relation (∗), we infer that π′′L maps TfFepimorphically onto f × TLL. Consequently, the triple (K,F, τK), where K = kerπ′′L =v ∈ TF | π′′L(v) ∈ F × 0 = v ∈ TF | DπL(v) = 0 and τK = τF K

, is a vector subbun-dle of (TF,F, τF), see [64], Chap.3, Cor.8.4. The kernel K consists exactly of those tangentvectors of TF which are tangent to some fibre of πL. Because these fibres are mapped byπP diffeomorphically onto the point rows in P according to Corollary (1.15), the kernelK consists of those tangent vectors that are mapped by Dπp to vectors of TP which aretangent to point rows.

Before we proceed, we have to interpose a few facts on (topological) vector bundles.Let ξ = (E,B, π) be an n-dimensional vector bundle. We write E(ξ)0 = E0 for the totalspace of ξ with the zero section removed. It is possible to define on E0 an equivalencerelation ∼ by calling two vectors v1, v2 ∈ E0 equivalent, iff they lie in a common fibre ofπ and span the same one-dimensional subspace there. We write PE for the quotient spaceE0

/∼. Let π′ : E0 → PE be the canonical projection. Then the bundle Pξ = (PE,B, π),

where π is given by π = (π π′)E0

, is called the projective bundle associated to ξ (see [64],Chap.16, (2.1)). Providing PE with the quotient topology, the fibres of π are compact.Thus, the total space PE is compact if and only if the base space B is compact, cf. [23]§,§4,Lemma 2. If ξ1 = (E1, B1, π1) and ξ2 = (E2, B2, π2) are vector bundles, then every bundlemorphism g : ξ1 → ξ2 induces a bundle morphism g between the projective bundles Pξ1and Pξ2, since g is linear on every fibre of π1. We will need the following lemma on vectorbundles.

(2.1) Lemma. Let ξi = (Ei, Bi, πi), (i = 1, 2) be two vector bundles, where the base

spaces Bi are (topological) manifolds and dimE1 = dimE2 = n. If g : ξ1 → ξ2 is a bundle

morphism such that g0 := g (E1)0is injective, then the mapping g0 : (E1)0 → (E2)0 is an

open map. If, in addition, B1 is compact and B2 is connected, then g0 is a homeomorphism.

Proof. By invariance of domain (see [19], Chap.IV, (19.9)), the maps g and g0 are open(note that the total spaces Ei are topological manifolds as well). Passing to the projectivebundles Pξi associated to ξi, we obtain an injective bundle morphism g : Pξ1 → Pξ2.According to the remark above, the total space PE1 is a compact (n − 1)-dimensionaltopological manifold, while PE2 is a connected manifold of the same dimension. Becausethe restriction g0 is an injection, the induced map g is injective on PE1. Since PE1 iscompact and PE2 is connected, we infer that g maps PE1 onto PE2. In order to show that


g0 is surjective, let v2 ∈ (E2)0 and denote by v2 the one-dimensional subspace of E2 suchthat v2 ∈ v2. Since g is a surjection, there is an element v1 ∈ PE1 with g(v1) = v2. Bylinearity, the map g is a surjection, too.

For the rest of this section, we fix some point p ∈ P , and we set B := π−1P (p) =

p×Lp, G := π−1L (Lp), and KB = K

B. Via restriction, we obtain from the last diagram

TpP DπP TFB

]JJJJJ

ϑ

⊆

KB

JJJJJ

τB

^p

τP

? πP

B

τF

?

where τB = τF KB. Using the terminology of [64], the next lemma states that ϑ is an

effective Gauss map of the vector bundle (KB,B, τB) into TpP , cf. [64], Chap.3, §5 and[23], §3, p.22.

(2.2) Lemma. The restriction ϑ0 := DπP (KB)0: (KB)0 → TpP \ 0 is a smooth

homeomorphism with ϑ0(τ−1B (p, L) \ 0) = TpL \ 0 for every line L ∈ Lp.

Proof. Being the restriction of a smooth map to a submanifold of TF, the map ϑ0 issmooth.

1) In order to show that ϑ0 is injective, let v1, v2 ∈ (KB)0 with ϑ0(v1) = ϑ0(v2). Sincetwo distinct point rows through p intersect transversally by Corollary (1.13), and sincev1, v2 6= 0, there is a flag (p, L) ∈ F such that v1, v2 ∈ T(p,L)F. By Corollary (1.15), the fibreπ−1

L (L) is mapped by πP diffeomorphically onto the point row L ⊂ P . Hence, the tangentfibre Dπ−1

L (TLL) is mapped injectively into TP by DπP . From T(p,L)F ⊆ Dπ−1L (TLL) and

DπP (v1) = ϑ0(v1) = ϑ0(v2) = DπP (v2), we conclude that v1 = v2.

2) The vector bundles (TpP, p , τP TpP) and (KB,B, τB) have compact connected

manifolds as bases and dim TpP = 2l = l + l = dim B + dim τ−1B (b) = dim KB holds for

every b ∈ B. By part 1), the map ϑ is a bundle morphism such that ϑ0 is an injection.Thus, the map ϑ0 is a homeomorphism by Lemma (2.1).

3) Since ϑ0 is a bijection by 2) which is linear on the fibres of τB and because ofdimR τ

−1B (p, L) = dimR TpL, it suffices to show that ϑ0(τ−1

B (p, L) \ 0) ⊆ TpL \ 0


holds for every line L ∈ Lp. Let (p, L) ∈ B and select a vector v ∈ τ−1B (p, L) \ 0. By

definition of KB, the vector v is tangent to the fibre π−1L (L). Since this fibre is mapped

diffeomorphically into P by πP , the image of v under the map DπP lies in the tangentspace TpL \ 0. This completes the proof of the Lemma.

We call the Gauss map ϑp = ϑ0 the characteristic map of S at the point p, and werefer to the restriction DπP K

as the (global) characteristic map of S.

(2.3) Definition. Let V be a real vector space of dimension 2m. A family X of m-

dimensional subspaces of V is called a spread in V, if and only if (i) and (ii) hold:

(i)⋃

X = V

(ii) X1 ⊕X2 = V for every pair of distinct elements X1, X2 of X.

Every point row L ∈ Lp is a closed submanifold of P by Theorem (1.6) and theirtangent spaces TpL at p are l-dimensional subspaces of TpP . Since two distinct linesK,L of Lp are transversal by Corollary (1.13), the tangent space TpP decomposes asTpP = TpK ⊕ TpL. We call Sp := TpL | L ∈ Lp the tangent spread of S at p. We willoften write SL instead of TpL. By the remark preceding Lemma (2.2) and by Lemma (2.2)itself, we get

⋃Sp = TpP , and thus we have indeed:

(2.4) Proposition. The set Sp is a spread in TpP .

Every spread of a vector space V defines an (affine) translation plane, see [1] or [25].Thus, Proposition (2.4) shows that the tangent space TpP is an affine translation planeAp which is defined by the spread Sp. In fact, we even have the following theorem.

(2.5) Theorem. The tangent translation planes Ap of a smooth stable plane S are locally

compact connected topological affine translation planes.

Since every locally compact connected affine translation plane is coordinatized by alocally compact connected ternary field (see [100], 7.15, 7.16, or [44], 4.2), the theoremabove yields the following result (compare also [84]).

(2.6) Corollary. The projective closures Pp of the tangent planes Ap are compact con-

nected projective translation planes.

We will base the proof of Theorem (2.5) on the first Linearization Theorem below.Recall that O is the domain of the intersection map ∧.

(2.7) Lemma. The inverse image G := π−1L (Lp) is a submanifold of F.


Proof. By Theorem (1.6), the line pencil Lp is a submanifold of L, and thus P × Lp is asubmanifold of P × L. Since G ⊆ P × Lp, it suffices to show that G is a submanifold ofP × Lp. In order to verify that, we will again utilize Lemma (1.5). Let (q, L) ∈ P × Lp.Choose some point o ∈ P \ L and let U and V be connected open neighborhoods of q andL, respectively, such that x ∨ o | x ∈ U×V ⊆ O holds. Considering as in Theorem (1.14)the smooth retraction

f : U × V→ P × Lp : (r,K) 7→ ((r ∨ o) ∧K,K)

onto G ∩ (U × V), we conclude that G ∩ (U × V) is a submanifold of U × V, see [24], 5.13.Since G \B is diffeomorphic to P \ p via the projection (r,K) 7→ r, we infer by Theorem(1.3), part d) that all those local submanifolds G ∩ (U × V) are of a uniform dimension.Hence, by Lemma (1.5) we infer that G is a submanifold of P × Lp.

In the next corollary we use the notion of a (smooth) microbundle, see [66], Essay IV,§1 for a definition.

(2.8) Corollary. The quadruple (G,B, κ, ι), where ι : B → G is the inclusion map and

κ : G→ B : (q, L) 7→ (p, L), is a smooth l-dimensional microbundle.

Proof. Clearly, we have κ ι = 1lB. Let (p, L) ∈ B and choose some line L′ ∈ LP \ L.Select neighborhoods U ⊆ P of p and U ⊆ L of L according to Lemma (1.11). The setV := p × (U ∩ Lp) is an open neighborhood of (p, L) in B and W := (U × U) ∩ G is anopen neighborhood of (p, L) in G. The map

h : V× (U ∩ L)→W ∩ κ−1(V) : ((p,K), q) 7→ ((q ∨ q′) ∧K,K),

where q′ ∈ L′ corresponds to q2 in Lemma (1.11), is a diffeomorphism by Lemma (1.11)(iii).This map h satisfies the relations

κ h((p,K), q) = κ((q ∨ q′) ∧K,K) = (p,K)

andh((p,K), p) = (p,K).

Since U ∩L is diffeomorphic to Rl by Lemma (1.11)(ii), this shows that h is a smooth localtrivialization of (G,B, κ) and hence (G,B, κ, ι) is a smooth microbundle.

By the Kister-Mazur theorem (see [67], [76], and [116]), there is an open neighborhoodW of B in G such that ξ = (W,B, κ

W) is a locally trivial Rl-bundle with zero section ι

and structure group Diff(Rl) consisting of all origin-preserving diffeomorphisms of Rl.


Since OlR is a deformation retract of Diff(Rl) (see [119]), we may reduce the structuregroup of ξ to OlR. In this way, the bundle ξ becomes a smooth l-dimensional vectorbundle. Moreover, we have dim G = dim B + dim Rl = 2l, and thus the set W is a tubularneighborhood of B ([72], III, (2.4)). By [72], (3.2), any two tubular neighborhoods of B

in G are isomorphic. In particular, the bundle ξ is isomorphic to the normal bundle νB ofB in G, see [72], (2.3). We will denote this isomorphism by β1. Note that ξ and νB areisomorphic smooth vector bundles, hence β1 induces a diffeomorphism between their totalspaces, cp. [72], p. 48, Ex. Providing G with a Riemannian metric, we may represent νB asthe complementary bundle to TB in TG

B. The restriction πL B

: B→ L is an embeddingand thus DπL TB

: TB→ TL is an injection. Recalling that K = v ∈ TF | DπL(v) = 0,we thus have T(p,L)B ∩ K(p,L) = 0 for every line L ∈ Lp. In particular, no vector ofK(p,L) is orthogonal to ν(p,L)B. Since both νB and KB are l-dimensional vector bundlesover B, orthogonal projection in each fibre gives rise to a bundle isomorphism β2 betweenνB and KB. According to Lemma (2.2), the composition

β : W \Bβ1- E(νB)0

β2- E(KB)0ϑ0- TpP \ 0

is a smooth homeomorphism. The following two theorems are the main result of thischapter. They state that a smooth stable plane can be linearized in some neighborhood ofeach point.

(2.9) First Linearization Theorem. There exists an open neighborhood U of p in P

and a smooth homeomorphism λ : U \p → TpP \0 such that λ(L∩U \p) = TpL\0holds for every line L ∈ Lp.

Proof. Since W is an open neighborhood of B in G, there is an open neighborhood U ofp in P such that W = (U × Lp) ∩ G. The map α : U \ p → W \ B : q 7→ (q, p ∨ q) isa smooth bijection having pr1 : (q, L) 7→ q as a smooth inverse map. The compositionλ := β α : U \ p → TpP \ 0 is a smooth homeomorphism. For every line L ∈ Lp wehave α(L ∩ U \ p) = (L ∩ U \ p , L) and the composition β2 β1 maps the latter setbijectively onto τ−1

B (p, L) \ 0. Hence, we have λ(L ∩ U \ p) = TpL \ 0 according toLemma (2.2).

(2.10) Lemma. Let h be a homeomorphism of R := Rn \ 0 onto itself, where n ≥ 2,

and let S := R ∪ 0 ∪ ∞ ≈ Sn be the two-point compactification of R. Then there is a

homeomorphism ϕ of S onto itself such that ϕR

leaves every vector subspace minus the

zero vector invariant and such that ϕ H : S → S is a homeomorphism, where HR

= h

and ϕ H(0) = 0.

Proof. Let Σ1 ⊆ R be an (n−1)-sphere. Then Σ2 := h(Σ1) is an (n−1)-sphere too. By thegeneralized Jordan curve theorem (see [19], Chap. IV, §19, e.g.) the complements S \ Σi


consist of exactly two connected components Ki, Li, where we may assume that 0 ∈ Ki.Since n ≥ 2, the sets K∗

i := Ki \ 0 and L∗i := Li \ ∞ are the connected componentsof R. Thus we have h(K∗

1 ), h(L∗1) ∈ K∗2 , L

∗2. Suppose that h(K∗

1 ) = L∗2. The map

ϕ : S → S : x 7→

0 if x =∞∞ if x = 0||x||−2x otherwise

is a homeomorphism of S onto itself which leaves every vector subspace minus the zerovector invariant. Extending the map h by H(0) = ∞ and H(∞) = 0 to S, it remains toshow that H : S → S is continuous at the points 0 and ∞. Choose a sequence (xn)n∈N inS that converges to 0. Since spheres of positive dimensions are locally connected, we mayassume that this sequence lies in K1. The sequence (h(xn))n∈N is thus contained in thecompact set L2 ∪Σ2. In particular, the sequence (h(xn))n∈N has an accumulation point inL2 ∪Σ2. Since h : R→ R is a homeomorphism, every accumulation point of the sequenceabove must be ∞. Hence, the map H is continuous at the point 0. Analoguously, oneproves that H is continuous at ∞ and that H−1 is continuous. For the case h(K∗

1 ) = K∗2 ,

we take ϕ = idS and the proof runs as before.

From Theorem (2.9) and the previous lemma we immediately get the second Lin-earization Theorem.

(2.11) Second Linearization Theorem. There exists an open neighborhood U of p in

P and a homeomorphism λ : U → TpP such that λ(L ∩ U) = TpL holds for every line

L ∈ Lp.

In order to prove Theorem (2.5), we have to check that the tangential spread Sp iscompact with respect to the Grassmann topology, see [84] and [114], 64.4d.

(2.12) Corollary. The tangential spread Sp is a compact subset of the Grassmann man-

ifold Gl(R2l).

Proof. Let (TpLν)ν∈N be a sequence of spread elements of Sp which converges to an l-dimensional vector subspace V of R2l. We have to show that V ∈ Sp. Let q ∈ V \ 0.Then there are vectors qν ∈ TpLν \ 0 such that the sequence (qν)ν∈N converges to q inR2l. By Theorem (2.9), the sequence λ−1(qν) ⊆ U \ p converges to λ−1(q), and henceλ−1(qν) ∨ p = Lν converges to L := λ−1(q) ∨ p in L. Thus we have

q = λ(λ−1(q)) ∈ λ(L ∩ U \ p) = TpL \ 0

and we conclude that V is contained in TpL. Since both subspaces V and TpL have thesame dimension l, the claim follows.


(2.13) Corollary. For every point p ∈ P the map σ : Lp → Sp : L 7→ TpL is a

homeomorphism.

Proof. Clearly, the map σ is a bijection. For q ∈ TpP \ 0 we denote by Sq the uniquespread element of Sp that contains q. Since the affine tangent plane Ap is a topologicalplane by (2.5), the join map τ : TpP \0 → Sp : q 7→ Sq is a continuous surjection. Hence,the composition σ ∨p = τ λ is continuous on U \ p. Since ∨p is an identification map,we conclude that σ is continuous, too. Because the line pencil Lp is compact, the map σ

is in fact a homeomorphism.

Since every line pencil of an 2l-dimensional (affine or projective) locally compacttopological translation plane is homeomorphic to the l-sphere Sl, the preceding corollaryyields the following result, cp. also [114], 52.3.

(2.14) Corollary. For every point p ∈ P the line pencil Lp is homeomorphic to an

l-sphere.

26 Chapter 3: Regular Stable Planes

CHAPTER 3

Regular Stable Planes

In this chapter we study tangent translation planes Ap of a smooth stable plane in thecase where the local characteristic map ϑp defined in the last chapter is a diffeomorphism. Itturns out that then the plane Ap is a smooth affine translation plane. For the treatment ofshift planes in Chapters 9 and 10 we need to generalize p-regularity to p-semiregularity. Inthis case the tangent plane Ap is not necessarily smooth but nevertheless its coordinatizingquasifield is still a smooth quasifield.

(3.1) Definition. A smooth stable plane S = (P,L,F) is called regular at some point p, orp-regular for short iff the characteristic map ϑp at p has maximum rank everywhere. Theplane S is called regular iff its global characteristic map has maximum rank everywhere.This is equivalent to requiring that S is p-regular for every point p ∈ P . Clearly, if S isp-regular, the smooth homeomorphism λ of the Linearization Theorem (2.9) has maximalrank everywhere and hence is a diffeomorphism.

The notion of regularity for smooth projective planes has been introduced by S. Breit-sprecher [21]. The four classical projective Moufang planes are regular planes. Breit-sprecher claimed that these are the only regular smooth projective planes. This conjectureis not yet confirmed.

According to Sard’s theorem (see, e.g., [24], §6 or [61], Chapt. 3, Theorem 1.3), theregular values of the characteristic map ϑp are dense in TpP . Let x ∈ (KB)0 be someregular point lying in some fibre F = τ−1

B (b). Choose some neighborhood U ⊆ K around xsuch that ϑp is regular on U . Then U ∩F spans the vector space F , and since ϑp is linearon every fibre of τB, we conclude that ϑp is regular on F \ 0. Let F ′ be another fibre ofτB and choose L,L′ ∈ Lp with ϑp(F ) = TpL and ϑp(F ′) = TpL′, cf. Lemma (2.2). Assumethat there is a (smooth) collineation γ of S which fixes p and maps L′ to L, cp. Definition(4.1). Considering γ = γF as a diffeomorphism of the flag set onto itself, the derivativeDγF maps F ′ isomorphically onto F , while Dp(γ−1

P ) maps TpL isomorphically onto TpL′.From πL γF = γL πL we infer that KB is invariant under ϑ′p := Dp(γ−1

P ) ϑp DγF.Since we have πP γF = γP πP , this implies that ϑ′p = ϑp. Hence the characteristic mapϑp is regular on F ′ \ 0, because ϑp is regular on F \ 0 by hypothesis. Thus we haveproved the following criterion.

Chapter 3: Regular Stable Planes 27

(3.2) Proposition. If the stabilizer Γp of the automorphism group Γ of S acts transitively

on the line pencil Lp, then S is p-regular.

We turn to the investigation of the tangent translation plane Ap of a p-regular stableplane. We will show that Ap is a smooth affine plane. In order to prove this, we haveto verify that the tangent spread Sp is smooth, cf. J. Otte [93], (6.12), i.e. the projectionπ∨ : TpP \ 0 → Sp : v 7→ π∨(v), where π∨(v) is the unique spread element of Sp

containing v, is a smooth map, [93], (5.12). J. Otte [93], (5.15), shows that a smoothspread is a smoothly embedded submanifold of the Grassmannian Gl(TpP ).

Select an open neighborhood U of p in P as in the Linearization Theorem (2.9).Fix a nonzero point v in TpP . We will check that the projection π∨ is smooth on someneighborhood V of v. Using the notation of Chapter 2, we choose an open subset W of(p, x ∨ p) in B = p × Lp such that the restriction ξW = (KW ,W, τW ) of the vectorbundle (KB,B, τB) to W , where τW = τF W

, is a trivial bundle. Then we can choosea family X = (Xi)li=1 of linearly independent vector fields on W having values in K.We put U := u ∈ TpP \ 0 | u ∈ SL ∈ Sp and L ∈W. Since the tangent plane Ap is atopological plane by Theorem (2.5), the joining map ∨ is continuous, and hence the set Uis open in TpP . Clearly, U is a neighborhood of v. Set α : P \ p → F : x 7→ (x, p ∨ x)and let π2l,l : Vl(TpP )→ Gl(TpP ) : (v1, . . . , vl) 7→ 〈v1, . . . , vl〉 be the (smooth) projectionfrom the Stiefel manifold Vl(TpP ) of l-frames in TpP onto the Grassmannian Gl(TpP ).Then we may write the projection π∨ on U as the composition

π∨ : Uα λ−1

- p ×WX-

l∏i=1

KB

∏li=1 ϑp- Vl(TpP )

π2l,l- Sp ⊆ Gl(TpP )

of smooth mappings. (Recall that the inverse map λ−1 is smooth since S is p-regular andnote that v ∈ π∨(v) for v ∈ U by Lemma (2.2) and Theorem (2.9)). Hence we have provedthe following theorem.

(3.3) Theorem. The tangent translation plane Ap = (TpP, Sp) at the point p is a smooth

affine plane, if S is a p-regular stable plane.

The proof of the theorem above also can be used for the non-regular case. In thegeneral situation it shows that the projection π∨ is at least continuous which implies thatthe tangent translation plane is a topological plane. However, basing the proof of Theorem(2.5) on R. Lowen’s characterization of spreads belonging to locally compact translationplanes yields a much simpler verification.

(3.4) Corollary. For every point p ∈ P the map σp : Lp → Sp : L 7→ SL is a smooth

homeomorphism. If S is p-regular, then σp is a diffeomorphism.

28 Chapter 3: Regular Stable Planes

Proof. Locally we may write the map σp as the composition

σp : WK 7→ p ×K- p ×W

X-l∏i=1

KB

∏li=1 ϑp- Vl(TpP )

π2l,l- Sp

of smooth mappings. If S is p-regular, then ϑ−1p is smooth. Since ϑ−1

p τB is constant onevery element of Sp, it induces a smooth map σ′p between Sp and B = p ×Lp. Thus theinverse map σ−1

p can be represented as

σ−1p : Sp

σ′p- BπL- Lp

which finishes the proof.

In order to apply the results of this Chapter to shift planes we have to weaken thedefinition of a p-regular stable plane.

(3.5) Definition. A smooth stable plane S = (P,L,F) is called p-semiregular for p ∈ P , ifthe characteristic map ϑp has maximum rank everywhere except possibly at the elementsof one single fibre F of τB. The fibre F is called the singular fibre of S at p. The lineS = σ−1

p ϑp(S) is called the singular line of S at p. Note that call F and S singular evenif ϑp happens to have maximum rank on F .

By what we have shown so far, it is clear that for p-semiregular planes the mapσ−1p : Sp → Lp : SL 7→ L is a diffeomorphism between Sp \ ϑp(F ) and Lp \ (πL τB(F )).

Although the tangent plane Ap may fail to be a smooth affine plane if S is only semiregularat p, the set Sp \ ϑp(F ) still constitutes a smooth L-set by using Grassmann coordinates,see the Appendix and Lemma (1.8) of Otte [93]. This is equivalent to saying that thequasifield associated to the spread Sp is smooth (but not necessarily smooth at infinity).From [93], (1.9), we infer that the following lemma holds.

(3.6) Lemma. Let S = (P,L,F) be a p-semiregular smooth stable plane of dimension n

with singular fibre F . Let the spread Sp be coordinatized by Grassmann coordinates using

Sp \ ϑp(F ) as Y and some other spread element as X (see the Appendix), and denote by

Mp the set of Grassmann coordinates of the elements of Sp \ ϑp(F ). Then the evaluation

map εx : Mp → Rl : M 7→Mx is a diffeomorphism for every element x ∈ Rl \ 0.As in the case of p-regular planes we have a similar criterion for a plane being p-

semiregular.

(3.7) Proposition. If the stabilizer Γp of the automorphism group Γ of S acts transitively

on Lp \ Y for some line Y ∈ Lp, then S is p-semiregular and Y is the singular at p.

Chapter 4: Collineations 29

CHAPTER 4

Collineations

The main topic of this chapter is the investigation of continuous collineations (i.e.automorphisms) of smooth stable planes. We show that every continuous collineation ofsuch a plane is in fact a smooth collineation. This implies that the group Γ of all continuouscollineations is a Lie group with respect to the compact-open topology. In particular, thesmooth structure of a smooth stable plane are uniquely determined by the underlying(topological) stable plane.

(4.1) Definition. Let S = (P,L,F) and S′ = (P ′,L′,F′) be two smooth stable planes.A collineation γ : S → S′ is a pair (γP , γL) of continuous bijections γP : P → P ′ andγL : L → L′ such that (γP × γL)(F) ⊆ F′. We write γF instead of (γP × γL). If S′ = S,then γ is called a collineation of S. A collineation γ is called smooth, if the mappings γPand γL are smooth. Most of the time, we will just use the symbol γ instead of γP , γL orγF.

Since the point rows of a stable plane S = (P,L,F) are submanifolds of P by Theorem(1.6), we may apply the notion of smoothness to the restrictions γP L

of a collineationγ : S→ S′ to some point row L.

(4.2) Proposition. Let γ : S → S′ be a collineation. If there exists a line L ∈ L such

that the restriction γP Lis smooth, then γ is a smooth collineation.

Proof. Let L ∈ L such that γP Lis a smooth mapping.

1) We first show that γP Kis smooth for every line K ∈ L. Fix a line K ∈ L \ L

and let u be some point on K. Choose some point v ∈ L and select another pointp ∈ (u ∨ v) \ u, v which exists by (S3). Then, by axiom (S1), the projectivity

η = η−1p,L ηp,K : K → L : x 7→ (x ∨ p) ∧ L

is defined on some open neighborhood U of u in K. Moreover, by Corollary (1.7), themap η : U → η(U) is a diffeomorphism and η(U) is an open subset of L. Since γ is a

30 Chapter 4: Collineations

collineation, we have the commuting diagram

Uη - η(U)

γP (U)

γP? η′- γP (η(U))

γP?

where η′ is the projectivity of S′ which corresponds to η, i.e. we have η′ : γP (K)→ γP (L) :x 7→ (x ∨ γP (p)) ∧ γL(L). Hence, the restriction γP U

: U 7→ γP (U) is smooth and thusγP K

is smooth as well.

2) In virtue of Lemma (1.8) and (1.9), a collineation γ is smooth if and only if one ofthe maps γP and γL is smooth. Hence, in order to prove the proposition, we only haveto check that γL is smooth. Let M ∈ L. Choose two distinct points p, q on M , and letK ∈ Lp \ M and L ∈ Lq \ M. By Lemma (1.9) there is an open neighborhood UM

of M in L such that νK,L : UM → K × L is a diffeomorphism onto some open subset ofK × L. Setting a′ := γ(a) for a ∈ P ∪ L, we have the diagram

UMνK,L- K × L

U′M ′

γL? ν′K′,L′- K ′ × L′

δ?

where ν′K′,L′ : U′M ′ → K ′ × L′ is the diffeomorphism given by Lemma (1.9) with respectto S′ and δ = γP K

× γP L. This diagram commutes, because for N ∈ UM we have

(γP K× γP L

) νK,L(N) = (γP (K ∧N), γP (L ∧N))

= (γL(K) ∧ γL(N), γL(L) ∧ γL(N))

= (K ′ ∧N ′, L′ ∧N ′)

= ν′K′,L′ γL(N).

By what we have proved in 1), this implies that γL is smooth at M . Since M ∈ L canbeen chosen arbitrarily, the claim follows.

The following main theorem shows that there is, in fact, no difference between collinea-tions and smooth collineations.

(4.3) Theorem. Every collineation γ : S → S′ between two smooth stable planes is a

smooth collineation.

Proof. By Proposition (4.2) we have to check that the restriction γP Lis smooth for

some line L ∈ L. Fix a point o incident with L. By [79], (1.21), there is a quadrangle

Chapter 4: Collineations 31

♦ = o, u, v, e in P with u ∈ L, and there is an open neighborhood U of o in L such that aternary operation τ : U ×U ×U → L is defined in terms of join and intersection. Thus τ issmooth. Setting x+y := τ(1, x, y) for x, y ∈ U , where 1 = (v∨e)∧L, we get a smooth localH-space H = (L,U, o,+) in the sense of [15], (1.2). Since the image γ(S) = S′ is a stableplane, the quadrangle ♦ is mapped by γP onto a quadrangle ♦′ = o′, u′, v′, e′ ⊆ P ′. Asbefore, we obtain a smooth local H-space H′ = (L′, U ′, o′,+′) on the point row L′ = γP (L).Since γ is a collineation and since the operations + and +′ are defined in purely geometricalterms, the map γP L

is a continuous homomorphism between the smooth local H-spacesH and H′. By [15], (1.8), we conclude that γP L

is smooth on some neighborhood of o.By varying the point o on L, this shows that the map γP L

is smooth on every point ofL. This completes the proof.

Let Γ be the group of all (continuous) collineations of a smooth stable plane S =(P,L,F). We will refer to Γ as the automorphism group of S. The compact-open topologiesof Γ on P and on L coincide. In this topology, the group Γ is locally compact, and Γ acts asa topological transformation group on P as well as on L by [79], (2.9). In virtue of Theorem(4.3), the group Γ consists of diffeomorphisms of P and of L. Since L is connected (see(1.3)), we get the following result from [88], Chap.V, Th.2.

(4.4) Corollary. The automorphism group Γ of a smooth stable plane S = (P,L,F) is

a smooth Lie transformation group on the manifold P of points, the manifold L of lines,

and the manifold F of flags with respect to the common compact-open topology.

As another immediate consequence we have

(4.5) Corollary. A (topological) stable plane S admits at most one smooth structure on

the set P of points and on the set L of lines, respectively, such that S becomes a smooth

stable plane.

32 Chapter 5: Central Collineations

CHAPTER 5

Central Collineations

Central and axial collineations of topological stable planes behave quite different com-pared to their relatives in projective planes. For example, a central collineation of a stableplane must not have an axis, because you can remove the points of an existing axis and geta new stable plane by restriction. However, we show that for smooth stable planes most ofthese distinctions disappear.

Basic Concepts

(5.1) Definition. Let S = (P,L,F) be a smooth stable plane. A collineation γ of S

is called a central collineation with center p iff γ fixes every line through p. Dually, acollineation γ is called axial with axis L iff every point on L is a fixed point of γ.

Note that it is not necessary to require that a central or axial collineation be continuousor even smooth. In fact, any such collineation is continuous by Lowen [79], (3.2), and thusis smooth by Theorem (4.3). In contrast to the projective case, a central collineation of astable plane need not have an axis and an axial collineation may have no center. Moreover,a central collineation γ may fix every point of an open subset of some line A rather thanevery point of A. Such a central collineation is called semi-axial and A is referred to asits semi-axis, see Lowen [81], §3. As usual, we denote by Γ the collineation group of thesmooth stable plane S.

(5.2) Notation. For any point c ∈ P we denote by Γ[c] the subgroup of Γ consisting ofall central collineations γ ∈ Γ with center c. The group Γ[A] for some line A is defineddually. Moreover, we use the abbreviations Γ[c,A] := Γ[c]∩Γ[A] and Γ[c,c] :=

⟨⋃A∈Lc

Γ[c,A]

⟩.

The group Γ[A,A] is defined dually. If (c, A) is a flag, the collineations of Γ[c,A] are calledelations, otherwise they are called homologies. Note that unlike in the projective case,where every central (axial) collineation has an axis (a center), it is not clear whether ornot the generating sets

⋃A∈Lc

Γ[c,A] and⋃c∈A Γ[c,A] form subgroups of Γ. Nevertheless, as

we will see later, this turns out to be true for the groups Γ[c,c] (see Corollary (5.22)).

Chapter 5: Central Collineations 33

Now we turn to local versions of the aforementioned groups. Let (c, A) be a flag. Wedefine Γs[c,A] as the set of all central collineations with center c and semi-axis A that fixevery point of some open neighborhood O of p in A. We refer to the elements of Γs[c,A]

as semi-elations. Clearly, the set Γs[c,A] of semi-elations forms a group. As in the case of

elations we put Γs[c,c] :=⟨⋃

A∈LcΓs[c,A]

⟩and Γs[A,A] :=

⟨⋃c∈A Γs[c,A]

⟩. If, on the other hand,

(c, A) is an anti-flag, we define Γs[c,A] to be the set of all axial collineations with axis A thatfix every line c ∨ x for x ∈ A. We call such collineations semi-homologies and we speak ofc as their semi-center. We will show, however, that every semi-homology (semi-elation) ofa smooth stable plane is in fact a homology (an elation); cp. Theorem (5.5) and Corollary(5.19).

(5.3) Theorem. For every point p ∈ P and for every line L ∈ L the derivation mappings

Dp : Γp → GL(TpP ) : γP 7→ DpγP and DL : ΓL → GL(TLL) : γL 7→ DLγL are continuous

homomorphisms.

Proof. The set P of points is locally connected by Lowen [79], (1.4) and (1.11), and thusevery connected component of P is an open subset of P . Hence the connected componentPp of p in P becomes a smooth stable plane via restriction. Moreover, the stabilizer Γpleaves Pp invariant and acts effectively on Pp. Applying [88], Theorem, p.208ff to (Pp, Γp),we conclude that the homomorphism Dp : Γp → GL(TpP ) : γP 7→ DpγP is continuous.Since the set L of lines is (arcwise) connected, the continuity of DL follows immediately.In order to show that the line space L is arcwise connected, we use the fact that everyline is locally arcwise connected (Lowen [79], (1.12)). Hence every line pencil Lp is locallyarcwise connected as well (Lowen [79], (1.7)) and thus Lp is arcwise connected, since it isconnected (Lowen [79], (1.14)). Now choose two lines K 6= L and fix two distinct pointsk ∈ K and l ∈ L. Then there are paths from K to k∨ l and from k∨ l to L. Concatenatingthese two paths gives a path from K to L in L. This proves the arcwise connectedness ofL as well as the theorem.

Let p ∈ P . Every collineation γ ∈ Γp acts on the line pencil Lp and thus induces anaction on the collection Sp := TpL | p ∈ Lp of l-dimensional subspaces of the tangentspace TpP . This action is given by Dpγ(TpL) = Tp(Lγ) for L ∈ Lp. Thus, the transforma-tion groups (Γp,Lp) and (Dp(Γp), Sp) are equivalent via the equivariant homeomorphismσp : Lp → Sp : L 7→ TpL. Since Dpγ is a linear map on TpP which acts on the tangentialspread Sp, the group DpΓp consists of automorphisms of the tangent translation planeAp = (TpP, Sp) that fix the origin 0 of TpP .

(5.4) Lemma. If γ ∈ Γ fixes three lines L1, L2, L3 ∈ Lp for some point p ∈ P , then the

set Fγ of fixed lines of γ acting on Lp is a homology m-sphere S with m ≥ 1.


Proof. By Corollary (2.14) the line pencil Lp is homeomorphic to Sl. Assume that γ actsnon-trivially on Lp. The image Dpγ fixes the three different lines TpLi of the tangenttranslation plane Ap. By Hahl [50], (2.1), the group of automorphisms of Ap that fix thethree lines TpLi induces a non-trivial compact Lie group Φ on Sp. Thus the collineationDpγ generates some torus subgroup T of Φ or it has finite order r. Assume that γ generatessome torus group T. Then we have |FT| ≥ 3 since |Fγ | ≥ 3. Thus, by Floyd [34], (4.1) and[33], (5.2), the fixed point set FT of a torus group T acting on a (homology) sphere withfixed points is a homology m-sphere with m ≥ 1. Since γ (topologically) generates T, weinfer that Fγ ≈ F (Dpγ) :=

S ∈ Sp

∣∣ SDpγ = S

= FT is a homology m-sphere. If, on theother hand, the order r of Dpγ is finite, we let r = ps11 · . . . · p

sk

k be the decomposition ofr into its prime powers, and we put ri := ps11 · . . . · p

sii and ϕi := (Dpγ)ri . By Liao [77],

the fixed point set Fk−1 of ϕk−1 is a homology sphere of positive dimension (rememberthat |Fγ | ≥ 3 by hypothesis). Because ϕk−i−1 acts on Fk−1, we inductively conclude thatF (Dpγ) is a homology m-sphere with m ≥ 1. Hence the claim of the lemma follows.

The next theorem is in the spirit of R. Baer’s characterization of quasi-perspectivitiesof projective planes (Baer [2]); see Stroppel [120], Theorem 5 for an extension of Baer’stheorem to stable planes.

(5.5) Theorem. If γ ∈ Γ fixes an open subset of some line pencil Lc, then γ is a central

collineation with center c. In particular, every semi-homology is in fact a homology.

Proof. By Lemma (5.4), the set Fγ of fixed lines of γ acting on Lc is a homology m-sphereS. Since γ fixes an open subset U of Lc, we have m = dimU = dim Lc = l by Lowen [82],Theorem 11c. We will show that S = Lc. The following part of the proof is due to LinusKramer. Let L be some interior line of S and choose some neighborhood U ⊂ S aroundL. Denote by ι : S → Lc the inclusion map and consider the diagram of homology groupsover Z2

Hl (S) −→ Hl (S, S \ L) ←− Hl (U,U \ L)yι∗ y y(iU

)∗

Hl (Lc) −→ Hl (Lc,Lc \ L) ←− Hl (U,U \ L)

By invariance of domain, the restriction ιU

is a homeomorphism and thus the inducedmap (i

U)∗ is an isomorphism. Since S and Lc are Z2-orientable, the horizontal arrows

on the left hand side are isomorphisms, while the horizontal arrows on the right hand siderepresent excision isomorphisms. In particular, the induced map ι∗ is an isomorphism,too. Suppose that ι is not surjective. Choose some line K ∈ Lc \ S. Then ι factors asι : S → Lc \ K → Lc in contradiction to Hl (Lc \ K) ∼= Hl (Rl) = 0. Thus we haveproved that Fγ = S = Lc. Now, the second statement follows from Corollary (1.7).


(5.6) Theorem. For every point p ∈ P the kernel of the mapping Dp consists of central

collineations whose semi-axes, if they exist, are incident with p. Dually, for every line

L ∈ L the kernel of the mapping DL consists of axial collineations whose centers, if they

exist, are lying on L.

Proof. Let p ∈ P and γ ∈ Γp. If γ acts non-trivially on the line pencil Lp, then Dpγ

acts non-trivially on the tangential spread Sp and hence γ 6∈ ker Dp. Thus we may assumethat γ is a central collineation (with center p). Moreover, suppose that γ has a semi-axisA not incident with p. Then A is an axis of γ according to Theorem (5.5). We have toverify that γ is not contained in the kernel of Dp. Indeed, fix two distinct points u, v onA, and put Ku := p ∨ u and Kv := p ∨ v. The triangle p, u, v defines a smooth localloop T in the sense of [13] on a neighborhood V ⊆ Ku of p, see Lowen [79], (1.21). Sinceγ fixes its axis A pointwise, the restriction of γ to U is a homomorphism of T . We nowconstruct a coordinate system (h, U) of P around p such that γ

U ∩Kuis a linear map,

which shows that Dpγ 6= 1l. In order to define h, we use two charts hKu and hKv of thelines Ku and Kv at p. Applying Lemma (1.11) to these charts we get a chart h of P aroundp satisfying h

Ku= hKu

. By the results of J.Kozma, [73] and [74], there is a so-calledcanonical coordinate system hKu for the local loop T , and with respect to this chart therestriction γ

U ∩Kuis locally a linear map. Taking an arbitrary chart hKv

on Kv aroundp, this proves the first part of the theorem.For the dual assertion, we assume that γ ∈ ker DL is an axial collineation with axis Land semi-center p 6∈ L. By Theorem (5.5), the point p is in fact a center of γ. The map∨u : V → V ⊆ Lu : x 7→ x ∨ u is a diffeomorphism between V and an open neighborhoodV of p∨ u (see Corollary (1.7)) that transfers the smooth local loop structure T on V to asmooth local loop structure T on V. Clearly, the collineation γ is a homomorphism of Tagain. Since γ commutes with ∨u, the map γ induces a homomorphism of T and we mayproceed as before.

(5.7) Proposition. Let (p, L) be a flag. Then Γs[p,L] is contained in the kernel of the map

Dp, and Γ[p,L] ≤ ker DL.

Proof. Let γ ∈ Γs[p,L] for some flag (p, L). Since γ fixes every line of Lp, the derivativeDpγ fixes every element of the tangential spread Sp. According to H. Hahl [56], (2.1) and(2.3), we may select two spread elements V,W ∈ Sp and a basis B := v1, . . . , v2l ofTpP such that TpL = 〈v1, . . . , vl〉, V = 〈vl+1, . . . , v2l〉, and W = 〈v1 + vl+1, . . . , vl + v2l〉.Since Dpγ leaves the subspaces TpP , V , and W invariant, the matrix representation ofDpγ with respect to B is just

(A 00 A

), where A ∈ GLlR. Because γ is the identity on some

neighborhood of p in L, its derivative Dpγ is the identity on TpL, i. e., we have A = 1l. Weomit the verification of the dual statement.


(5.8) Corollary. We have Γs[p,p] ⊆ ker Dp and Γ[L,L] ⊆ ker DL for every point p ∈ P and

every line L ∈ L.

Since every central collineation of a projective plane has an axis and vice versa, thelast two results enable us to characterize the kernels of Dp and DL for smooth projectiveplanes in purely geometrical terms.

(5.9) Corollary. If S is a smooth projective plane, the kernel of Dp (of DL) consists

exactly of all elations with center p (with axis L).

We will generalize the above result to smooth stable planes in the next section. If acentral or an axial collineation γ 6= 1l of a stable plane S fixes some triangle ∇ := o, u, v,then γ has both a center and an axis. Moreover, the center of γ must be contained in ∇and the axis must be the line of this triangle opposite to the center, cf. Lowen [79], (3.4).Thus, by Theorem (5.6), the map Do is a faithful continuous representation of Γ∇:

(5.10) Corollary. The stabilizer Γ∇ of a triangle ∇ in a smooth stable plane P is a linear

Lie group.

In the case of smooth projective planes, Theorem (4.3) together with Theorem (2.10) of[13] implies the compactness of the stabilizer of a quadrangle (recall that every quadrangleof a projective plane determines a ternary field, and that the stabilizer of a quadrangle canbe regarded as the automorphism group of the associated ternary field).

(5.11) Corollary. The stabilizer Γ♦ of a quadrangle ♦ in a smooth projective plane S is

a compact Lie group.

(5.12) Corollary. Let Σ := Γp,L. If (p, L) is a flag, then Σq is a linear Lie group for every

point q ∈ P \ L. If (p, L) is not a flag, then Σ itself is a linear Lie group.

Another corollary which can easily be derived from Theorem (5.6) deals with thestructure of homology groups of stable planes (whereas the structure of elation groups isthe subject of the next section). Beforehand we need a lemma.

(5.13) Lemma. Let Λ be a Lie group whose maximal torus subgroups are at most one-

dimensional. Then every one-parameter subgroup of Λ is closed in Λ. Consequently, every

subgroup of Λ is closed.

Proof. Let P ≤ Λ be a one-parameter subgroup. Suppose that P is not closed. Then, byHochschild [62], Chap.XVI, Prop. 2.3, the closure P of P is compact. Since P is abelian,we conclude that P is contained in a torus subgroup T of Λ. Thus we have dim P = 1 by


hypothesis. Arcwise connectedness of P and P now implies that P = P, and hence P is aclosed subgroup of Λ after all. The last statement follows from [62], Chap.XVI, Theorem2.4.

(5.14) Corollary. If p 6∈ L, then the homology group Γ[p,L] can be embedded as a

closed subgroup of the homology group of the classical projective plane of the respective

dimension.

Proof. By Theorems (5.3) and (5.6), the derivation map Dp : Γ[p,L] → Aut(Ap) is acontinuous injection, and since Γ[p,L] fixes every line through p, its image under Dp fixesevery element of the tangential spread Sp. Thus DpΓ[p,L] is a group of homologies of thetangent translation plane Pp with the origin o as its center and the line L∞ at infinity asits axis. By Hahl [47], (3.2), the group Aut(Pp)[o,L∞] is isomorphic to the multiplicativegroup of the kernel of the quasifield Q which is associated with the spred Sp. Since thekernel of Q is one of the classical fields R, C, or H according to Buchanan and Hahl [26],we can apply Lemma (5.13) in order to conclude that Dp is a closed mapping.

Recall that in the topological case it is not known whether the preceding corollary istrue or not. Even for compact projective planes only a weaker result is known, see Hahl[52], (3.5). For stable planes of dimension at most 4, however, there is a proof of thiscorollary which is due to H.-P. Seidel [115].

The Structure of Elation Groups

In this section we will elucidate the structure of the semi-elation groups Γs[A,A] andΓs[c,c]. We will show that the connected components of these groups are simply connectedsolvable closed subgroups of the automorphism group Γ of the smooth stable plane S. Forthe classical projective planes P2K (K ∈ R,C,H,O) and, more generally, for affine orprojective translation planes, these groups are isomorphic to vector groups. However, noproof of this claim is known even for compact connected projective planes. The mostsophisticated results on the structure of the (semi-)elation groups in the projective caseare due to H. Hahl [52], (3.7), and to M.Luneburg [87], VII, Satz 7.

For the investigation of the structure of semi-elation groups, we start with a lemmawhich is well-known for affine and projective planes.

(5.15) Lemma. If Γs[c,c] contains elations with different axes, then Γs[c,c] is commutative.

The dual statement holds for the group Γs[A,A].


Proof. It suffices to show that any two elements of the generating set⋃A∈Lc

Γs[c,A] commute.Let γ ∈ Γs[c,A] and γ′ ∈ Γs[c,A′] with A 6= A′. Then we have γ′−1γγ′ ∈ Γs[c,A] and thusthe commutator [γ′, γ] of γ and γ′ lies in Γs[c,A]. With the same type of argument it iseasily checked that [γ′, γ] ∈ Γs[c,A′]. Since Γs[c,A] ∩ Γs[c,A′] = 1l by Lowen [79], (3.4), weconclude that γ commutes with γ′. If, on the other hand, we have γ, γ′′ ∈ Γs[c,A], we chooseγ′ ∈ Γs[c,A′] with A 6= A′. Since γγ′ 6∈ Γs[c,A], we already know that γγ′ commutes with γ′′.By associativity, we finally conclude that γ and γ′ commute. Since the dual statement of[79], (3.4) is trivially true, the second part of the Lemma is proved similarly.

It is not obvious that Γs[c,A] is a closed subgroup of Γ, since at first sight it seems tobe possible that there is a sequence (γn)n∈N in Γs[c,A] such that

⋂n∈N On = c, where

On = p ∈ A | pγn = p. However, the next proposition shows that in fact this cannothappen.

(5.16) Proposition. For every flag (c, A) the group Γs[c,A] is a closed subgroup of Γ.

Proof. If dim S ≤ 4, then Γs[c,A] = Γ[c,A] according to [81], (3.4), and the assertion is obvious.Hence we may assume that dim S ≥ 8. Let γ ∈ Γs[c,A]. By definition, the collineation γ

fixes some neighborhood O of c in A pointwise. By [81], (3.3) every element of Γs[c,A] fixessome connected component of A \ c pointwise. We will show that there is a commonconnected component which is fixed pointwise by any γ ∈ Γs[c,A]. Being a manifold, the lineA is locally connected as well as a Cantor manifold, i.e. any connected component of Acannot be separated by a set of dimension at most dimL− 2, cp. [83], Theorem 11. Thuswe may choose O to be connected, and since dimO = dimL ≥ 4, we infer that O \ cis still connected. Let C denote the connected component of A \ c containing O \ c.Since every element of Γs[c,A] fixes C ∪ c pointwise, the assertion follows.

The impact of the last proposition on the maximal compact subgroups of Γs[c,A] is asfollows.

(5.17) Corollary. If (c, A) is a flag, then Γs[c,A] does not contain any non-trivial compact

subgroup.

Proof. Since Γs[c,A] is closed in Γ by Proposition (5.16), the group Γs[c,A] is a Lie group. Anon-trivial compact subgroup K of a Lie group is either finite or contains a torus subgroup.In either case, such a group K contains non-trivial elements of finite order if K 6= 1l. ButΓs[c,A] does not contain any non-trivial element of finite order, see [81], (3.2). Hence theclaim follows.

By Lowen [81], (3.18) we obtain another important consequence.


(5.18) Corollary. If Γs[c,A] 6= 1l for some flag (c, A), then every line of S different from

A is homeomorphic to Rl or to Sl. Moreover, we have P \ A ≈ R2l, and every line of the

smooth stable plane (P \A,L \ A) is diffeomorphic to Rl.

The next lemma collects the results of Propositions (3.2) to (3.4) of Lowen’s paper[81]. Note that for smooth stable planes we need not assume that dim S ≤ 4. In order toformulate the next lemma we need the notion of coaffine points. A point p ∈ P is calledcoaffine if and only if exactly one line L of Lp is not compact. In this case, the line L iscalled pointwise coaffine, see Lowen [81], (1.4) and (1.6).

(5.19) Lemma. Let γ ∈ Γs[c,A] \ 1l for some flag (c, A). Then the following statements

hold:

a) For every point x ∈ P \A, the iterated images xγn

converge to c as n→∞.

b) The smooth stable plane S is either projective, or the line A is pointwise coaffine.

c) If γ fixes some point x ∈ P \A, then γ is the identity.

Proof. In order to prove part a), we assume that (xγn

)n∈N has an accumulation pointy ∈ P \c. Then there is a collineation δ which is a cluster point of γnn∈N with respectto the compact-open topology such that xδ = y, see [81], Prop. 3.8. By Weil’s lemma,either 〈γ〉 ≤ Γ is relatively compact or 〈γ〉 is discrete. Because Γs[c,A] is a closed subgroupof Γ which has no non-trivial compact subgroup (Proposition (5.16) and Corollary (5.17)),we conclude that 〈γ〉 is discrete with respect to the induced topology. In particular, theset γnn∈N has no cluster point at all, a contradiction. Thus it remains to verify that(xγ

n

)n∈N accumulates at c. For that, choose some point p ∈ A \ c which is fixed byγ. Since the line pencil Lp is compact by [79], (1.17), the sequence (xγ

n ∨ p)n∈N has acluster line L ∈ Lp. For the sake of simplicity we will assume that this sequence convergesto L in Lp. Fix some point q ∈ L \ p. By the stability axiom (S1), almost all linesxγ

n ∨ p intersect the line c ∨ q, and by continuity of the intersection map ∧, the sequence(qn)n≥k of these intersection points converges to q. Hence, from the first part of our proofwe conclude that q = c. This finishes part a). For part b) we follow Lowen’s proof [81],(3.2): choose some point p ∈ A which is fixed by γ. If L ∈ Lc \ A is non-compact, thereis some line K ∈ Lp \ A which does not intersect L. By stability, however, the imagesKγn

and Lγn

= L do intersect for sufficiently large n ∈ N, because Kγn

converges to Aby part a) and because L intersects A. Thus, the lines K and L must intersect, too. Thiscontradiction shows that the only possible non-compact line of Lc is the line A. If A iscompact, the smooth projective plane S is projective by [81], (1.3). If, on the other hand,the line A is not compact, then A is pointwise coaffine, see [81], (1.6). This proves part b).If γ fixes some point x ∈ P \ A, then γ has a semi-axis A′ 6= A through x by [81], (3.3).According to [79], (3.4), this implies that γ is the identity.


Using part b) of the proposition above and the remark before (3.5) in [81], we get thefollowing important corollary.

(5.20) Corollary. Every semi-elation is in fact an elation, i.e. Γs[c,A] = Γ[c,A] for every flag

(c, A). In particular, we have Γs[c,c] = Γ[c,c] for every point c and Γs[A,A] = Γ[A,A] for every

line A.

Note that in the topological (non-smooth) case there are stable planes which admitproper semi-elations, see Lowe [78], Example 6. Hence, Lowe’s example of a shear planeis a non-smooth stable plane, i.e. there are no smooth structures on P and L which turnS = (P,L) into a smooth stable plane.

We note several other consequences of Lemma (5.19), part b). The first one is basedon the characterization of pointwise affine lines (for a definition see Lowen [81], (1.6)),while the second one follows from Lowen [81], (3.5).

(5.21) Corollary. If Γs[c,A] 6= 1l for some flag (c, A), then the smooth stable plane S has a

compact line.

(5.22) Corollary. If Γs[c,A] 6= 1l for some flag (c, A), then every central collineation has an

axis.

(5.23) Corollary. For every point c ∈ P the group Γ[c,c] can be written as the union

Γ[c,c] =⋃A∈Lc

Γ[c,A].

Proof. Let c ∈ P . If Γ[c,A] = 1l for every line A ∈ Lc or if S is a projective plane, nothingremains to be proved. Thus, by Corollary (5.22), every element of Γ[c,c] has an axis andit suffices to verify that the product of two elations in Γ[c,c] is an elation again. So letα ∈ Γ[c,A] and β ∈ Γ[c,A′], where A 6= A′, and denote by L the axis of the product αβ. ByLemma (5.19), part b), the lines A and A′ are pointwise coaffine. In particular, the pointc is coaffine and thus at least one of the two lines A and A′ has to be compact (see Lowen[81], (1.6)). Let A be a compact line. Then A meets L in some point p ([79], (1.15)) andfrom pβ = pαβ = p together with Lemma (5.19), part b), we conclude that p ∈ A∧A′ = c.Thus the axis L of αβ goes through c and αβ is an elation with center c.

According to Lowen [81], (3.2), the last corollary implies that Γ[c,c] contains no non-trivial element of finite order. This result is false for (topological) stable planes. H. Lowehas given an example of a shear plane ([78], Example 6) that admits a group of semi-elationsisomorphic to the compact group Spin3R.

(5.24) Corollary. For every point c ∈ P the elation group Γ[c,c] is torsion-free.


Our proof of the preceding corollary mainly relies on the fact that a central collineationhas an axis in the presence of compact lines. Unfortunately, the dual of this is not true.Thus we are forced to look for another way of proving the dual of Corollary (5.24). In orderto do so, recall that the groups Γs[c,A] are closed in Γ by Proposition (5.16). We will showthat this is also true for the elation groups Γ[c,c] and Γ[A,A]. We prove two preparatoryresults.

(5.25) Proposition. Let α be an element of prime order p of the closure Υ := Γ[c,c] of

Γ[c,c]. Then Υ \ 〈α〉 contains no element of order p. In particular, Υ contains at most one

involution and every torus subgroup of Υ is at most one-dimensional.

Proof. Let β ∈ Υ be an element of order p. Since the groups Γ[c,A] for c ∈ A do not containany non-trivial element of finite order (see [81], (3.2)), we may assume that α and β donot have an axis through c. Moreover, by Corollary (5.17), we infer that Γ[c,c] 6= Γ[c,A]

for every line A through c. Hence, any two elements of Γ[c,c] commute by Lemma (5.15).According to Theorem (5.6) and Corollary (5.9), no collineation of Υ possesses a semi-axesnot incident with c. In view of [81], (3.3), the elementary abelian group E := 〈α, β〉 actsfreely on L \ c for every line L ∈ Lc. Choose some line L ∈ Lc. By Corollary (5.18),the line L is homeomorphic to either Rl or Sl. If necessary, this action can be extendedto an action of the one-point compactification L∪ ∞ ≈ Sl by setting ∞ε =∞ for everyε ∈ E, since E acts on L as a group of homeomorphisms. Now we may apply a theoremof P.A. Smith (see [118], p. 407 or [114], 55.27(i)) in order to deduce that E has periodiccohomology. Hence, the group E must be cyclic by [29], p. 262. This proves our proposition.

The dual statement of Proposition (5.25) can be proved even for the groups Γ[A] of axialcollineations instead of Γ[A,A]. Moreover, this dual result can be verified for (topological)stable planes. For a proof see Stroppel [120], (8). However, if we focus on reflections(i.e. involutory central or axial collineations), we are able to prove a stronger result, seeTheorem (5.33) of the next section, cp. Lowen [81]. We are now able to verify that theelation groups Γ[c,c] and Γ[A,A] are closed subgroups.

(5.26) Proposition. For every point c and for every line A the elation groups Γ[c,c] and

Γ[A,A] are closed subgroups of Γ.

Proof. We will concentrate on the groups Γ[c,c], since the following proof can easily bedualized (recall that the dual statement of Proposition (5.25) is true as well). It sufficesto show that the connected component (Γ[c,c])1 is closed in Γ, for this implies that Γ[c,c] islocally compact and hence is closed in Γ. Put Υ := (Γ[c,c])1. If (Γ[c,c])1 = 1l, then (Γ[c,c])1

is closed in Γ. So let us suppose that dim(Γ[c,c])1 ≥ 1. If Υ 6= (Γ[c,c])1 then there is a non-closed one-parameter subgroup P of (Γ[c,c])1, see [62], Chap. XVI, Theorem 2.4. In view of


Proposition (5.25) this contradicts Lemma (5.13). Hence we infer that Υ = (Γ[c,c])1 andthe proof is finished.

We are now heading for the proof that Γ[A,A] is a torsion-free group. We divide thisproof into two separate parts.

(5.27) Proposition. Let (c, A), (c′, A) and (c, A′) be flags. If γ ∈ Γ[c,A] and γ′ ∈ Γ[c′,A] ∪Γ[c,A′], either γγ′ = 1l or γγ′ has an infinite order.

Proof. Let γ′ ∈ Γ[c,A′]. For A = A′ the claim follows from [81], (3.2). So let us supposethat A 6= A′ and that (γγ′)k = 1l holds for some integer k. Since A 6= A′, the collineationsγ and γ′ commute due to Lemma (5.15), and we get 1l = (γγ′)k = γkγ′

k. Note alsothat in this case we have γγ′ 6= 1l. We conclude that γk ∈ Γ[c,A] ∩ Γ[c,A′]. By [79], (3.4),this intersection is just the trivial group, and thus k = 0 follows by Corollary (5.17). Ifγ′ ∈ Γ[c′,A], the assertion is proved analoguously, for Lemma (5.15) is self-dual and thedual statement of [79], (3.4) is trivially true.

(5.28) Theorem. Every non-trivial element of Γ[A,A] has an infinite order.

Proof. Let γ ∈ Γ[A,A] \ 1l. Because Γ[A,A] is generated by the groups Γ[c,A] for c ∈ A, theelement γ can be written as a product γ = γ1 · . . . · γn with γi ∈ Γ[ci,A] for some pointsci ∈ A. If n = 2, the assertion follows from Proposition (5.27). Hence, by induction on n

we may assume that γ = δγn, where δ ∈ Γ[A,A] has infinite order and does not act triviallyon the line pencil Lcn . Suppose that (δγn)k = 1l for some integer k. In view of Lemma(5.15) and Proposition (5.16), we may assume that Γ[A,A] is commutative. Hence we getδk = γ−kn ∈ Γ[cn,A]. Put ε := δk, choose some line L ∈ Lcn

with Lδ 6= L, and select somepoint x ∈ L\cn. Since the sequence (xε

m

)m∈N converges to cn by Proposition (5.19), weinfer from the stability axiom (S1) that the lines xε

m ∨ xδ meet the axis A for sufficientlylarge integers m. Hence, let p := (xε

m ∨xδ)∧A for some suitable integer m. From L 6= Lδ

we obtain that p 6= cn. Because the line xεm ∨ xδ meets the axis A, the equation

(xεm

∨ xδ) ∧A = p = pδ−1

= (xεmδ−1

∨ x) ∧A

shows that K := xεmδ−1 ∨ x is a fixed line of εmδ−1. From

x(εmδ−1)ki

= xεmkiδ−ki

= xεi(mk−1)

we conclude that the sequence (x(εmδ−1)ki

)i∈N converges to cn. But, on the other hand, thepoints x(εmδ−1)ki

lie on the fixed line K which shows that this sequence cannot convergeto cn since cn 6∈ K, a contradiction. Hence ord(γ) =∞ and the theorem is proved.


Our last theorem shows that the connected component Γ1[c,c] of Γ[c,c] is a simply con-

nected group without non-trivial compact subgroups. Decomposing Γ1[c,c] into its solvable

radical and a semi-simple Levi complement Ψ, this implies that Ψ is either trivial or iso-morphic to a direct product of quasi-simple factors each of which is isomorphic to theuniversal covering group Ω of SL2R. In order to show that Γ1

[c,c] is solvable, it remains toexclude the case that Γ1

[c,c] contains a subgroup isomorphic to Ω.

(5.29) Proposition. If Γ contains a non-linear subgroup Ω, then Ω[c,c] 6= Ω for any point

c ∈ P and dually Ω[A,A] 6= Ω for any line A ∈ L.

Proof. Assume that there is a point c ∈ P such that Ω[c,c] = Ω. Since Ω is not linear,it is not abelian and Lemma (5.15) shows that we have Ω[c,c] = Ω[c,A] for some lineA ∈ Lc. Let p ∈ A \ c. Because Ω contains no elation with center p, the mappingDp : Ω → GL(TpP ) is a (continuous) injection (see Theorem (5.6) and Proposition (5.7))which is a contradiction. Hence, we have Ω[c,c] 6= Ω. The proof of the dual statement runsalong the same lines.

Since the universal covering group of SL2R is not a linear group, we may summarizethe information on the groups Γ1

[c,c] and Γ1[A,A] as follows.

(5.30) Corollary. The elation groups Γ1[c,c] and Γ1

[A,A] for arbitrary points c ∈ P and lines

A ∈ L are simply connected solvable closed subgroups of Γ.

We close this section with a generalization of Corollary (5.9) by determining the kernelsof the differentiable mappings Dp in the stable case.

(5.31) Theorem. For every point p and every line L we have ker Dp = Γ[p,p], and ker DL

has no non-trivial compact subgroup.

Proof. Let p ∈ P . By corollaries (5.8) and (5.20) we only have to show that kerDp ⊆ Γ[p,p].The kernel of Dp consists of central collineations with center p, see Theorem (5.6). If kerDp

contains a closed non-compact subgroup (e.g. if Γ[p,p] 6= 1l, Corollary (5.24)) then everycentral collineation has an axis as in the projective case, see Lowen [81], (3.5) and Corollary(5.22), and the claim follows from Theorem (5.6). Hence we may suppose that ker Dp iscompact. Providing the manifold P of points with a suitable Riemannian metric ρ suchthat kerDp becomes a group of isometries with respect to ρ (see [18], Chap.VI, (2.1)) thisshows that kerDp = 1l by [60], Chap.I, (11.2), and hence the inclusion ker Dp ⊆ Γ[p,p] holdstrivially. Thus we have proved that ker Dp = Γ[p,p]. Moreover, the preceding argumentationshows that a maximal compact subgroup of kerDL has to be trivial.


Reflections

In this section we study involutory central and axial collineations which are calledreflections for short. We transfer the results of Lowen [81], p. 303–306 to the smooth case.As in the previous section it turns out that these results hold without the restriction tolow dimensions. Before we start with the dual statement of [120], (8), we prove a lemmaabout the eigenspaces of linearized involutions.

(5.32) Lemma. For x ∈ P ∪ L let α ∈ Γx be an involution. If Ξ+ and Ξ− denote

the positive and negative eigenspaces of Dxα, respectively, either dim Ξ+ = dim Ξ− =12 dimP = l or x is an axis (if x ∈ L) or a center (if x ∈ P ) of α.

Proof. Let x ∈ L. By Theorem (5.31) we have Dxα 6= 1l. If dim Ξ+ > l, then TxLp∩Ξ+ 6=0 for every point p ∈ x. Now, Dxα acts on the set of tangent spaces TxLp and so Dxα

leaves invariant every subspace TxLp. The same argument shows that Dxα acts trivially onthe partial spread TxLp | p ∈ x of tangent spaces if dim Ξ− > l. This, however, impliesthat in both cases x is an axis of α. Being an involution, the map Dxα is diagonalizablewith ±1 as its only eigenvalues. Hence we conclude that dim Ξ+ = l = dim Ξ− if x is noaxis. For x ∈ P the proof runs the same way.

(5.33) Theorem. The group Γ[c] contains no pair of commuting reflections.

Proof. Let α, β ∈ Γ[c] be two different commuting reflections. Then none of the collineationsα, β and αβ is an elation by Corollary (5.17). If α fixes some point p 6= c, then α has anaxis A ∈ Lp with c 6∈ A ([81], (3.3)). Since α and β commute, this means that E := 〈α, β〉is a subgroup of Γ[c,A]. Thus the group E acts freely on Lp \ A, p ∨ c, [79], (3.4). By atheorem of P.A. Smith, see [118], p. 407 or [114], 55.27(i), we conclude that E has periodiccohomology and hence must be a cyclic group ([29], p. 262), a contradiction. Thus, for therest of the proof we may assume that for a fixed line L ∈ Lc the group E acts freely onL \ c. In particular, we infer that the derivation map DL : E → GL(TLL) is injective.According to Lemma (5.32) we may write

DLα =(−1l

1l

)and DLβ =

(X Y

1l

)with respect to a suitable basis of TLL, where all entries are l-dimensional square matrices.Since DLα and DLβ commute, we get Y = 0, and since the negative eigenspace of β has


dimension l, we obtain X = −1l. Hence, we have DLα = DLβ and thus DL(αβ) = 1l, whichis a contradiction to the injectivity of DL on E.

Now we turn to Lowen’s results on reflections which he verified for stable planes ofdimensions at most 4. Note that no proofs of the following two results exist even in thecase of compact topological projective planes.

(5.34) Lemma. If α, β ∈ Γ[c] are different reflections, then αβ is an elation. In particular,

the smooth stable plane S is either projective or the axis of αβ is pointwise coaffine.

Proof. Let α, β ∈ Γ[c] be two different reflections and set E := 〈α, β〉 = 〈αβ〉 · 〈α〉. Fix someline L ∈ Lc. Assume that αβ acts non-trivially on L. Then α and β induce on L differentinvolutions. Denoting by N the kernel of the action of E on L, we put Φ := 〈αβ〉

/(N ∩ 〈αβ〉).

Now we are going to prove that Φ is infinite. By Lemma (5.32), we may choose a basis ofTLL such that

DLα =(−1l

1l

)and DLβ =

(X Y

1l

).

From (DLβ)2 = 1l we infer that X2 = 1l and (X + 1l)Y = 0. Setting M := (DLα)(DLβ) weget

Mn =(

(−X)n ([n+12 ]− [n2 ]X)Y

1l

).

If Mn = 1l, then n must be even and thus

Mn =(

1l n2 (1l−X)Y

1l

)= 1l

holds. Hence we have (1l − X)Y = 0. Combining this result with (1l + X)Y = 0 we getY = 0. But this implies that DLα and DLβ commute and we obtain DLα = DLβ as in theproof of Theorem (5.33). We conclude that αβ acts trivially on L which is a contradiction.Hence the group Φ is infinite.By Weil’s lemma (cf. [62], Chapt.16, (2.3)) the group Φ is either discrete or compact. If Φ

is discrete, the orbits of 〈αβ〉 on L have at most two ends. One of these ends is the center c,while the other end is either ∞ or it lies on a semi-axis of αβ, see [81], (3.8) and (3.9). Byconjugation, the reflection α induces inversion on 〈αβ〉, and thus α interchanges the endsof the orbits in question. Since α fixes the point c, both ends most coincide. In particular,the point c lies on a semi-axis A of αβ. By Lemma (5.19), the axis A is pointwise coaffineand αβ is an elation by Corollary (5.20). If, on the other hand, Φ is compact, it contains amaximal torus subgroup T of positive dimension t (recall that Φ is infinite). If t = 1, thenα centralizes the unique involution ω of T (note that α normalizes 〈αβ〉). Since α doesnot commute with β by Theorem (5.33), we have α 6= ω. If t > 1, the group T possesses a


pair of commuting involutions. But according to Theorem (5.33), the group Γ[c] containsno commuting pair of involutions. Thus, the group Φ cannot be compact and the lemmais proved.

As an immediate consequence of the last lemma we obtain the following result which canalso be deduced in a less direct way from Corollaries (5.14) and (5.17).

(5.35) Corollary. For every pair (c, A) ∈ P × L the groups Γ[c,A] contain at most one

reflection.

The last two results of this section transfer Proposition 3.22 and Corollary 3.23 of Lowen[81] to smooth stable planes.

(5.36) Proposition. Let α ∈ Γ[c,A] and β ∈ Γ[a] be reflections, where a ∈ A. Then either

(αβ)2 ∈ Γ[a,A] \ 1l and A is pointwise coaffine, or αβ is an involution with axis a∨ c, and

β has an axis B ∈ Lc.

Proof. Put L := a∨ c. Clearly, the product (αβ)2 is an element of Γ[a,A]. If (αβ)2 6= 1l, theassertion follows from Lowen [81], (3.2) and Lemma (5.19). Hence, let αβ be an involution.

Since A 6= L is the (unique) axis of α, we get the representation DLα =(−1l

1l

)from

Lemma (5.32). Moreover, we have DLβ =(

1l YX

), because a is the center of β. Recalling

that α and β commute, we infer that Y = 0. The line L cannot be an axis of β, for βis a reflection. Consequently, we derive from Lemma (5.32) that X = −1l and hence

DL(αβ) =(−1l

−1l

)follows. Clearly, this implies that L is an axis of αβ. The last

claim follows from Lemma (5.19), part c), because β fixes the point c.

As in Lowen’s paper [81], the last result implies the following corollary.

(5.37) Corollary. Let α ∈ Γ[a,A] and β ∈ Γ[b,B] be reflections, where a ∈ B and b ∈ A.

Then αβ is a reflection with axis a ∨ b.

Chapter 6: The Structure of Collineation Groups 47

CHAPTER 6

The Structure of Collineation Groups

Basic Results

In order to study the collineation group Γ of a smooth stable plane we focus onstabilizers of points and lines. For this investigation, we utilize a theorem of H. Hahlabout collineation groups of compact translation planes which is presented in a slightlymodified version in the Appendix. Moreover, we have to note some facts about semi-simplesubgroups of a Lie group. Let Ψ be a connected semi-simple subgroup of the connected Liegroup ∆. Since Ψ need not be closed in ∆, we consider Ψ with respect to its Lie topologyτLie which turns Ψ into a Lie group. A basis of τLie is given by the arc components ofopen sets in the induced topology, see [35], (3.2) and (4.2). Note that this topology maydiffer from the induced topology of ∆. In any case, the Lie topology τLie is finer than theinduced topology. Since Ψ and (Ψ, τLie) do have the same one-parameter subgroups ([35],(3.2(5))), the topology τLie is not the discrete topology. In particular, this is true for everyalmost simple factor Σ of Ψ. Every non-trivial normal subgroup of an almost simple Liegroup is contained in its center and thus is countable. Since the connected component ofthe identity of a topological group is a normal subgroup, and because Σ has a non-trivialone-parameter subgroup, this implies that (Σ, τLie) is connected as well. Being a productof almost simple factors, we infer that (Ψ, τLie) is still connected.

A Levi subgroup Ψ of a Lie group is a maximal semi-simple analytic subgroup. Inparticular, such a subgroup Ψ is connected by definition. Note, however, that this differsfrom the definition of a Levi subgroup given in [90], p.287, where connectedness is notassumed. We start with the examination of flag stabilizers.

(6.1) Proposition. If the connected Lie group ∆ of collineations of a smooth stable plane

S fixes some flag (p, L), every Levi subgroup Ψ of ∆ is a compact group and the derivation

mappings Dp : Ψ→ GL(TpP ) and DL : Ψ→ GL(TLL) are injective.

Proof. We have mentioned already that the Lie topology τLie of Ψ is finer than theinduced topology of Ψ in ∆. Hence, the homomorphism Dp : (Ψ, τLie)→ GL(TpP ) is stillcontinuous. We will simply write DpΨ for the image of (Ψ, τLie) under the map Dp. Since

48 Chapter 6: The Structure of Collineation Groups

Ψ is semi-simple and connected, its image DpΨ is closed in GL(TpP ) by [32] or [36], andDpΨ acts completely reducibly on the tangent space TpP , cp. [90], Chap. 5, §2, Corollary2. Moreover, the group DpΨ leaves the l-dimensional subspace SL ∈ Sp invariant. Thusthere is a DpΨ–invariant complement W of SL in TpP . Let g denote the (real) Lie algebraof DpΨ.Assume that the group DpΨ is not compact. Then there is a non-trivial subalgebra a of g

which is R–diagonalizable, i. e. all operators adx for x ∈ a can be expressed by diagonalmatrices with respect to a suitable basis of g, see [90], Chap. 5, §4, p. 268–270. Moreover,the subalgebra a of g is ρ′-diagonalizable for every representation ρ′ : g → gl(V ), i. e.every element ρ′(x) for x ∈ a is represented by a diagonal matrix with respect to somefixed basis of V , see [90], Chap. 5, p. 277. We denote by ρ the representation of DpΨ onW and we put ρ′ : g → gl(W ) as the associated representation of the Lie algebra g ofDpΨ. Since a 6= 0 is ρ′–diagonalizable, there is a one-dimensional subspace R(v1, . . . , vl)of a which is mapped by the exponential mapping onto the one-parameter subgroupP = (exp tv1, . . . , exp tvl) | t ∈ R of ρ(DpΨ). In particular, the group P is not relativelycompact, because (v1, . . . , vl) 6= 0. By Weil’s lemma, we infer that P is a closed subgroup ofGL(W ) which fixes the one-dimensional subspaces Wi := (0, . . . , 0, exp tvi, 0, . . . , 0). SinceP acts on Sp, the group P fixes the unique spread element S of Sp that contains W1, say.Thus, the one-parameter group P fixes the two lines SL and S. Because DpΨ is semi-simple and leaves SL invariant, we have det ρ(g)

SL= det ρ(g)

S= 1 for every g ∈ P, and

consequently P is relatively compact by Theorem (A.1) of the Appendix, contrary to ourassumption. Hence, the group DpΨ is compact.In order to verify that Dp Ψ

is an injection, we recall that no almost simple factor of Ψ

lies in the kernel of Dp according to Corollary (5.30) and Theorem (5.31). Thus ker(Dp Ψ)

is contained in the center Z(Ψ) of Ψ. Since Z(Ψ) is discrete with respect to τLie, weconclude that Ψ is a covering group of DpΨ ∼= Ψ

/ker(Dp Ψ

). Recalling that Ψ is semi-

simple, this implies that (Ψ, τLie) is a compact group too, and we infer that the centerZ(Ψ) is finite, see [18], 0.6.10 or [114], 94.29. Because the induced topology on Ψ is coarserthan τLie, this shows that Ψ is compact as well and hence is closed in ∆. By Corollary(5.30), this shows that ker(Dp Ψ

)∩Z(Ψ) = 1l and so Dp Ψis injective. Finally, the map

DL : Ψ→ GL(TLL) is injective by Theorem (5.31).

The proof of the last proposition also shows the compactness of Levi complements ofcertain stabilizers in the case of compact projective translation planes:

(6.2) Corollary. Let P be a compact connected translation plane with translation axis

L∞, and let W be some line of P through the origin o. Let Φ be the stabilizer of the

collineation group of P that fixes both lines W and L∞ as well as the origin o. Then every

Levi complement of Φ is compact.


(6.3) Corollary. The center Z of a semi-simple subgroup Ψ of ∆ does not contain any

elation.

Proof. Assume that 1l 6= ζ ∈ Z is an elation with center c and axis A. Since ζ fixes nopoint outside of A according to Lemma (5.19) the group Ψ leaves invariant the axis A.Analogously, the group Ψ has to fix the center c (for otherwise we would have again a fixedpoint outside A). Hence the semi-simple group Ψ fixes the flag (c, A). By Proposition(6.1) this implies that Ψ is compact and thus has a finite center. In particular, the elationζ has finite order which is a contradiction to Corollary (5.24).

(6.4) Corollary. Let ∆ fix some flag (p, L) and let Ψ be a Levi subgroup of ∆. Then

Υ := Ψ n ∆[p,p] is a linear group and Ψ is a maximal compact subgroup of Υ.

Proof. By Proposition (6.1), the Levi subgroup Ψ is compact and linear. By [62], Chapt.18,(4.2), the group Υ is linear if and only if the radical ∆[p,p] is simply connected, which istrue by Corollary (5.30). Since Ψ is compact and ∆[p,p] contains no non-trivial compactsubgroup, we conclude that Ψ∩∆[p,p] = 1l, and hence Υ is a semi-direct product of Ψ and∆[p,p].

Now we turn to stabilizers of points.

(6.5) Corollary. If ∆ fixes some point of S, then every Levi subgroup Ψ of ∆ is a linear

group.

Proof. Let ∆ = ∆p for some point p and choose a Levi complement Ψ of ∆. Considerthe homomorphism Dp : Ψ → GL(TpP ). Assume that Ψ is not a linear group. Then thecenter Z(Ψ) is infinite and the kernel kerDp ⊆ Z(Ψ) is non-trivial. By Theorem (5.31) andCorollary (5.23) we have ker Dp = ∆[p,p] =

⋃A∈Lp

Γ[p,A]. Let ζ ∈ ker Dp \1l have L ∈ Lp

as its axis and choose some point q ∈ L \ p. If q∆ ⊆ p∨ q, then ∆ fixes the flag (p, p∨ q)and the claim follows from Proposition (6.1). If, on the other hand, we have q∆ 6⊆ p ∨ q,then r = qδ 6∈ p ∨ q for some δ ∈ ∆ implies that

(r ∨ q)ζ = (qδ ∨ q)ζ = qδζ ∨ qζ = qζδ ∨ qζ = qδ ∨ q = r ∨ q,

i. e. the line K = r ∨ q is fixed by ζ. But this is impossible, since K 6∈ Lp and ζ ∈ ∆[p,L].Hence Ψ is a linear group.

Applied to smooth projective planes, this corollary shows that a non-classical com-pact projective Moulton plane cannot be turned into a smooth projective plane. For aintroduction into Moulton planes the reader is referred to [100], §4 and [114], §34. The


first proof of this fact is due to D. Betten [3]. However, he gives an entirely different proofthat is not based on the collineation group of a Moulton plane.

(6.6) Corollary. Let P be a compact connected projective Moulton plane not isomorphic

to the real projective plane. Then there are no smooth structures on the set of points and

the set of lines such that P becomes a smooth projective plane.

Proof. The collineation group Γ of P is isomorphic to the product RΩ, where Ω is theuniversal covering group of PSL2R. The group Ω is not a linear group, see e.g. [90],Table 10. The group Γ fixes an antiflag (p0, L0). Thus P cannot be turned into a smoothprojective plane according to Corollary (6.5).

Remark. By removing the line L0 of the non-classical Moufang plane P we obtain a locallycompact affine plane A having the same collineation group as P. Arguing as in the proofof Corollary (6.6) we obtain an example of a locally compact connected affine plane whichcannot be turned into a smooth plane.

The next theorem shows that every Levi subgroup of a flag stabilizer of a smoothstable plane is classical, i.e. it is isomorphic to some subgroup of one of the flag stabilizersof the classical planes.

(6.7) Theorem. Let ∆ be a connected subgroup of the stabilizer of some flag (p, L). If

dim S ≤ 4, then ∆ is solvable. If dim S = 8 or 16, then every Levi subgroup Ψ of ∆ is

locally isomorphic to some closed subgroup of (Spin3R)3 or Spin8R, respectively.

Proof. Let Ψ be a Levi subgroup of ∆. By Proposition (6.1), the group Ψ is compactand the restrictions of Dp and DL to Ψ are injections. Since ∆ fixes the line L, the imageDp∆ leaves the corresponding tangent space SL invariant. Being a semi-simple group, ∆

acts completely reducibly on TpP ∼= R2l and thus leaves invariant some complement U ofSL in TpP . Every quasi-simple factor Σ of DpΨ acts almost effectively on at least one ofthe subspaces SL and U . Thus, if dim S ≤ 4, we conclude that Σ = 1l, since a compactquasi-simple group does not have any linear representation of dimension less than 3. HenceΨ is solvable if dim S ≤ 4. If DpΨ acts almost faithfully (i.e. with discrete kernel) on thesubspace W , then Ψ is locally isomorphic to some subgroup of SO(W ) ∼= SOlR (recallthat Ψ is compact). Thus, the assertion follows, because SO4R is locally isomorphic toSpin3R× Spin3R.Hence, let us assume that the kernel N of the action of DpΨ on W is not discrete. ThenN is a non-trivial semi-simple group having a quasi-simple factor Nι. Since Nι fixes everyelement of W , we conclude that Nι fixes the unique spread element S := S0∨w ∈ Sp foran arbitrary chosen point w ∈ W \ 0. If S 6= W , there is a S′ := S0∨w′ ∈ Sp withw′ ∈ W \ 0 different from S which again is fixed by Nι. By a suitable choice of some


basis e1, . . . , e2l of TpP , we may assume that SL = 〈e1, . . . , el〉, S = 〈el+1, . . . , e2l〉 andS′ = 〈e1 + el+1, . . . , el + e2l〉. Since Nι acts trivially on SL and leaves both S and S′

invariant, we infer that the action of Nι on SL coincides with the action on S, i.e. Nι actstrivially on TpP which is a contradiction. Thus we have shown that S = W and thatNι fixes at most two spread elements of Sp. The group DpΨ commutes with Nι and thusacts on the set SL, S of fixed elements of Nι with respect to its action on Sp. SinceDpΨ fixes SL, it has to fix S too. Now the assertion follows from the classification of 8-and 16-dimensional translation planes due to H.Hahl [49], (2.1), (1.6), (1.7) and (2.9) andthe fact that the group Spin8R contains every simply connected, compact, semi-simple Liegroup of dimension at most 9 as a subgroup.

The Stabilizer of Three Concurrent Lines

Let S = (P,L) be a smooth stable plane of dimension n = 2l = 2k, and let ∆ bea closed connected subgroup of its collineation group Γ. In this section we are going toconsider the subgroup Θ of ∆ which fixes three given lines K, L, and M through somepoint p. Our aim is to derive upper bounds for the dimensions of Θ. We start with twolemmas that contain useful formulas for dim ∆. The first lemma is due to Halder [59].

(6.8) Lemma. Let G be a locally compact Lindelof group acting on a separable metric

space M . For every element a ∈M with dim aM <∞ we have

dimG = dim aG + dimGa,

where GA denotes the stabilizer of G at a and aG is the orbit of a under G.

(6.9) Lemma. For q ∈ P we have

dim ∆q = dim kerDq + dim Dq∆q

and

dim Dq∆q ≤ dim A0,

where A is the connected component of the identity of Aut(Aq), the collineation group of

the tangent translation plane Aq, and Dq : ∆q → A0 is the derivation map. Moreover, we

have ker Dq = ∆[q,q].


Proof. According to Theorem (5.3) the stabilizer Λ := ∆q acts continuously on the linearLie group A0 (cp. Appendix) via composition Λ×A0 → A0 : (λ, α) 7→ Dq(λ)α. By Lemma(6.8) this implies

dim Λ− dim Λ1l = dim 1lΛ = dim DqΛ ≤ dim A0,

where the inequality follows from the monotonicity of the small inductive dimension. FromΛ1l = kerDq we conclude that

dim Λ− dim kerDq = dim DqΛ,

and hence the Lemma is proved.

We continue with a remark on the structure of elation groups Γ[p,p]. In case of acompact connected topological translation plane such groups are isomorphic to Rn, wheren is the dimension of the plane. For arbitrary compact connected topological planes itis known that Γ[p,p] is abelian and the connected component Γ1

[p,p] is isomorphic to somevector group Rk if there exist elations with different axes in Γ[p,p], see H. Salzmann [107],§1(1), H. Luneburg [86], I, 4.9, and M.Luneburg [87], I, Z1 and Z2. If, on the other hand,there do not exist elations with different axes, little is known in the topological case. Forsmooth stable planes, however, it can be shown that Γ1

[p,p] is always a simply connectedsolvable group, see Corollary (5.30).

(6.10) Proposition. If S is a smooth stable plane of dimension n with Γ as its collineation

group, and if dim Γ[p,p] = m > l for some point p ∈ P , then Γ[p,p] is commutative and its

connected component Γ1[p,p] is isomorphic to Rm.

Proof. A nontrivial elation γ ∈ Γ[p,A] fixes no point outside its axis A according to Lemma(5.19). Because the group Γ[p,A] is a closed subgroup of the Lie group Γ, we may applyLemma (6.8) in order to get

dim Γ[p,A] ≤ dimxΓ[p,A] + dim(Γ[p,A])x ≤ dimxΓ[p,A] ≤ dimL = l,

where x ∈ P \ A and L = p ∨ x. Hence there are two distinct lines A and A′ through p

such that Γ[p,A] 6= 1l 6= Γ[p,A′]. According to Lemma (5.15), this forces the group Γ[p,p] tobe commutative, and since Γ1

[p,p] is a simply connected Lie group according to Corollary(5.30), this shows that Γ1

[p,p]∼= Rm.

For n-dimensional smooth stable planes, the existence of an n-dimensional group Γ[p,p]

of elations has similar consequences as for n-dimensional compact topological projectiveplanes.


(6.11) Theorem. Let S = (P,L) be a smooth stable plane of dimension n. If there is

a point p such that dim Γ[p,p] = n, then S is an almost projective translation plane, i.e.

there is a compact connected (not necessarily smooth) projective plane P = (P ′,L′) with

translation axis W such that P = P ′ \ C, where C 6= W is some closed subset of W , and

L \ W ⊆ L′ ⊆ L.

Proof. Proposition (6.10) shows that Γ[p,p]∼= Rn, and [121], (3.6) yields that S is an almost

projective topological translation plane. Finally, we have C 6= W for S has a compact lineby Corollary (5.21), and there are no compact lines in a locally compact affine plane.

As an immediate consequence of the last two results the Theorem of Otte (see theIntroduction) together with Lemma (6.9) provides the following corollary.

(6.12) Corollary. If ∆ is a locally compact connected collineation group of a smooth

stable plane S = (P,L) of dimension n which fixes some point p ∈ P , then either S is an

almost projective translation plane or dim ∆ < n+ dim Dp∆ holds. If, in addition, S is a

smooth projective plane, then either S is classical or we have dim ∆ < n+ dim Dp∆.

We denote by A the group of continuous collineations of the tangent translation planeAp. Moreover, we put B := ASK ,SL,SM

as in the Appendix. Since the group Θ fixes thethree lines K,L,M , the image DpΘ fixes the three distinct lines SK , SL, and SM . Thuswe have Dp : Θ → B. By Theorem (A.1) of the Appendix, the group B has a compactsubgroup M of codimension at most 1, and M is a subgroup of SOlR. Now we are able tostart the investigation of dim Θ.

(6.13) Theorem. Let dim S = 2. Then either S is an almost projective translation plane

or dim Θ ≤ 2. If S is a smooth projective plane, either S ∼= P2R or dim Θ ≤ 2.

Proof. Since dim B ≤ 1 for l = 1, we infer that dim DpΘ ≤ 1. From Corollary (6.12) weconclude that either S is an almost projective translation plane (if S is a smooth projectiveplane, then S is even classical by Otte’s Theorem), or that dim Θ < 2+dim DpΘ ≤ 3 holds.

Before we move on to higher dimensional smooth stable planes, we will formulate arather elementary lemma which nevertheless will play an important role in our proofs.

(6.14) Lemma. Let (p,N) be some flag of S which is fixed by ∆. If α ∈ ∆[p] \ ∆[p,p]

centralizes the elation group ∆[p,N ], then ∆[p,N ] = 1l.

Proof. Let δ ∈ ∆[p,N ] \ 1l. Then α possesses an axis A (which of course does not passthrough p) by Corollary (5.22). Moreover, there is a point q ∈ A \ (A ∧ N) and a pointr ∈ (p ∨ q) \ q such that rδ = q. Thus we have

rα = rδαδ−1

= qαδ−1

= qδ−1

= r


which implies that α = 1l, see Lowen [79], (3.4) and [81], (3.4). This contradiction showsthat we have ∆[p,N ] = 1l.

If S is not an almost projective translation plane, we may suppose that dim∆[p,p] < n.If dim ∆[p,N ] = 0 for some line N ∈ Lp, then dim ∆[p,p] ≤ l by [114], 61.11. Sincethe classical compact projective planes have a collineation group that is transitive on(degenerated) quadrangles, we obtain from Lemma (6.9)

dim ∆ ≤ dim ∆[p,p] + dim Aut(Ap)SK ,SL,SM≤ l + dclass(n)− 2l = dclass(n)− l,

where dclass(n) denotes the dimension of the stabilizer of three concurrent lines in theclassical projective plane of dimension n. Hence we may always assume in our proofs that,say,

1 ≤ ∆[p,K] < l

holds.


or dim Θ ≤ 4. If S is a smooth projective plane, either S ∼= P2C or dim Θ ≤ 4.

Proof. According to the remark before Theorem (6.13), the image DpΘ is contained inthe group B. The group B has a compact subgroup M of codimension at most 1, and M isisomorphic to some subgroup of SO2R. From this we infer that DpΘ is a subgroup of C×.Furthermore we have either dim DpΘ ≤ 1 (which gives dim Θ < 4 + 1 = 5 by Corollary(6.12)) or DpΘ ∼= C×. In the latter case, the group DpΘ has a central involution ω = −1l.Clearly, this involution fixes every element of the tangent spread Sp. Hence, the involutionω is a reflection with center 0. Choose an element α ∈ D−1

p (ω). Since ω = −1l fixes everyline SH for H ∈ Lp, the collineation α fixes every line H through p. Thus, α is a centralcollineation with center p. According to Corollary (5.22), the existence of elations ensuresthat α has an axis A. The axis A does not pass through p, for otherwise we would haveDpα SA

= 1l which contradicts the fact Dpα = ω = −1l. By our general assumption we

have dim ∆[p,K] = 1 and thus ∆1[p,K]

∼= R holds according to Proposition (6.10). Sincethe connected group Θ fixes the line L, it acts on ∆1

[p,K] via conjugation. Because Θ isconnected, this action gives rise to a homomorphism Θ→ (GL1R)1 ∼= R>0. The homologyα is an involution, too, since we have α2 ∈ ker Dp = ∆[p,p] and ∆[p,p] ∩ ∆[p,A] = 1l byLemma (5.19). Hence α acts trivially on ∆1

[p,K]. Since Θ is connected, it acts also triviallyon ∆[p,K]. Now Lemma (6.14) shows that ∆[p,K] = 1l, and we finally obtain the inequalitydim Θ ≤ dim ∆[p,p] + dim DpΘ ≤ 2 + 2 = 4.

Before we are going to investigate the 8-dimensional case, we need to prove two morelemmas on Lie groups.


(6.16) Lemma. Let G be a Lie group and let H be a connected analytic solvable subgroup

of G. Then the closure H of H in G is a connected solvable Lie subgroup of G. If H 6= H,

then dimH < dimH.

Proof. By Hochschild [62], XVI, Theorem 2, there is an abelian Lie subgroup A of G suchthat H = H · A. Since the direct product H × A is solvable and because H is a quotientof the latter group, we conclude that H is solvable, too. If dimH = dimH, then H isan open subgroup of H. Clearly, the group H is connected, since H is connected. Hence,H = H follows, because an open subgroup is always closed.

The next lemma is a consequence of Richardson’s classification of compact connectedeffective Lie transformation groups G on S4 with dimG ≥ 2, see [98] or [114], 96.34.

(6.17) Lemma. The group SO4R has no closed subgroup isomorphic to SO3R× SO2R.


or dim Θ ≤ 11. If S is a smooth projective plane, either S ∼= P2H or dim Θ ≤ 11.

Proof. The image DpΘ is a subgroup of the stabilizer B, where the latter group is isomor-phic to some closed subgroup of GO4R ∼= SO4R× R according to Theorem (A.1).Assume that Θ is solvable. Then DpΘ is a solvable (not necessarily closed) subgroup ofB. The maximal dimension of a closed solvable subgroup of GO4R is 3. Hence we havedim DpΘ ≤ 3, and dim Θ < 8 + dim DpΘ ≤ 11 follows from Corollary (6.12).Thus we may suppose that Θ contains a nontrivial Levi-subgroup Ψ and that dim DpΘ ≥ 5holds. By Proposition (6.1), the group Ψ is compact and Dp Ψ

is a closed injection. Inparticular, this shows that DpΨ is a Levi subgroup of DpΘ. According to the structure ofGO4R, this implies that Ψ ∼= DpΨ is isomorphic to one of the groups

SU2C, SO3R, SO4R.

Let√

DpΘ denote the radical of DpΘ. Then we have DpΘ = DpΨ ·√

DpΘ and dim DpΘ =dim DpΨ + dim

√DpΘ. If DpΨ is locally isomorphic to SO3R, then dimDpΘ ≥ 5 implies

that dim√

DpΘ ≥ 2. Thus we infer that a maximal connected solvable subgroup H ofDpΘ has dimension at least 3. Together with the first part of the proof this shows that infact dim H = 3. By Lemma (6.16), a connected solvable subgroup of maximal dimensionis closed, whence the group DpΘ = DpΨ · H is closed in B. Recalling that B is a closedconnected subgroup of GO4R, we moreover get that H ∼= T2×R and

√DpΘ ∼= T×R. Let

ω denote the (unique) involution of√

DpΘ. Since√

DpΘ contains exactly one compactsubgroup T ∼= T, and because

√DpΘ is a normal subgroup of DpΘ, we infer that T lies in

the center of DpΘ. In particular, the involution ω is central in DpΘ. By Lemma (6.17), we


conclude that Ψ 6∼= SO3R. In other words, we have Ψ ∼= SU2C and thus DpΘ ∼= U2C× R.If DpΘ ∼= SO4R, we get either DpΘ ∼= SO4R or DpΘ ∼= GO4R (and in both cases the groupDpΘ is again a closed subgroup of B).Each of the groups

U2C× R, SO4R, GO4R

has (up to equivalence) a unique 4-dimensional irreducible real representation ρ and pos-sesses a unique central involution ω with ρ(ω) = −1l. Moreover, neither one of these groupsnor one of its Levi subgroups has a nontrivial real representation of dimension less then4. Furthermore, we have shown that Ψ is isomorphic to SU2C or to SO4R, and its center〈α〉 ∼= Z2 is mapped onto 〈ω〉, whence the collineation α is a homology with center p andsome axis A, see Lemma (5.19) as well as Lowen [79], (3.4), and [81], (3.4). By the remarkabove, the group Ψ acts trivially on ∆1

[p,K]∼= Rk, since we have k ≤ 3 by our general

assumption. Now we use Lemma (6.14) in order to conclude that ∆[p,K] is trivial. Thisfinally gives dim Θ ≤ dim ∆[p,p] +dim DpΘ ≤ 4+dim GO4R ≤ 4+7 = 11 and the theoremis proved.

For 16-dimensional stable planes we will need the following information about sub-groups of the orthogonal group SO8R, cp. Hahl [49], 2.8, and [114], 95.12.

(6.19) Lemma. Let K be a closed connected subgroup of SO8R which does not contain a

subgroup isomorphic to SO5R. Then either K is isomorphic to one of the groups Spin7R,

U4C, SU4C, or G2(−14), or dim K ≤ 13 holds. Moreover, we have K 6∼= U4C, if rkK ≤ 3.

Proof. By hypothesis, the group SO8R is not contained in K. In particular, we havedim K < dim SO8R. Since dim SO8R = 28, this implies that dim SO8R

/K ≥ 7, or, equiv-

alently, that dim K ≤ 21 holds, see [88], Chap. VI, Theorem 2. Thus we have to check,which compact groups of dimension at most 21 are subgroups of SO8R. According to theLevi decomposition we may write K = Ψ ·T, where Ψ is a semi-simple compact group andT ∼= Tr is a central torus subgroup. We assume first that K is quasi-simple. Using theclassification of quasi-simple Lie groups, see for example J. Tits, [123], or [14], and recallingthat K does not contain a subgroup isomorphic to SO5R, we infer that K is isomorphic toone of the groups

Spin7R, SU4C, G2(−14), Spin5R, PSU3C, SU3C, SU2C, SO3R.

Note that a group locally isomorphic to SU3H is a group of dimension 21, but such agroup does not have a faithful representation of dimension less than 12 and hence is not asubgroup of SO8R. The same argument excludes the group PSU4C from being a subgroupof SO8R. Since the groups that appear on the right hand side of G2(−14) have dimension


at most 10, the lemma is proved in the case of a quasi-simple group K. The following tableshows the dimensions of real irreducible representations of dimension at most 8, togetherwith their centralizers, cp. J. Tits [123] or [14].

group Spin7R SU4C G2(−14) Spin5R PSU3C SU3C SU2C SO3Rdimension 8 8 7 8 8 6 4,8 3,5,7centralizer R C R H R C H R

Since a semi-simple Lie group is completely reducible, we get the following centralizers inGL8R

Spin7R SU4C G2(−14) Spin5R PSU3C SU3C SU2C SO3RR C R2 H R C×GL2R H×GL4R or H R×GLmR

where we have m = 1, 3, 5.This shows that a closed subgroup of SO8R of dimension at least 8 which does not

contain SO5R is isomorphic to one of the groups

Spin7R,U4C, SU4C, G2(−14),

(Spin5R× Spin3R)/〈−1l〉, (Spin5R× T)

/〈−1l〉, Spin5R,

PSU3C, U3C× T, U3C, SU3C× T, SU3C,

SO4R× SO4R, SU2C× SO4R,

SO3R× SO4R.

Noting that the torus rank of U4C is 4, this proves the lemma.

Remark. Since a compact connected topological projective plane does not admit a groupΓ ∼= SO5R of collineations, see M. Luneburg [87], II, Korollar 1, or [114], 55.40, we haveexcluded this case in the lemma above.


or dim Θ ≤ 30. If S is a smooth projective plane, either S ∼= P2O or dim Θ ≤ 30.

Proof. According to Theorem (A.1) the group DpΘ is a subgroup of GO8R ∼= SO8R× R.No compact connected topological projective plane admits a group SO5R of collineations(see M. Luneburg [87], II, Korollar 1 or [114], 55.40). This rules out the case DpΘ ∼= SO8R.By Proposition (A.3) of the Appendix, the group DpΘ has torus rank at most 3. Thuswe may apply Lemma (6.19) to the closure H of DpΘ and we obtain that a maximalcompact subgroup K of H is either isomorphic to one of the groups Spin7R or SU4C, orthat dim K ≤ 14 holds. If dim K ≤ 14, then dim DpΘ ≤ dim H ≤ dim K + 1 ≤ 14 + 1 = 15


(again by Theorem (A.1)), and dimΘ < 16 + dim DpΘ ≤ 31 follows by Corollary (6.12).So we may assume that K is isomorphic to one of the quasi-simple groups Spin7R or SU4C.Now choose some Levi subgroup Ψ of Θ. Since Ψ is compact and because the restrictionDp Ψ

is an injection by Proposition (6.1), we get Ψ ∼= K as in the proof of Theorem (6.18).Both groups Spin7R and SU4C possess a central involution ω which is mapped onto −1lby the map Dp, and both groups do not have nontrivial real representations of dimensionless than 8 (see [123] or [14]). Hence we can apply Lemma (6.14) once again in order toget dim ∆[p,p] ≤ 8. This implies that

dim Θ ≤ dim ∆[p,p] + dim DpΘ ≤ 8 + dim K + 1 ≤ 9 + dim Spin7R = 9 + 21 = 30,

and the theorem is proved.

Summing up the results proved in this section we get the following theorems.

(6.21) Theorem. Let S be a smooth stable plane of dimension n with Γ as its group of

continuous collineations. Let Θ be a closed subgroup of Γ that fixes three lines through

some point p. Then either the plane S is an almost projective translation plane or we

have dim Θ ≤ dclass(n)− l, where dclass(n) denotes the dimension of the stabilizer of three

concurrent lines in the classical projective plane of dimension n.

(6.22) Theorem. Let P be a smooth projective plane of dimension n with Γ as its group

of continuous collineations. Let Θ be a closed subgroup of Γ that fixes three lines through

some point p. Then either the plane P is classical or we have dim Θ ≤ dclass(n)− l, where

dclass(n) denotes the dimension of the stabilizer of three concurrent lines in the classical

projective plane of dimension n.

59

Part Two

Global Theory

Chapter 7: Solvable Collineation Groups 61

CHAPTER 7

Solvable Collineation Groups

We continue the study of collineation groups with solvable groups ∆. As in thepreceding chapter we determine upper bounds for the dimensions of ∆. Our proofs rely onthe fact that up to duality a solvable collineation group of a compact connected topologicalprojective plane has a global fixed p, i.e. ∆ = ∆p holds. This, of course, is not valid forstable planes, whence this chapter is placed in the global part.

Let P = (P,L) be a smooth projective plane of dimension n = 2l = 2k. By theresults of Chapter 2, the tangent spaces TpP (p ∈ P ) and TLL (L ∈ L) together with thetangential spreads Sp and SL induced by the line pencil Lp and the point row L are locallycompact affine translation planes Ap and AL, respectively. Their projective closures aredenoted by Pp and PL and in both cases we put L∞ to be the line at infinity (which isalso the translation line). We start with a proposition on compact connected topologicaltranslation planes.

(7.1) Proposition. Let P be a compact connected translation plane of dimension 2k,where 1 ≤ k ≤ 4, and let L∞ be some translation axis. Let K,L 6= L∞ be two different

lines such that o = K ∧L does not lie on L∞. If A is a closed connected solvable subgroup

of the stabilizer ∇ of the triangle defined by K,L,L∞, then dim A ≤ 2 (if k = 1) and

dim A ≤ 2 + r (if k ≥ 2), where r ≤ k is the dimension of a maximal torus subgroup of A.

Proof. By Hahl [47], (4.2), the connected component ∇1 of ∇ possesses a maximal compactsubgroup K of codimension at most 2. Since a compact connected solvable Lie group isisomorphic to some torus group Tr, we infer that a closed solvable subgroup A of ∇1 hasdimension at most r+ 2. According to M. Luneburg [87], II, Satz 2 (see also [114], 55.37),we have r ≤ k. If k = 1, then P is isomorphic to the projective plane over the reals (seeSalzmann [100], (7.24)), and hence dim A ≤ 2 follows, because A is a subgroup of the groupof all regular real diagonal matrices.

(7.2) Corollary. Let P be a smooth projective plane of dimension 2k, let K,L be two

different lines, and put p = K ∧ L. If A is a closed connected solvable subgroup of the

stabilizer of K and L, then dim A ≤ A[p,p]+2 (if k = 1) and dim A ≤ A[p,p]+2+r (if k ≥ 2),

62 Chapter 7: Solvable Collineation Groups

where r ≤ k is the dimension of a maximal torus subgroup of A. If dim DpA ≥ 2, 4, 5, 6(for k = 1, 2, 3, 4), then dim A[p]

/A[p,p]

≥ 1.

Proof. We consider the derivation mapping Dp : A → A0, where A0 < Aut(Ap) is thestabilizer of the origin 0. Using Corollary (5.9) and Proposition (7.1) together with Lemma(6.9) we get for k = 1

dim A = dim ker Dp + dim DpA ≤ dim A[o,o] + 2

anddim A = dim ker Dp + dim DpA ≤ dim A[o,o] + r + 2

for k ≥ 2. From Hahl [47], (4.2), we know that DpA is a subgroup of a group R · Hl · Tl,where R ∼= R, Hl is a closed solvable subgroup of the homology group fixing the origin 0and the line at infinity L∞, and Tl ∼= Tlog2 l, see M. Luneburg [87], II, Satz 1. Moreover,we have Hl ≤ R× for k = 1, 4 and Hl ≤ C× for 2 ≤ k ≤ 3, see Buchanan–Hahl [26]. Inparticular, we infer that dim A[p]

/A[p,p]

≥ 1 holds, if dim DpA ≥ 2, 4, 5, 6 for k = 1, 2, 3, 4,respectively.

The following result deals with the structure of groups of central collineations. Thisproposition is also true in the case of compact connected topological projective planes (seeHahl [52]). Nevertheless we will give a proof which uses the special situation of smoothness.

(7.3) Proposition. Let P be a smooth projective plane and let p ∈ P . Let ∆ 6= 1l be

a connected locally compact collineation group of central collineations of P with center

p. Then the derivation mapping Dp maps ∆ onto a closed subgroup Υ of Aut(Pp), and

∆ = ∆[p,A] n ∆[p,p]∼= Υ n ∆[p,p] for some line A 6∈ Lp.

Proof. By Corollary (5.14), the image Υ = Dp∆ is a closed (connected) subgroup of H×.If Υ 6= 1l, then Υ contains a nontrivial central element ω. If ∆[p,p] = ker Dp = 1l, thenDp maps ∆ isomorphically (as a topological group) onto Υ and ∆ consists entirely ofhomologies. Moreover, the collineation δ := D−1

p (ω) is a homology of P with some line Aas its axis. Because δ is central in ∆, we infer that δ fixes every line of the orbit A∆. From∆ = ∆[p] we infer that the orbit A∆ does not contain any line incident with the centerp. This implies that A∆ = A, whence A is the axis of every homology of ∆, i.e. we have∆ = ∆[p,A]. Now let ∆[p,p] 6= 1l, and assume that ∆ 6= ∆[p,p] (otherwise there is nothingleft to prove). Fix a collineation δ ∈ D−1

p (ω). Then δ is a homology, since Dp(δ) 6= 1l.Again, we denote by A the axis of δ. Because the point p does not lie on the line A, we inferfrom ∆[p,p] 6= 1l that A is moved by ∆. By Hahl [52], dual of theorem (1.3), this impliesthat A∆ = A∆[p,p] . Now let γ ∈ Υ and choose ε ∈ D−1

p (γ). Then there is some (uniquelydetermined) elation τ ∈ ∆[p,p] such that Aε = Aτ . We conclude that ετ−1 ∈ ∆[p,A] holds.


Thus we have proved that for every γ ∈ Υ there exists exactly one ε ∈ D−1p (ω)∩∆[p,A]. In

particular, the restriction Dp : ∆[p,A] → Υ is a (closed) isomorphism, i.e. the groups ∆[p,A]

and Υ are isomorphic as topological groups. This proves the proposition.

The next lemma shows that a solvable Lie group cannot act transitively on an n-sphereif n ≥ 2, cf. also [114], 96.20, 96.23, or 96.28.

(7.4) Lemma. Let G be a solvable Lie group acting on an n-sphere S. If n ≥ 2, then G

does not act transitively on S.

Proof. Assume that G acts transitively on S. Let K be a maximal compact subgroup ofG. Then the connected component K1 of K is isomorphic to an r-dimensional torus groupT which acts transitively on S as well, see [88], Chap. V, Theorem 5.6. Thus we haveT

/Tx ≈ S for x ∈ S. Consider the following part of the long exact homotopy sequence

π1(Tx) −→ π1(T ) −→ π1(Sn).

This sequence reduces to Zr −→ Zs −→ 1l, because of π1(Tx) ∼= π1((Tx)1) ∼= Zr andn ≥ 2. This means that the morphism Zr → Zs is a surjection and hence we conclude thatr = s. This implies that T

/Tx is trivial, which is a contradiction. Hence G does not act

transitively on S.

By M.Luneburg [87], (G2), p. 22, the following four cases cover all possible situationswhich can occur for a closed connected solvable collineation subgroup ∆ of P.

(I) ∆ fixes some flag (p, L) and dim ∆[p,L] ≥ 1.(II) ∆ fixes some antiflag.

(III) ∆ fixes some Baer subplane.(IV) Every minimal connected commutative normal subgroup N of ∆ has some point

or line orbit xN which generates a subplane of P.

In the following theorems we will study each of these cases separately.

(7.5) Theorem. Let P be a smooth projective plane of dimension 2k. Assume that

the solvable group ∆ fixes some flag (p, L) such that t := dim ∆[p,L] ≥ 1 holds. Put

dI(1) = 3, dI(2) = 8, dI(3) = min 16, 13 + t, and dI(4) = min 29, 22 + t. Then either

the dimension of ∆ is at most dI(k), or P is isomorphic to the classical plane of dimension

2k.

Proof. Let K ∈ Lp \L. If dim ∆[p,p] = 2l, then P is a dual translation plane, whence P isclassical by Otte’s Theorem. Thus we may assume that dim ∆[p,p] < 2l. Let Dp : ∆K → A0


be the derivation mapping. By Lemma (6.8) and Corollary (7.2) we have

dim ∆ = dimK∆ + dim ∆K = dimK∆ + dim kerDp + dim Dp(∆K)

≤ l + dim ∆[p,p] + r + 2 ≤ l + (l + t) + r + 2 (∗)

< 3l + k + 2 ≤

5 if k = 110 if k = 217 if k = 330 if k = 4

Furthermore, we obtain from inequality (∗)

dim ∆ ≤ 2l + t+ k + 2 ≤ 13 + t if k = 3

22 + t if k = 4

which proves the theorem for k ≥ 3. Now we turn to the cases k = 1, 2. Choose somepoint b 6= p on L. Recalling that ∆ fixes the point p, we get ∆[b,b] = ∆[b,L]. Thus

dim ∆ = dim b∆ + dim ∆b ≤ l + dim kerDb + dim Db(∆b) ≤ l + dim ∆[b,L] + dim Db(∆b)

holds.

Case k = 1. According to M.Luneburg [87], III, Satz 1, there is a minimal connectedclosed abelian normal subgroup N of ∆ with N = N[p,L]. In particular, we have dim N = 1,and hence ∆ is (p, L)-transitive (note that L \ p is connected). If dim ∆[b,L] = 1, then Lis a translation axis of P and P is classical by Salzmann [100], (7.25). Thus we may assumethat dim ∆[b,L] = 0, whence dim ∆ ≤ l+dim Db(∆b). If we suppose that dim ∆ = 4, the lastinequality shows that dim Db(∆b) ≥ 3. Since Db(∆b) is a solvable subgroup of A0

∼= GL2R,we infer that Db(∆b) contains the group λ1l | λ > 0 which consists of homologies (withcenter 0 and axis L∞). By Lemma (7.3) the plane P has a one-dimensional group H

consisting of homologies with center b. Since H is commutative, there is a line K throughp such that H = H[b,K]. Since dim ∆[p,p] = 1 and dim ∆ = 4, the same argumentationshows that there is a nontrivial homology δ having p as its center and some line M as itsaxis. Since we may choose the point b arbitrarily on L \ p, we may assume that b does


not lie on M 6= L. Then (∆[b,K])δ = ∆[bδ,K] with bδ 6= b.

Figure 3.

Applying [114], Lemma 33.3 to H and ∆[bδ,K] 6= 1l, we get a (p,K)-transitive trans-lation group ∆[p,K]. Consequently, we have dim ∆[p,p] = 2, contrary to our assumptions.This shows that dim ∆ ≤ 3 if P is not isomorphic to P2R.

Case k = 2. If dim∆[p,p] ≤ 2, then dim ∆ ≤ 8 follows immediately by (∗). Hencewe may suppose that dim ∆[p,p] = 3 and thus ∆ is (p,K)-transitive for some line K ∈ Lp.Assume moreover that ∆ is 9-dimensional. From

9 = dim ∆ = dim kerDp + dim Dp∆ = dim ∆[p,p] + dim Dp∆ = 3 + dim Dp∆

we conclude that dim Dp∆ = 6. Thus the translation tangent plane Ap has an automor-phism group of dimension at least dim Dp∆+dim T = 10, where T ∼= R4 is the translationgroup of Ap. According to the classification of 4-dimensional compact translation planes(see [114], Chapt. 73) this implies that Ap is isomorphic to the affine complex plane A2C.In particular, we have Dp∆ ≤ GL2C. Since Dp∆ fixes the spread element SL, we have upto conjugacy

Dp∆ ≤ Υ :=(

a b

c

) ∣∣∣∣ a, b, c ∈ C, ac 6= 0≤ GL2C

and from dim Dp∆ = 6 = dim Υ we conclude that Dp∆ = Υ, since both Dp∆ and Υ areconnected. Thus

(Dp∆)[p] =(

a

a

) ∣∣∣∣ a ∈ C×

is two-dimensional, and hence dim ∆[p]

/∆[p,p]

≥ 2 follows. By Lemma (7.3) this impliesthat ∆ contains a (commutative) two-dimensional group H consisting of homologies withcenter p and some axis M . Since K \ p,K ∧M is connected, this shows that H is(p,M)-transitive. Using the dual statement of Pickert [96], 3.1.11, p. 67, we infer that P

is (K ∧M,M)-transitive. Since p 6= K ∧M this implies that P is (K,K)-transitive, see


Pickert [96], 3.1.12.1, p. 68. In particular, the plane P is a translation plane. Hence P isclassical by Otte [93] or by Grundhofer, Hahl [45]. This finishes the proof of the theorem.

(7.6) Lemma. Let P be a smooth projective planes of dimension n = 2l = 2k with k ≥ 2.

If the solvable group ∆ fixes some point p, and if dim ∆[p,p] = 0, then

dim ∆ ≤ 2k−1 + rk ∆ + 2 ≤ 2k−1 + k + 2 ≤

6 if k = 29 if k = 314 if k = 4

.

Proof. By Lemma (7.4), the group ∆ is not transitive on Lp. Hence either the group ∆

fixes some line K ∈ Lp or there is a non-trivial orbit K∆ of dimension at most l − 1. IfK∆ = K, we may apply Corollary (7.2) in order to get

dim ∆ = dim ∆K = dimK ′∆K + dim ∆K,K′ ≤ l + k + 2

for any line K ′ ∈ Lp \ K. Now let K∆ be a non-trivial orbit of dimension at mostl − 1. Since ∆ is solvable, there are normal subgroups P and Π of ∆ with Π < P andKΠ = K 6= KP such that P

/Π is a minimal normal subgroup of ∆

/Π. Then P

/Π is

isomorphic to one of the abelian groups T, R, or R2. Hence, the centralizer Ξ := Cs(P/Π) =

ξ ∈ ∆∣∣ ∀ ρ ∈ P : Πρξ = Πρ

has codimension at most 2 in ∆ and contains the group Π.

For Kξ = K we have Kρξ = Kξ−1ρξ ∈ KΠρ = Kρ, whence ΞK acts trivially on the orbitKP. In particular, the stabilizer ΞK fixes a second line K ′ ∈ Lp. Applying Proposition(7.1) and Corollary (7.2) to ΞK = ΞK,K′ , and using Lemma (6.8) as well as a theorem ofK.Nagami [89], (2.1), we get

dim ∆ ≤ dim Ξ + 2 ≤ dimKΞ + dim ΞK + 2 ≤ (l − 1) + (r + 2) + 2 ≤ l + k + 3.

If dimKΞ ≤ l−2 or if r ≤ k−1, the computation above shows that dim ∆ ≤ l+k+2 holds,and the lemma is proved. Hence we may assume r = k and dimKΞ = l − 1. Since thekernel of the tangent translation plane Ac is isomorphic to some closed connected subgroupof the the multiplicative group of the quaternions (see T.Buchanan and H. Hahl [26]), andbecause r = k ≥ 2, there is a torus group T ∼= T of ΞK that acts faithfully on Lp. ByE. Floyd [33], Theorem 5.2, the dimension of the fixed point set of an element δ ∈ T oforder 3 is of codimension at least 2. In particular, the collineation δ acts non-trivially onKΞ, because the latter set has codimension 1. If we choose K∗ ∈ KΞ in such a way thatK∗δ 6= K∗, then dimK∗T = 1. Consequently, the torus rank of ΞK∗ is at most r−1 (recallthat r is the torus rank of Ξ is r). Let ξ ∈ Ξ with Kξ = K∗. From

ΞK∗ = ΞKξ = (ΞK)ξ = (ΞK,K′)ξ = ΞK∗,K′ξ


we see that we can apply Proposition (7.1) to ΞK∗ ; we get

dim ΞK∗ ≤ (r − 1) + 2 ≤ (k − 1) + 2 = k + 1.

This finally implies that

dim ∆ ≤ dim Ξ + 2 ≤ dimK ′Ξ + dim ΞK′ + 2 ≤ (l − 1) + (k + 1) + 2 = l + k + 2,

and the proof is completed.

We turn to the antiflag case.

(7.7) Theorem. Let P be a smooth projective plane of dimension 2k. Assume that the

solvable group ∆ fixes some antiflag (p, L). Put dII(1) = 3, dII(2) = 6, dII(3) = 9, and

dII(4) = 14. Then either the dimension of ∆ is at most dII(k), or P is isomorphic to the

classical plane of dimension 2k.

Proof. Using Corollary (7.2) and noting that ∆[p,p] is trivial, we get

dim ∆ = dim kerDp + dim Dp∆ = dim ∆[p,p] + dim Dp∆ = dim Dp∆.

Case k = 1. Since Dp∆ is a solvable subgroup of A0∼= GL2R, where A0 is the stabilizer

of the origin 0 in Aut(Ap), we conclude that dim Dp∆ ≤ 3.

Case k ≥ 2. For k ≤ 3 we could give a similar proof as in the case k = 1 using theclassification of 4-dimensional and 8-dimensional translation planes. However, the classifi-cation of 16-dimensional compact translation planes does not give us useable results as inthe previous cases. Instead, we use Lemma (7.6). We can apply this lemma, because ∆

contains no elation with center p, since ∆ fixes the antiflag (p, L).

(7.8) Theorem. Let P be a smooth projective plane of dimension 2k. Put dIV (1) = 2,

dIV (2) = 4, dIV (3) = 9, and dIV (4) = 14. If every minimal connected commutative closed

normal subgroup N of ∆ has some orbit xN (x ∈ P ∪ L) which generates a subplane E of

P, either the dimension of the solvable group ∆ is at most dIV (k), or P is isomorphic to

the classical plane of dimension 2k.

Proof. By M. Luneburg [87], Anhang B, Satz 3, the group ∆ fixes some point or some line.Up to duality we may assume that ∆ fixes some point p. Consider the derivation mapDp : ∆→ A0.

(a) dim ker Dp = 0: If dim kerDp > 0, there is a minimal connected commutative closednormal subgroup M in kerDp (namely a suitable commutator subgroup of (kerDp)1, sinceker Dp is a solvable group). By Corollary (5.9), we have ker Dp = ∆[p,p]. Hence the normal


subgroup M cannot have an orbit (neither in P nor in L) that generates a subplane of P.This contradicts the assumptions of the theorem, and thus we have dim kerDp = 0.

From (a) and Lemma (6.9) we obtain

dim ∆ = dim kerDp + dim Dp∆ = dim Dp∆.

(b) dim ∆[p] = 0: Clearly, ∆[p] is a normal subgroup of ∆. Since the connected componentof the identity ∆1

[p] is a characteristic subgroup of ∆[p], the connected group ∆1[p] is a

normal subgroup of ∆, too. Assume that ∆1[p] 6= 1l. According to part (a), the group ∆1

[p]

contains no elations, i.e. ∆1[p] consists of homologies only. Since ∆ is solvable and because

locally compact groups of homologies are isomorphic to some closed connected subgroup ofH× (see Corollary (5.14)), we conclude that ∆1

[p] is commutative (in fact it is a subgroupof C×). Thus, the group ∆ has a minimal connected commutative closed normal subgroupM consisting entirely of homologies. As in part (a), we infer that M cannot have an orbitthat generates a subplane of P. This contradiction shows dim ∆[p] = 0.

Case k = 1. If dim Dp∆ ≥ 3, then Dp∆ contains the subgroup λ · 1l | λ ∈ R, λ > 0. ByLemma (7.3), this implies that dim ∆[p] ≥ 1 which contradicts part (b). Thus dimDp∆ ≤ 2follows.

Case k = 2. The group ∆ cannot act transitively on the line pencil Lp ≈ Sl by Lemma(7.4). So let L∆ be an orbit of dimension less than l. If L is a fixed line of ∆ then we havedim ∆ ≤ 2 + dim ∆L,K for every K ∈ Lp \ L. If, on the other hand, L is not fixed by ∆,we choose K ∈ L∆ \ L and we get again dim ∆ ≤ 2(l − 1) + dim ∆L,K = 2 + dim ∆L,K .From part (b) and Corollary (7.2) we obtain

dim ∆ = dim Dp∆ ≤ 2 + 2 = 4.

Cases k = 3, 4. As in the previous theorem we apply Lemma (7.6) in order to prove theassertion.

The remaining case that may occur for solvable groups can be deduced from the threepreceding theorems by means of induction if we recall that the kernel of the action of ∆

on a Baer subplane is at most one-dimensional (zero-dimensional in the case dim P = 4),see M. Luneburg [87], I, V.5.

(7.9) Theorem. Let P be a smooth projective plane of dimension 2k ≥ 4. Assume that

the solvable group ∆ leaves some Baer subplane B invariant. Put dIII(2) = 3, dIII(3) = 9,

and dIII(4) = 17. Then either the dimension of ∆ is at most dIII(k), or P is isomorphic

to the classical plane of dimension 2k.


Summing up the theorems of this section we get the following table.

dim P flag antiflag Baer subplane otherwise2 3 3 — 24 8 6 3 48 min 16, 13 + t 9 9 916 min 29, 22 + t 14 17 14

By considering the automorphism groups of the four classical planes it can easily beseen that the upper bounds in every case excluding the ”otherwise” case are sharp bounds.

70 Chapter 8: Subplanes of Smooth Projective Planes

CHAPTER 8

Subplanes of Smooth Projective Planes

Let P = (P,L,F) be a projective plane and let P′ = (P ′,L′,F′) be a subplane ofP, i.e. P ′ ⊆ P , L ∈ L′ ⊆ L if and only if L is incident with at least two points of P ′,and F ⊆ F′. We will call elements (i.e., points, lines and flags) of P′ inner elements ofP. An element of P which is not inner is called an outer element. A subplane P′ of atopological (smooth) projective plane P is called closed (smooth) if P ′ is a closed subset(a closed smooth submanifold) of P . For a smooth submanifold U of a smooth manifoldM we require that the inclusion mapping ι : U → M is a smooth embedding. Clearly, asubplane P′ of a topological plane P is topological with respect to the induced topologies.Thus, by [114], Prop. 41.7, we have the following lemma.

(8.1) Lemma. Let P = (P,L,F) be a compact projective plane and let P′ = (P ′,L′,F′)be a subplane of P. The following statements are equivalent.

(i) P′ is a closed subplane.

(ii) L′ is a closed subset of L.

(iii) F′ is a closed subset of F.

(iv) The set of inner points of an inner line L is closed in the point row L.

(v) The set of inner lines through an inner point p is closed in Lp.

An analoguous result for the smooth case is also true.

(8.2) Lemma. Let P = (P,L,F) be a smooth projective plane and let P′ = (P ′,L′,F′)be a subplane of P. The following statements are equivalent.

(i) P′ is a smooth subplane.

(ii) L′ is a smooth submanifold of L.

(iii) F′ is a smooth submanifold of F.

(iv) The set I of inner points of an inner line L is a closed smooth submanifold of the

point row L.

(v) The set I of inner lines through an inner point p is a closed smooth submanifold

of Lp.

Proof. Let P′ be a smooth subplane of P. Since the intersection of two smooth submanifoldsis again a smooth submanifold, we infer that statement (iv) holds. The same argument

Chapter 8: Subplanes of Smooth Projective Planes 71

shows that (ii) implies (v). Since the perspectivities mapping a point row to a line pencilare diffeomorphisms (see Corollary (1.7)), statements (iv) and (v) are equivalent. Sincethe line space L′ is locally diffeomorphic to the product of two line pencils (see Lemma(1.12)(i))), assertion (v) implies statement (ii). Dually, (i) follows from (iv). Assuming(i) and (ii), we get from Theorem (1.14) that F′ is a closed smooth submanifold of theproduct manifold P ′ × L′. Thus F′ = F ∩ (P ′ × L′) is a closed smooth submanifold ofF. Conversely, let p be an inner point and let I ⊆ Lp be the set of inner lines. Since theprojections πP : F → P and πL : F → L are submersions (Corollary (1.15)), the inverseimage B := π−1

P (p) = p × Lp is a closed smooth submanifold of F which is mappeddiffeomorphically onto Lp by πL (Corollary (1.16)). The subset B′ := B ∩ F′ is a closedsmooth submanifold of B, because F′ is a closed smooth submanifold of F. Applying thediffeomorphism πL B

and noting that πL(B) = I holds, this implies (v).

The next proposition shows that for smooth projective planes the classes of closedsubplanes and of smooth subplanes in fact coincide.

(8.3) Proposition. A subplane P′ = (P ′,L′,F′) of a smooth projective plane P =(P,L,F) is a smooth subplane if and only if it is a closed subplane.

Proof. Let P′ be a smooth subplane of P. Then the point space P ′ of P′ is compact (see,e.g, [114], 41.7)), whence it is a closed subset of P . Conversely, we may assume by Lemma(8.1) that the set I of inner points of an inner line L is a closed (and connected) subsetof the point row L. Hence I is a (not necessarily smoothly embedded) submanifold of L.According to Lemma (8.2) we have to show that I is a smooth submanifold of L. For thatpurpose we show that the inclusion map ι : I → L has constant rank. Fix some pointx ∈ I and denote by r the rank of ι at x. Let y ∈ I. Choose a projectivity ρ : L → L

that maps y to x and that leaves I invariant. Such a projectivity exists, because P′ isa subplane of P. The map ρ is a diffeomorphism by Corollary (1.7). Hence we haverky ι = rkx(ι ρ) = rkx(ρ ι) = rkx ι = r, which proves the proposition.

The following easy lemma will be the key for proving that a smooth subplane of asmooth projective plane induces a closed subplane of the tangent plane Ao at some innerpoint o. Recall that any two lines of a smooth projective plane always meet transversally,see [21] or Corollary (1.13).

(8.4) Lemma. Let P = (P,L,F) be a smooth projective plane and let P′ = (P ′,L′,F′)be a smooth subplane of P. Then every outer point row L meets P ′ transversally.

Proof. The point set P ′ is a smooth submanifold of P by hypothesis. Let p′ be some pointof P′ and let L ∈ Lp′ be an outer line. The point rows of P′ through p′ cover the point

72 Chapter 8: Subplanes of Smooth Projective Planes

space P ′ and every tangent vector v ∈ Tp′P ′ is contained in a (unique) subspace Tp′L′ forsome line L′ ∈ L′ through p′. Choose a tangent vector v ∈ Tp′P belonging to both L andP ′. Then there is a line L′ ∈ L′ such that v ∈ Tp′L′. Since L and L′ are different, theyare transversal by Breitsprecher, [21], see also Corollary (1.13). This shows that v = 0 andhence P ′ is transversal to L.

(8.5) Proposition. Let P = (P,L,F) be a smooth projective plane and let P′ =(P ′,L′,F′) be a smooth subplane of P. For every inner point o of P the incidence struc-

ture A′o = (ToP ′, S′o) defines a closed subplane of the tangent plane Ao = (ToP, So), where

S′o = ToK | K ∈ L′o. The subplane A′o is a Baer subplane of Ao if and only if P′ is a

Baer subplane of P.

Proof. We have to check that ToK ∩ ToP ′ | K ∈ L′o is a spread on ToP ′. Since anytwo tangent spaces ToK intersect transversally by [21], it remains to verify that ToP ′ iscovered by

⋃K∈L′o

ToK. Let x ∈ ToP ′ \ 0. Then there is a unique tangent space ToMof some line M ∈ Lo with x ∈ ToM (remember that So = ToK | K ∈ Lo is a spread ofToP ). Thus we have x ∈ ToK ∩ ToP ′ and so x ∈

⋃K∈L′o

ToK, because outer lines aretransversal to P ′ according to Lemma (8.4). In order to show that A′o is a subplane of Ao

we have to prove that ToK ∈ S′o if and only if ToK ∩ P ′ 6= 0. But this follows againfrom the fact that exactly the outer lines of P through o are transversal to P ′. Since P ′ isa closed smooth submanifold of P and because o is an inner point, the tangent subspaceToP ′ is closed in ToP . Hence we have shown that A′o is a subplane of Ao which is closed byLemma (8.1). A subplane of a topological plane of dimension 2l is a Baer subplane if andonly if the dimension of the subplane is l, see e.g., Salzmann [106], 1.4 or [114], 41.11. Thusthe last part of the proposition follows from dim ToP ′ = dimP ′ and dim ToP = dimP .

Chapter 9: Flat Projective Planes 73

CHAPTER 9

Flat Projective Planes

Let P = (P,L,F) be a compact 2-dimensional (or flat) topological projective planewith ∆ as its collineation group. By Salzmann [100], the group ∆ is a locally compacttransformation group with respect to the compact-open topology on both the set P ofpoints and the set L of lines. Moreover, every collineation of P is a continuous map onboth P and L. If dim ∆ ≥ 5, then P is isomorphic to the real projective plane P2R and∆ ∼= PSL3R, see [100], (5.5). In case 2 ≤ dim ∆ ≤ 4 all flat projective planes are known,[100], [37]— [43], [97]. If dim ∆ = 4, then P isomorphic to a proper (i.e. non-isomorphic toP2R) Moulton plane and ∆ ∼= Ω ·P, where Ω is the universal covering group of PSL2R andP ∼= R, [100], (5.6). It has already been proved in Corollary (6.6) that a proper Moultonplane cannot be turned into a smooth flat projective plane.

Thus we may concentrate on the case where ∆ is 3-dimensional. If dim ∆ = 3,there are three families of flat projective planes. One of these families consists of theskew hyperbolic planes. Such a plane admits a collineation group H that is isomorphicto PSL2R. The group H fixes no point and no line. If ∆ has fixed elements, then ∆ issolvable and the number f of fixed points (and of fixed lines, too) equals dim ∆

/∆′, where

∆′ denotes the commutator subgroup of ∆. Moreover, the only possible values for f are1 and 2. If f = 1, then P is isomorphic to some skew parabola plane. In case f = 2, theplane P is defined over some Cartesian field. See Salzmann [100], §5 for more details.

Throughout this chapter let P = (P,L,F) be a smooth flat projective plane with ∆ asits collineation group. We assume that dim ∆ = 3. The group ∆ is a Lie transformationgroup with respect to the compact-open topology on both the set P of points and the setL of lines, see Betten [3], Satz 1 and Corollary (4.4). Moreover, every collineation of P iscontinuous and thus is even a smooth map on P and L by [3], Satz 2 or Theorem (4.3),cp. [15].

Our main theorem which we are going to prove in this chapter is as follows.

(9.1) Theorem. Every smooth flat projective plane that admits a collineation group of

dimension at least three is isomorphic to the real projective plane.

74 Chapter 9: Flat Projective Planes

Skew Hyperbolic Planes

The skew hyperbolic planes have been found by H. Salzmann in 1962, see [99] and[114], §35 for a detailed description of these planes. Here we will focus on those propertiesof skew hyperbolic planes which are necessary to verify that the only skew hyperbolicplane that can be turned into a smooth projective plane is the projective plane over thereal numbers.

Let P2R = (P,L) be the real projective plane represented by the real vector space R3.We consider the indefinite quadratic form q(x, y, z) = x2 + y2− z2 on R3 and we take H asthe connected component of the hyperbolic motion group PO3(R, q). Then H is isomorphicto PSL2R and leaves the quadric Q :=

Rv

∣∣ v ∈ R3 \ 0 , q(v) = 0

invariant. Fix a realnumber r. The set Pr of points of the skew hyperbolic plane Pr = (Pr,Lr) is just thepoint set P of P2R, possibly with a different smooth structure on it. In order to describethe set of lines Lr, we put

Lr := R(x, 0, 1) | − 1 ≤ x ≤ 1 ∪R(x, y, z)

∣∣ y2 = r2(x2 − z2), rxy ≥ 0

andLr := L ∈ L | |L ∩Q| ≤ 1 ∪ Lηr | η ∈ H .

The plane Pr is isomorphic to the real projective plane if and only if r = 0. With re-spect to the affine point set PA =

(x, y, 1)

∣∣ (x, y) ∈ R2

of Ar the quadric Q is justthe unit circle C and the affine part Ar of the line Lr consists of three parts: thestraight line segment [−1, 1] × 0 and the two semi-hyperbolas

(x, y)

∣∣ y = r√x2 − 1

and

(x, y)

∣∣ y = −r√x2 − 1

, see Figure 4.

Figure 4.

In order to describe the lines through the point p = R(1, 0, 1) we take a look at the stabilizerHp. This stabilizer contains a closed one-parameter subgroup

N :=

νs =

2− s2 2s −s2−2s 2 −2ss2 −2s 2 + s2

∣∣∣∣∣∣ s ∈ R


which acts sharply transitive on the line pencil Lp through p excluding the vertical lineY =

R(x, y, x)

∣∣ (x, y) ∈ R2 \ (0, 0). Thus we may parametrize the set Lp \Y by the

map χ : s 7→ Lνsr : R→ Lp \ Y . Figure 5 shows several lines of this line pencil.

Figure 5.

Similarly, the lines through the origin o = R(0, 0, 1) can be obtained as the orbit of Lrunder the rotation group

O :=

ωs =

cos s − sin ssin s cos s

1

∣∣∣∣∣∣ s ∈ R

.

Let X± :=R(x,±y, y)

∣∣ (x, y) ∈ R2 \ (0, 0)

= R(sin t,± cos t, cos t) | t ∈ (−π, π] bethe horizontal lines through the points R(0,±1, 1). We consider the projectivities ρ1 =[X−, o,X+] and ρ2 = ρ21ρ22 = [X+, p][p,X−] = [X+, p,X−] as well as their compositionρ = ρ2 ρ1.

Assume now that P = Pr is a smooth projective plane. Then every projectivity is adiffeomorphism by [21] and Theorem (1.7). In particular, this is true for the projectivity ρ.We will show that this implies r = 0, whence Pr is isomorphic to the real projective plane.The smooth structures of Pr and Lr are given by the smooth structure of one single line(or of one single line pencil) and the set of all perspectivities, see [21]. Since every line is


diffeomorphic to the circle S1, we may choose the ordinary smooth structure D on the lineX− which is induced by the ordinary smooth structure DP2R of the point set of P2R. Notethat the smooth structure on the point set Pr may not agree with the smooth structureDP2R of P , the point set of P2R, although they are diffeomorphic to each other (accordingto the classification of surfaces). The point is that this diffeomorphism may not map pointrows onto point rows. Being a diffeomorphism between X− and X+, the projectivity ρ1

carries the smooth structure of X− to the smooth structure of X+. In terms of projectivecoordinates this map is given by

X− → X+ : R(sin t,− cos t, cos t) 7→ R(− sin t, cos t, cos t),

because the line Lr is symmetric with respect to the origin o and every line through o canbe written as Lωr for some rotation ω ∈ O. In particular, the smooth structure on the lineX+ coincides with the induced smooth structure of DP2R on X+.We are going to determine the projectivity ρ2. Put wr := R(xr, yr, zr) = ρ−1

2 (R(0,−10, 1)).Then we have wr 6= R(1, 1, 1), because the line Y is the unique line joining R(1, 0, 1) withR(1, 1, 1). We will show that ρ2 is not differentiable at wr unless r = 0. As we havementioned before, the line pencil Lp \ Y is parametrized by the map χ. With respect tothis parametrization, the inverse map ρ−1

21 can be written as

s 7→ R

2− s2 (1− αr(s+ 1))− 2 r s√

1− αr(s+ 1)2

2 s (1− αr(s+ 1)) + 2 r√

1− αr(s+ 1)2

−s2 + (2 + s2)αr(s+ 1)− 2 r s√

1− αr(s+ 1)2

t

in some neighborhood of 1, where

αr(t) =−1 + 4 t2

√r2 (1 + r2) + t4

1 + 2 t2 + 4 r2 t2 + t4.

From

α′r(t) =4 t

(1 + 2 r2 + 2 (t4 − 1)

√r2 (1 + r2) + 2 t2 + t4 + 2 r2 t4

)(1 + 2 t2 + 4 r2 t2 + t4)2

we infer that α′r(2) 6= 0, whence the map αr : R → R is a local diffeomorphism in someneighborhood of 1 for every real parameter r. Moreover, an easy calculation shows that

αr(2) =15 + 16

√r2 (1 + r2)

25 + 16 r2< 1

for any r ∈ R. Thus the map s 7→√

1− αr(s+ 1)2 is a local diffeomorphism around 1 andwe infer from the representation of ρ−1

21 that this perspectivity is a local diffeomorphism in


some neighborhood of 1 with respect to the parametrization χ. Now we investigate the per-spectivity ρ22 which we will consider as a mapping R→ X− by using the parametrizationχ again. In some neighborhood of 1 we get the formula

ρ22 : s 7→ R

2− s2 (1− βr(1− s))− 2 r s√

1− βr(1− s)22s (1− βr(1− s)) + 2 r

√1− βr(1− s)2

s2 − (2 + s2)βr(1− s)− 2 r s√

1− βr(1− s)2

t

,

with

βr(t) =

−1−4 t2

√r2 (1+r2)+t4

1+2 t2+4 r2 t2+t4 if t ≥ 0

αr(t) if t ≤ 0.

Obviously, both components of βr are differentiable at t = 0 and we have

β′r(t) =

4 t

(1+2 r2+2 (1−t4)

√r2 (1+r2)+2 t2+t4+2 r2 t4

)(1+2 t2+4 r2 t2+t4)2

if t ≥ 0

α′r(t) if t ≤ 0.

Since both parts of β′r vanish at t = 0, the map βr is differentiable at t = 0. Fromβr(0) = −1 and limt→∞ βr(t) = 1 = limt→−∞ βr(t), we infer that βr has a local minimumat t = 0 for any r ∈ R. Hence the map δ : s 7→

√1− βr(1− s)2 is not differentiable at

s = 1, because the limits on the left and on the right of its differential quotients do notagree at s = 1:

lims→1+

δ(s)s− 1

> 0 but lims→1−

δ(s)s− 1

< 0.

In particular, none of the coordinate functions fr, gr and hr of ρ22 is differentiable unlessr = 0. However, in order to check that ρ22 is not differentiable at s = 1, we have tostudy the affine representation of ρ22, because non-differentiability cannot be read off thehomogeneous coordinates. Thus we study the behaviour of the quotient

γr : s 7→ fr(s)hr(s)

.

Putting

f ′r(1±) := lims→1±

fr(s)− fr(1)s− 1

we obtain

f ′r(1±)− h′r(1±) = lims→1±

(fr(s)− fr(1))− (hr(s)− hr(1))s− 1

= lims→1±

2(1 + βr(1− s))1− s

= lims→1±

2β′r(1− s)

= 0.


Since fr(1) = 0 and hr(s) = 4 holds, and because the coordinate functions fr and hr arecontinuous wherever they are defined, the last equation yields

lims→1±

γ′r(s) = lims→1±

f ′r(s)hr(s)− h′r(s)fr(s)hr(s)2

= lims→1±

f ′r(s) lims→1±

hr(s)− fr(s)hr(s)2

=14

lims→1±

f ′r(s).

If r 6= 0 the coordinate function fr is not differentiable at s = 1, whence we have f ′r(1+) 6=f ′r(1−) and thus γ′r(s+) 6= γ′r(s−) follows. This shows that γr and hence ρ22 is notdifferentiable at s = 1 if r 6= 0. In particular, the skew hyperbolic plane Pr is not smoothfor r 6= 0. Since the real projective plane P0 is a smooth projective plane, see e.g. J.Otte,[93], (8.13), we have proved the following theorem.

(9.2) Theorem. The only smooth skew hyperbolic projective plane is the real projective

plane. The skew hyperbolic projective planes Pr 6∼= P2R are not even C1-differentiable.

Skew Parabola Planes

Let c, d ∈ R with 0 < c ≤ 1 < d and consider the skew parabola

f : R→ R : x 7→xd if x ≥ 0c(−x)d if x ≤ 0

.

An affine skew parabola plane Ac,d has point set R2 and the set of lines consists of thevertical lines la = a×R together with the images of the graph of f under the shift groupΣ = (x, y) 7→ (x+ s, y + t) | s, t ∈ R. The line defined by the graph of f is denoted byl0,0. We set ls,t = lσ0,0, where σ : (x, y) 7→ (x+ s, y + t). Clearly, we have

ls,t =

(x, y) ∈ R2

∣∣∣∣ y =

(x− s)d + t if x ≥ sc(s− x)d + t if x ≤ s

.

We denote by Pc,d the projective completion of Ac,d. With respect to this completion,we write l∞ for the line at infinity and (∞) ∈ l∞ as the common point of the verticalsla. The common point of ls,t | t ∈ R is denoted by (s). If (c, d) 6= (1, 2), the connectedcomponent ∆ of the collineation group of Pc,d is three-dimensional and can be describedas

∆ =(x, y) 7→ (rx+ s, rdy + t)

∣∣ s, t, r ∈ R, r > 0.


The group ∆ fixes the flag ((∞), l∞). The plane Pc,d is isomorphic to P2R if and only ifc = 1 and d = 2 (this gives the parabola model of the real projective plane). For moreinformation on skew parabola planes see [114], §36.

The planes Pc,d are compact topological projective planes. We now assume that Pc,d isa smooth projective plane. We will show that c = 1 and d = 2 follows, i.e. the only smoothskew parabola plane is the real projective plane. In fact we prove that the non-classicalskew parabola planes are not even C2-differentiable. We first define some C1-structureson certain parts of Pc,d. In order to do so, we use the following very useful lemma aboutopen orbits of Lie groups. A proof of this lemma can be found for example in [90], §1,Theorem 4.

(9.3) Lemma. Let G be a Lie group acting smoothly on a smooth manifold M such that

every element g ∈ G is a diffeomorphism of G. If there is an orbit mG that is open in

M , then the map G/Gm → mG : Gmg 7→ mg is a diffeomorphism, where G

/Gm is the

(unique) quotient manifold of G and mG is taken with the induced smooth structure of

M .

1. The smooth structure on the affine point set A = P \ l∞. The (closed) subgroupΣ of ∆ acts transitively on A. Thus the smooth structure on A ≈ R2 is the standard oneaccording to Lemma (9.3). In particular, every affine line possesses the standard smoothstructure as its induced smooth structure, i.e. the projections

χk = pr2 : la → R : (a, y) 7→ y

and

χs,t = pr1 : ls,t → R : q 7→x if q = (x, (x− s)d + t)x if q = (x, c(s− x)d + t)

are smooth charts.

2. The C1-structure on the line pencil L(0,0). We put

L(0,0) = ls,• | s ∈ R ∪ l0

with

ls,• =

(x, y) ∈ R2

∣∣∣∣∣∣∣∣ y =

(x− s)d − (−s)d if s ≤ x, s ≤ 0c(s− x)d − (−s)d if s ≥ x, s ≤ 0(x− s)d − csd if s ≤ x, s ≥ 0c(s− x)d − csd if s ≥ x, s ≥ 0

.

We define mappingsχ(0,0) : ls,• | s > 0 → R : ls,• 7→ s


and

χ∗(0,0) : L(0,0) \ l0,• → R :

ls,• 7→

1d (−s)

1−d if s < 0−1cd s

1−d if s > 0l0 7→ 0.

The map χ(0,0) is a C1-chart according to Lemma (9.3), because the stabilizer ∆(0,0)

which is the one-parameter subgroup P =ρt : (x, y) 7→ (tx, tdy)

∣∣ t > 0

acts transitivelyon ls,• | s > 0 and ρt maps ls,• to lst−1,•. We claim that χ∗(0,0) is a C1-chart. Themapping σp : Lp → Sp : L 7→ TpL is a smooth homeomorphism by Corollary (3.4). Letp = (0, 0). Using Grassmann coordinates (see the Appendix), we may assume that Tpl0has coordinate 0 and that Tpl0,• is used as the complement of Tpl0. Then the line Tpls,•has coordinate 1

d (−s)1−d if s < 0 and −1

cd s1−d if s > 0. Since the map ψ which assigns to

each spread element of Sp \ l0,• its Grassmann coordinate is a diffeomorphism, we inferthat

χ∗(0,0) = ψ σp L(0,0) \ l0,•is a smooth homeomorphism, too. It remains to verify that χ∗(0,0) is regular at l0. Thiswill be done in Section 6.

3. The C1-structure on l∗k := lk ∪ (∞) \ (0, 0). We set

χ∗k : l∗k → R :

(k, y) 7→ 1y

(∞) 7→ 0,

and we claim that the map χ∗k is a C1-chart. Since for any two lines ls and lt the projectivity[ls, ( 1

2 (s + t), lt] is just given by ls → lt : (s, y) 7→ (t, y), (s) 7→ (t), we may restrict ourattention to χ∗1. Let ρ1 be the perspectivity [L(0,0), l1]. We are going to prove thatπ1 := χ∗1 ρ1 χ∗−1

(0,0) : R → R is a C1-diffeomorphism. It is sufficient to show that π1 isdifferentiable and regular at 0 and that π′1 is continuous at 0. The perspectivity ρ1 assignsto each line ls,• the intersection ls,•∧ l1 which is given by the coordinates (1, y). The valuey is given by the equations

y =

(1− s)d − (−s)d = (−s)d((1− s−1)d − 1) if s ≤ 1c(s− 1)d − csd = csd((1− s−1)d − 1) if s ≥ 1.

Using the binomial formula (1 + z)d =∑k≥0

(dk

)zk, we obtain for |s| > 1

y =

d(−s)d−1

∑k≥1

1k

(d−1k−1

)(−s−1)k−1 if s < −1

−cdsd−1∑k≥1

1k

(d−1k−1

)(−s−1)k−1 if s > 1.

Putting lu,• = χ∗−1(0,0)(s) and v = χ∗1(1, y), we infer that

v−1 =

u−1

∑k≥1

1k

(d−1k−1

)(du)

k−1d−1 if 0 ≤ du < 1

u−1∑k≥1

1k

(d−1k−1

)(−cdu)

k−1d−1 if −1 < cdu ≤ 0.


For u = 0 both sums above converge and reduce to(d−10

)= 1. Writing ψ1(u) and ψ2(u)

for these sums, we get

π1 : u 7→ v = uψ1(u) if 0 ≤ du < 1u

ψ2(u) if −1 < cdu ≤ 0

and

π′1 : u 7→ v =

ψ1(u)+uψ′1(u)

ψ1(u)2 if 0 ≤ du < 1ψ2(u)+uψ′2(u)

ψ2(u)2 if −1 < cdu ≤ 0.

This proves that π1 is a local C1-diffeomorphism near 0.

4. The C1-structure on l∗0,0 := l0,0 ∪ p0 \ (0, 0). We are going to define a C1-chartχ∗0,0 : l∗0,0 → R by

χ∗0,0 : q 7→

x1−d if q = (x, xd)−1c (−x)1−d if q = (x, c(−x)d)

0 if q = p0.

We prove that this map is a C1-chart by showing that the composition π2 = χ∗0,0ρ2χ∗−1−1 ,

where ρ2 = [l−1, (−1), l0,0], is a local C1-diffeomorphism around 0. For p = (−1, y) thefirst coordinate x of ρ2(p) is given by

y =xd − (x+ 1)d if y 0c(−x)d − c(−x− 1)d if y 0

,

and we conclude as in Section 3 that π2 is indeed a local C1-diffeomorphism around 0.

5. The C1-structure in some neighborhood of (∞). Using the previously definedcharts, we may construct a coordinate system [ξ, η] in some neighborhood U of (∞) bysetting

ξ(p) = χ0,1((p ∨ (0, 0)) ∧ l0,1) and η(p) = χ∗0,0((p ∨ (1)) ∧ l∗0,0).

We are now going to compute dηdξ

∣∣∣lξ=0

which is defined as the derivative dηdξ

∣∣∣ξ=0

along

the line l ∈ L(−1,0). Since we have assumed that Pc,d is a C1-plane, the relation

liml→l−1

dηdξ

∣∣∣∣lξ=0

=dηdξ

∣∣∣∣l−1

ξ=0

, with l ∈ L(−1,0),

holds. As we will see, this relation forces d to be 2.

6. The computation of dηdξ

∣∣∣ξ=0

along l−1. For p ∈ U let ζ(p) = χ∗(0,0)(p∨ (0, 0)). This

yieldsξ(p) = χ0,1(χ∗−1

(0,0)(ζ(p)) ∧ l0,1).


Since x = χ0,1(ls,• ∧ l0,1) is given by

xd + 1 = (x− s)d − (−s)d if s < 0

c(−x)d + 1 = c(s− x)d − csd if s > 0,

we get for |xs | > 1 the relations

1 = d(−s)d−1x(1 + o(x

s))− xd if s < 0

1 = −cdsd−1x(1 + o(x

s))− c(−x)d if s > 0.

By definition of χ∗(0,0) this shows that α′(0) = 1, where α(t) = χ0,1(χ∗−1(0,0)(t)∧ l0,1), because

x converges to 0 as s tends to ∞. Hence we get for any line l that intersects l∞ in theneighborhood U

dξdζ

∣∣∣∣lζ=0

=d(α ζ)

dζ

∣∣∣∣lζ=0

= α′(0) = 1.

Furthermore, we derive from α′(0) 6= 0 that χ∗(0,0) is regular at l0. The proof of this facthas been postponed in Section 3.

We proceed by computing dηdξ

∣∣∣l−1

ξ=0. From what we have shown above, this expression equals

dηdζ

∣∣∣l−1

ζ=0. For p = (−1, y) ∈ l−1 with y 0 the line ls,• = p∨ (0, 0) is given by the equation

y = c(s+ 1)d − csd = csd((1 +1s)d − 1) = cdsd−1(1 + o(

1s)) = −1

ζ(1 + o(ζ)),

whereas the line p ∨ (1) = l1,t is given by

y = c(1 + 1)d + t = c2d + t.

Since the intersection (u, ud) = l1,t ∧ l∗0,0 can be expressed by

ud = (u− 1)d + t,

we get

y = ud−(u−1)d+c2d = ud(1−(1− 1u

)d)+c2d = dud−1(1+o(1u

))+c2d =d

η(1+o(η))+c2d.

Combining the equations for ζ and for η, we obtain

dηdζ

∣∣∣∣l−1

ζ=0

= limζ→0

η

ζ= limζ→0−d(1 + o(η))

1 + o(ζ)= −d.


7. The computation of dηdξ

∣∣∣ξ=0

along la,b ∈ L(−1,0). Now we consider some line

la,b ∈ L(−1,0). Clearly, we have b = −c(1 + a)d. Fix some point p = (x, y) ∈ la,b ∩ U andlet x > a > 1. Then (u, ud) = (p ∨ (1)) ∧ l∗0,0 satisfies the equation

ud = (u− 1)d − (x− 1)d + (x− a)d + b.

Thus we have for η = η(p)

(x− a)d − (x− 1)d + b = (u− 1)d − ud = dud−1(1 + o(1u

)) =d

η(1 + o(η)).

We use the binomial sum for the left hand of the equation above in order to get

xd−1(1− a)(1 + o(1x

)) =1η(1 + o(η)). (1)

The function ϑ of the line lϑ(p),• = (x, y) ∨ (0, 0) is determined by

(x− ϑ)d − cϑd = (x− a)d + b,

whence we get the following equation as above

(x− ϑ)d − (x− a)d = dxd−1(a− ϑ)(1 + o(1x

)) = cϑd + b = cϑd − c(1 + a)d.

If p converges to (a) on la,b, the line lϑ(p),• tends to some line l ∈ L(a) and we concludethat limx→∞ ϑ(p) = a. Thus we get

limx→∞

((x− ϑ)d − (x− a)d

)= limx→∞

dxd−1(a− ϑ) = cad − c(1 + a)d,

and from equation (1) we infer that

dϑdη

∣∣∣∣la,b

η=0

= limx→∞

ϑ− aη

=c((1 + a)d − ad

)(1− a)

d

holds. From ξ(p) = χ0,1(ll0,1∧ϑ(p),•) we derive the relation

c(−ξ)d + 1 = c(ϑ− ξ)d − cϑd,

because we have χ0,1(u, v) = u for (u, v) ∈ l0,1. Putting

F (ϑ, ξ) := c(ξ − ϑ)d − cξd − c(−ϑ)d − 1


and e = − limp→(a) ξ(p) we get by the implicit function theorem

dξdϑ

∣∣∣∣la,b

ϑ=a

= −∂F∂ϑ

(e, a)(∂F

∂ξ(e, a)

)−1

=(a+ e)d−1 − ad−1

(a+ e)d−1 − ed−1,

and thus

dξdη

∣∣∣∣la,b

η=0

=dξdϑ

∣∣∣∣la,b

ϑ=a

· dϑdη

∣∣∣∣la,b

η=0

=c((1 + a)d − ad

) ((a+ e)d−1 − ad−1

)(1− a)

d ((a+ e)d−1 − ed−1)(2)

holds. We have lima→∞ e = 0, because the intersection la,• ∧ l1,0 tends to (1, 0) as aconverges to ∞. Therefore, we have

((a+ e)d−1 − ad−1

)(1− a) = −(d− 1)ead−1

(1 + o(

1a))

((a+ 1)d − ad

)= dad−1

(1 + o(

1a))

((a+ e)d−1 − ed−1

)= ad−1

(1 +

e

a

)d−1

− ed−1 = ad−1

(1 + o(

1a)).

In order to determine the limit lima→∞ ead−1, we recall that e = −ξ((a)) holds. Conse-quently, the relation

c(−e)d + 1 = c(a− e)d − ad

holds which implies that

lima→∞

ead−1 =1cd

is true. Substituting the last equations in equation (2), we obtain

lima→∞

dξdη

∣∣∣∣la,b

η=0

= −cd(d− 1)cd2

= −d− 1d

.

This limit must coincide with dξdη

∣∣∣l−1

η=0= dζ

dη

∣∣∣l−1

η=0= −d, for the line line la,b converges to

l−1 if a tends to ∞. Hence we get d = 2.

Assuming that d = 2 holds, we are now going to prove that c = 1 follows. Consider theprojectivity ρ3 = [l0,0, (−1), l−1]. According to Section 1) the map π3 := χ0,0 ρ3 χ−1

−1 isnot only a C1-map; it is even smooth. Putting y = π3(x), the following relations hold:

x2 = (x+ 1)2 + y if x ≥ 0

cx2 = (x+ 1)2 + y if 0 ≥ x ≥ −1.


Thus we get

π′′3 (x) =

0 if x ≥ 02c− 2 if 0 ≥ x ≥ −1

and hence c = 1 follows. We summarize our result in the following theorem.

(9.4) Theorem. The only smooth skew parabola projective plane is the real projective

plane The skew parabola projective planes Pc,d 6∼= P2R are not even C2-differentiable.

Planes over Cartesian Fields

The last class of projective planes, whose collineation group is three-dimensional,consists of planes that are defined over Cartesian fields. Once again the reader is referredto [114], §37 for detailed information. We only provide the most important facts aboutthese planes. We define a multiplication ∗ on the set R of real numbers by setting

x ∗ y =

xy if x, y ≥ 0xαy if x ≥ 0, y ≤ 0xyβ if x ≤ 0, y ≥ 0|x|γ1c|y|γ2 if x, y ≤ 0,

where α, β, c are positive real numbers with α−1 + β−1 > 1 and δ = (α+ β − αβ)−1 > 0,and γ1 = αδ, γ2 = βδ holds. Then (R,+, ∗) is a Cartesian field R(α, β, c) and we denote byPα,β,c the projective plane that is coordinatized by this Cartesian field. Clearly, the planeP1,1,1 is the real projective plane. The affine plane over R(α, β, c) admits the collineations

ϕa,b,t : (x, y) 7→ (a(b ∗ x), ay + t) =

(abx, ay + t) if x ≥ 0(abαx, ay + t) if x ≤ 0.

The line pencil L(0,0) consists of the lines

lm = (mxµ, x) | x ≥ 0 ∪(−xµ+λ,−dx)

∣∣ x > 0, m > 0

andln = (nxν ,−x) | x ≥ 0 ∪

(−xν+λ, ex)

∣∣ x > 0, n > 0,

where the parameters λ, µ, ν, d, e satisfy the relations e/d = c, β = µ/ν and λ ∈ 0, 1,together with the lines X = R × 0 and Y = 0 × R. Obviously, the collineationsϕa,aµ−1,0 for a > 0 fix every line lm. Since in the smooth case, the set of fixed lines ofa collineation fixing at least three lines of some line pencil Lp is a homology-sphere, seeLemma (5.4), we conclude that ϕa,aµ−1,0 has to fix every line of L(0,0). This implies thatwe have µ = ν as well as d = e, and hence c = 1 and β = 1 follows. Finally, for α 6= 1,


the group ∆ = ϕa,b,t | a, b > 0, t ∈ R induces on L(∞) a group Φ ∼= R2, where (∞) isthe point at infinity of Y . The derivation map D(∞) : ∆ → Aut(A(∞)) is an equivariantmap between the action of ∆ on L(∞) and the action of D(∞)∆ on the line pencil throughthe origin of the tangent translation plane A(∞). Since a two-dimensional locally compacttranslation plane is isomorphic to the real affine plane, such an action is impossible. Thuswe have α = 1, and we have proved the following result.

(9.5) Theorem. Let P be a smooth flat projective plane defined over some Cartesian

field. If the collineation group of P is (at least) three-dimensional, then P is isomorphic to

the real projective plane.

Remark. D.Betten proved Theorem (9.5) in [3], Satz 5. Since the proof of Satz 5 of [3] isonly a sketch, we decided to include a verification of Theorem (9.5) in this manuscript.

Chapter 10: 4-dimensional Projective Planes 87

CHAPTER 10

4-dimensional Projective Planes

A compact topological projective plane P of dimension 4 is isomorphic to the complexprojective plane P2C, if it admits a 9-dimensional group Γ of continuous collineations.This result is due to H. Salzmann [102], cp. [114], 72.8. Since there are non-classical 4-dimensional translation planes having a collineation group of dimension 8 (Betten [5]),the critical (dimension) value gt(4) as described in the Introduction equals 8. Recall thatwe want to prove that the critical value gs(4) for smooth projective planes is at mostgt(4)−2 = 6. That gs(4) does not exceed 7 is a consequence of Otte’s Theorem on smoothprojective translation planes and a result of Salzmann [104], who proved that P or its dualplane is a translation plane if dim ∆ ≥ 8. Thus it remains to study compact topologicalprojective planes P = (P,L,F) with a (connected) 7-dimensional group ∆ of continuouscollineations. It turns out that there is only one topological non-translation plane Pf[0]with a 7-dimensional group of continuous collineations, [114], 74.27. This single plane is aso-called shift plane; shift planes are close relatives of translation planes.

Shift planes have been introduced by Dembowski-Ostrom, [31] in 1968. Their shiftplanes have been finite. In the topological setting shift planes only occur in dimensions 2and 4, see [114], 74.6. For an introduction to topological shift planes, the reader is referredto [114], Chapt. 74. Here, we will not only exclude the plane Pf[0] from the list of smoothprojective planes, we will also exclude some other types of shift planes. In this chapter,we are going to consider four-dimensional flexible shift planes, i.e. shift planes that admita group of collineations that have an open orbit in its flag space F. Since the dimensionof the flag space is 6, this implies that the dimension of the collineation group of a flexibleshift plane is at least 6.

All flexible shift planes are known explicitly. The two-dimensional flexible shift planesare exactly the skew parabola planes which admit a three-dimensional collineation group,see [114], §36, or Salzmann [100]. We have shown in the last chapter that these planes arenot smooth unless they are isomorphic to the projective plane over the reals. In dimension4, there are two exceptional flexible shift planes, a 2-parameter family and four morefamilies of flexible shift planes, see Betten and Knarr, [12], [68], [69].

88 Chapter 10: 4-dimensional Projective Planes

Four-dimensional shift planes.

(10.1) Definition. For a ∈ R4 we call σa : R4 → R4 : x 7→ x+a a shift map. Let f : R2 →R2 be an arbitrary map and consider its graph Gf :=

(x, f(x))

∣∣ x ∈ R2. Applying the

shift σ(u,v) to the graph Gf , we get the set L(u, v) :=(x, f(x− u) + v)

∣∣ x ∈ R2. More-

over, we put L(w) = w × R2 for w ∈ R2. Let A = R4 and L′ =L(u, v)

∣∣ u, v ∈ R2∪

L(w)∣∣ w ∈ R2

. If the pair (A,L′) defines an affine plane Af with respect to the natural

incidence, then Af is called a shift plane and the map f is called the generating func-tion of Af . In this case, we denote the projective closure of Af by Pf = (P,L), whereL = L′ ∪ L∞ and P = A ∪ L∞.

Clearly, the shift group Λ :=σa

∣∣ a ∈ R4

of a shift plane Af is a subgroup of thecollineation group of Af . The group Λ is even a normal subgroup of Γ, and Γ can bedecomposed as Γ ∼= Γo n Λ, where Γo is the stabilizer of some affine point o. Accordingto [114], 74.3, a map f is a generating function of a shift plane if and only if for eacha ∈ R2 \ (0, 0) the map x 7→ f(x + a) − f(x) is a bijection. Moreover, the plane Af

is a locally compact topological affine plane if and only if f is continuous. We denoteby (∞) the point on L∞ that corresponds to the parallel class

L(w)

∣∣ w ∈ R2. The

shift group Λ can be written as the direct product Λ = Ξ × Υ, where the subgroup Υ =σ(0,v)

∣∣ v ∈ R2

= Λ[(∞),L∞] is a linearly transitive translation group. In contrast, theother factor Ξ =

σ(u,0)

∣∣ u ∈ R2

does not contain any nontrivial translation.Topological affine shift planes Af defined over a differentiable generating function

f possess an associated ”infinitesimal plane”, the so-called derivative Df of Af . Thederivative Df is a topological affine translation plane of the same dimension as Af . Forflexible shift planes it turns out that the generating function f is always differentiable,and hence all such planes have a derivative. We will see in the next section that for asmooth 4-dimensional flexible shift plane Pf the derivative Df is isomorphic to the tangenttranslation plane Ap of any affine point p ∈ A.

The derivative of a shift plane. If f : R2 → R2 is a smooth map, we may consider itsderivative Df : (s, t) 7→ D(s,t)f . Let S be a spread in R4 that contains the vertical line L(0).Fix a homeomorphism h : R2 → S′ := S \ L(0). If Df = h, then f is called an integralfunction of h. If an integral function f of some spread S is a generating function of someprojective shift plane Pf , the translation plane Df defined by S is called the derivative ofPf . We will see that every flexible shift plane has a derivative Df , and we will present analternative way of describing Df .

The topological classification of flexible shift planes. Let Pf = (P,L) be a projectivetopological 4-dimensional flexible shift plane with generating function f , and let Γ be its


group of continuous collineations. Fix some affine point o of Pf . Then Pf is isomorphicto one of the planes defined in the theorems below, see [114], (74.22).

(10.2) Theorem. If the stabilizer Γo contains a subgroup Ξ ∼= R2, then up to isomorphism

the generating function f is one of the planar maps f[0] or f[−1] of R2 defined by

f[k](s, t) = (st− 13s

3 + ks, 12 (t2 + ks2)− 1

12s4).

The generating spread S0 which defines the derivative D0 = Df[k] of Pf[k] is given with

respect to Grassmann coordinates by

S0 = Y ∪(

t s− s

3

3 s2 + t

)∣∣∣∣ s, t ∈ R

;

see the Appendix for the description of spreads by Grassmann coordinates. Note that both

planes Pf[0] and Pf[−1] have the same derivative D0. The dimension of the group Γ is 7 for

the plane Pf[0] and 6 for Pf[−1] .

(10.3) Theorem. Let Γ1o be isomorphic to the group L2 := x 7→ ax+ b | a, b ∈ R, a 6= 0.

If Γ1o acts on the vertical line L(0) inducing only real homotheties, then Pf is isomorphic

to precisely one of the skew parabola planes Pg,k defined by the planar maps

f(s, t) =(

12s

2 + a(t)t2, st+ b(t)t2),

where

(a(t), b(t)) =

(− 12 , 0) if t ≥ 0

(g, k) if t < 0

with parameters g, k satisfying k ≥ 0, 2g + k2 ≤ −1 and (g, k) 6= (− 12 , 0). The derivative

Da of Pg,k is the translation plane which is defined by the spread

Sa = L(0) ∪(

s −tt s

) ∣∣∣∣ s, t ∈ R, t ≥ 0∪

(s −tat s

) ∣∣∣∣ s, t ∈ R, t ≤ 0,

where a = −12g+k2

(12 − g +

√(g + 1

2

)2 + k2

)2

6= 1. The stabilizer Γo contains the one-

parameter groupsP =

diag(r, r, r2, r2)

∣∣ 0 < r ∈ R,

N =

11

c 1c 1

∣∣∣∣∣∣∣ c ∈ R

,

whence the dimension of the collineation group Γ is 6.


If Γ1o does not act on the vertical line L(0) as a group of real homotheties, then the

derivative D of Pf is isomorphic to one of the translation planes constructed by D. Betten

in [8], Satz 1, Satz 2 and Satz 5.

(10.4) Theorem. If Γ1o∼= C×, then there exists a complex number w = (u+ iv) ∈ C such

that Pf is isomorphic to the plane Pw generated by f[w] : C → C : z 7→ z2|z|w−2. The

plane P2 is the complex plane. Let w 6= 2. Then the translation group Γ[L∞,L∞] of Pw

is 2-dimensional and Γ[L∞,L∞] = Γ[(∞),L∞] = Υ holds. Moreover, the group Γ1o consists of

the maps (x, y) 7→(cx, f[w](c)y

): C2 → C2, where c ∈ C×. Hence, we have dim Γ = 6 if

w 6= 2.

Theorems (10.2) and (10.3) have been proved by N. Knarr [68] and [69], whereasTheorem (10.4) is due to Betten and Knarr [12].

Smooth flexible shift planes. We are now going to prove that the only smooth 4-dimensional flexible shift plane is the complex projective plane. Note that in order toprove the Main Theorem (of the Introduction) it would only be necessary to exclude thesingle shift plane Pf[0] of Theorem (10.2). First, we show that the derivative of a smoothshift plane is isomorphic to every tangent translation plane Ap on TpP defined by thespread Sp = TpL | L ∈ Lp, see Chapter 2. Throughout this section, let Pf = (P,L) be asmooth 4-dimensional projective shift plane with f as its generating function.

(10.5) Proposition. Let Df be the derivative of Pf and let Ap be the tangent translation

plane at the affine point p ∈ A. Then Df is isomorphic to Ap as a topological affine plane.

Proof. According to Lemma (9.3), we may identify the affine point set of P \ L∞ withthe shift group Λ = Ξ × Υ ≈ R4. We may consider the generating function as a mapf : Ξ → Υ. Since Λ is transitive on the affine point set A of Pf , the tangent translationplanes at affine points are all isomorphic to, say, Po, where o is the origin of Λ ≈ R4. Theline pencil Lo is given by

Lo =L(−s,−f(s))

∣∣ s ∈ R2∪ L(0) ,

where L(−s,−f(s)) = (x, f(x+ s)− f(s)) | x ∈ Ξ. Setting ϕs(x) := f(x + s) − f(s),we get Doϕ = Dsf , which proves that ToL(−s,−f(s)) has Grassmann coordinates Dsf

with respect to the product Ξ × Υ. Hence, the spread So := ToL(−s,−f(s)) | s ∈ Ξ ∪0 × R2

coincides with the spread of the derivative Df . This shows that the topological

planes Df and Ao are isomorphic. Hence, the proposition is proved.


By [114], (74.22), the stabilizer Γo acts transitively on the line pencil Lo \ L(0) ofPf[k] . The same is true for the shift planes defined in Theorem (10.3), for it is easy to seethat the group Γo acts transitively on Lo \ L(0). Thus we can apply Proposition (3.7)in order to get the following result.

(10.6) Proposition. If Pf is isomorphic to one of the shift plane Pf[k] or Pg,k, then Pf

is semiregular at o with singular line L(0).

From Lemma (3.6) we get the following consequence.

(10.7) Proposition. Let Ap = (TpP, Sp) be a tangent plane of an 2l-dimensional smooth

stable plane S, and let Y ∈ Sp. We denote by L ⊆ End(Rn) the set of Grassmann

coordinates of Sp \ Y with respect to some given decomposition TpP = X ⊕ Y , where

X ∈ Sp \ Y . If S is semiregular at the point p with singular line Y , then the maps

εx,y = εx ε−1y : Rl → Rl are diffeomorphisms for any x, y ∈ Rl.

We are now going to prove that the projective shift planes Pf[k] and those of Theorem(10.3) are not smooth.

(10.8) Theorem. Let Pf be a smooth 4-dimensional projective shift plane isomorphic to

some plane Pf[k] or to some plane of Theorem (10.3). Then Pf is not a smooth projective

plane.

Proof. Assume that Pf is a smooth projective plane. We consider the tangent translationplane Ao. The plane Ao is isomorphic to the derivative Df by Proposition (10.5). Accord-ing to Proposition (10.6), the plane Pf is semiregular at o with Y as the singular line. ByProposition (10.7), the maps εx,y : R2 → R2 are diffeomorphisms for any x, y ∈ R2. Letus put x =

(10

)∈ R2 and y =

(01

)∈ R2.

First, we consider a shift plane Pf isomorphic to some plane Pf[k] (as defined in Theo-rem (10.2)). Then the derivative Df is isomorphic to the non-classical translation planedescribed by the spread S = S0. Putting

Ms,t =(

t s− s

3

3 s2 + t

)we get

εx(Ms,t) =(

t− s

3

3

)and εy(Ms,t) =

(s

s2 + t

),

and we obtain

εx,y(s, t) =(t− s2− s

3

3

).


From

Ds,tεx,y =(−2s −s21 0

)we infer that the map εx,y : R2 → R2 is not a diffeomorphism, because its differential isnot regular at the points

(0t

)for every t ∈ R. This contradiction shows that Pf is not a

smooth plane.Now let Pf be a shift plane as defined in Theorem (10.3). If Pf ∼= Pg,k, the derivative Df

of P is generated by some spread Sa with a > 0 and a 6= 1. In this case we get

εx(Ms,t) =

(st

)if t ≥ 0(

sat

)if t ≤ 0

and εy(Ms,t) =(−ts

),

where Ms,t is defined by

Ms,t =

(s −tt s

)if t ≥ 0(

s −tat s

)if t ≤ 0.

Thus we have

εx,y(s, t) =

(−ts

)if s ≥ 0(

−tas

)if s ≤ 0.

Clearly, the map εx,y is smooth if and only if a = 1. Hence, the plane Pf ∼= Pg,k is notsmooth. We omit the verification of the fact that the maps εx,y are not diffeomorphismsfor the remaining cases where the derivative of Pf is isomorphic to some translation planeof [8], Satz 1, Satz 2 and Satz 5.

Now we turn to the shift planes Pw. We will not give a complete proof of the factthat these planes are not smooth unless w = 2 (recall that P2 is the complex projectiveplane), because they admit only a collineation group of dimension six which is less thangt(n)− 1. Hence they do not come into play when we prove the Main Theorem. However,we will present the main ideas of the proof. Our approach will be the following: we usethe tangent translation planes Ap at points p ∈ L∞ rather than at affine points. It turnsout that the tangent planes at the points p ∈ L∞ are isomorphic to the complex affineplane. We prove this by showing that the collineation group of Ap (p ∈ L∞) has dimensiongreater that 8. Using the same method as for the skew parabola planes, which have beendiscussed in Chapter 9, it can be shown that the tangent planes at points on L∞ obtained


by this method are not classical unless w = 2. We are going to present a proof of thefirst fact, because it involves new ideas, and we will omit the second part in order to avoidrather lenghty computations similar to those of Chapter 9.

We want to prove that the dimension of the collineation group of Ap (p ∈ L∞) isgreater than 8. This is the motivation of the next proposition.

(10.9) Proposition. Let S = (P,L) be an n-dimensional smooth stable plane with

collineation group Γ, and let p ∈ P . If dim Γp/Γ[p]

= k, then dim Aut(Ap) ≥ k + n+ 1.

Proof. The derivation map Dp : Γp → A0, where A = Aut(Ap), maps Γp homomorphicallyto A0. The kernel kerDp is the group Γ[p,p] according to Corollary (5.31). Since a trans-lation plane always admits the group of real homotheties (with center 0) as a collineationgroup, we have

dim A0 ≥ dim A0/A[0]

+ 1.

The group Γ[p] of central collineations with center p are mapped by Dp to A[0], whence

dim A0/A[0]≥ dim Γp

/Γ[p]

= k

follows. Since Ap is an n-dimensional translation plane, we have A ∼= A0 n Rn and weconclude that

dim A = dim A0 + n ≥ dim A0/A[0]

+ n+ 1 ≥ k + n+ 1

holds.

(10.10) Proposition. If Pf is a shift plane isomorphic to some plane Pw, then the tangent

translation plane Ap (p ∈ L∞) of Pf is isomorphic to A2C, the complex affine plane.

Proof. The statement is trivial if Pw ∼= P2C, i.e. if w = 2. Thus we may assume that Pw

is not the complex projective plane. First, we consider the point p = (∞). Then Γ = Γp,and according to Proposition (7.3), the group Γ[p] can be written as

Γ[p]∼= Γ[p,A] n Γ[p,p]

for some line A 6= L∞. Since Γ[p,A] = 1l by [114], (74.9), and because we have Γ[p,p] =Γ[p,L∞], we obtain

dim Γ[p] = Γ[p,L∞] = 2.

Note that by Theorem (10.4) we only know that the dual assertion Γ[L∞,L∞] = Γ[p,L∞]

holds. This, however, does not matter, because the planes Pw are self-dual according to


[12]. By Theorem (10.4), the collineation group Γ of Pw has dimension 6 (note that wehave excluded the case Pw ∼= P2C). Using a theorem of Nagami [89], (2.1), we get

dim Γ/Γ[p]

= dim Γ− dim Γ[p] = 6− 2 = 4.

Due to Proposition (10.9), this implies that

dim Aut(Ap) ≥ 5 + 4 = 9,

whence Ap∼= A2C follows by Salzmann, [102], cp. [114], 72.8.

Now let p ∈ L∞ \ (∞). Since the shift group Λ acts transitively on L∞ \ (∞), we inferthat dim Γp = dim Γ − 2 = 4. We claim that dim Γ[p] = 0 holds. We know that Γ[p,p] istrivial, because Γ fixes the point (∞), and hence

Γ[p,p] = Γ[p,L∞] ⊆ Γ[L∞,L∞] = Γ[(∞),L∞]

follows. From the last equation we conclude that Γ[p,p] = 1l. By Proposition (7.3), there isa line A through (∞) such that we have Γ[p] = Γ[p,A]. Since the shift group acts transitivelyon the line pencil L(∞), we may choose the point p on L∞ \ (∞) in such a way thatthe line A goes through the origin o. Then the connected component Γ1

[p] is contained inΓ1o. Due to the description of Γ1

o given in Theorem (10.4) it is easy to see that Γ1o contains

a reflection as the only homology with axis A = L(0). (According to Theorem (74.28) of[114], even the stabilizer Γo contains no homology besides a reflection.) This implies thatdim Γ[p] = 0 holds. Now we can argue as in the first part of the proof in order to derivethat Ap

∼= A2C.

Salzmann proved in 1973 ([104]) that a four-dimensional projective plane P with an atleast 9-dimensional collineation group Γ is isomorphic to the complex projective plane, seealso [4], Satz 4 and [102]. If dim Γ ≥ 7, then P is a translation plane or a dual translationplane according to [101], [103] and [7], or P is isomorphic to the shift plane P0 defined inTheorem (10.2), see [11] and [85]. Hence, Theorem (10.8) together with Otte’s result onsmooth translation planes (see the Introduction) yields the following corollary.

(10.11) Corollary. Let P be a smooth four-dimensional projective plane. Then P is

either isomorphic to the complex projective plane or dim Aut(P) ≤ 6.


CHAPTER 11


In this chapter we are confronted with 8-dimensional Hughes planes. The first exam-ples of Hughes plane are due to D.R.Hughes. He constructed them from finite nearfields(see the book of Hughes-Piper, [63], IX.6). Ostrom [92] characterized these planes by theproperty that there is a desarguesian Baer subplane D such that every central collineationof D can be extended to a central collineation of the whole plane. Dembowski [30], gen-eralized Hughes’ construction in order to obtain infinite planes. In 1981, Salzmann, [108],introduced analogues of Hughes planes in the category of topological planes. It turns outthat for compact planes of dimension greater then 4 there are no Hughes planes in thestrict sense, since in these cases homologies of D cannot be extended to homologies of thewhole plane. On the other hand, there are non-classical planes P having a desarguesianBaer subplane D such that every collineation of D can be extended to P. This gives riseto the following definition.

(11.1) Definition. A compact connected topological projective plane P is called a Hughesplane if it contains a closed desarguesian Baer subplane D such that every collineation ofD can be extended to the whole plane P.

Clearly, a 2-dimensional projective plane cannot be a Hughes plane since such a planedoes not contain any closed proper subplane. The classical planes P2F over one of thedomains F = C,H,O are examples of Hughes planes. In case D ∼= P2R, Salzmann [103]has proved that P ∼= P2C, so the complex projective plane is the only 4-dimensionalHughes plane. Thus it remains to study 8- and 16-dimensional Hughes planes. Theeight-dimensional Hughes planes have been classified by Salzmann, [108], §4, while the 16-dimensional case is due to Hahl [54]. In both dimensions there is a single one-parameterfamily of (pairwise non-isomorphic) Hughes planes Pα. For a unified treatment of 8- and16-dimensional Hughes planes see [114], §86.

We will show that for 8-dimensional smooth Hughes planes the situation is exactlythe same as for 4-dimensional topological Hughes planes: they are always classical. In ourproof we will use the collineation groups of Hughes planes. It turns out that we only need


to consider the stabilizer of a suitable flag (p, L) in order to exclude non-classical Hughesplanes.

Collineation groups of Hughes planes. Let PF = (P,L,F) be a Hughes plane with aclosed Γ-invariant desarguesian Baer subplane DF = (P ′,L′,F′) ∼= P2F, where F ∈ C,H.We use the results and the notation of Chapter 8. In particular, we add a prime to thesymbols which address the Baer subplane. For example, we use P ′ for the set of pointsof the Baer subplane. Fix an outer point p and an outer line S through p. Let L be theunique inner line through p, and let o be the unique inner point on S, see Figure 6.

Figure 6.

We need the following facts from Hahl [54], (3.9) and [114], §86. There is a closedsubgroup Σ ∼= SL3F of Aut(PF) which induces the group PSL3F on the Baer subplane DF.The stabilizer ∆ := Σo,L induces on DF a group which is isomorphic to SL2F · F×. Thesubgroup Π := Σp,S fixes the unique inner line L through p and the unique inner pointo. The stabilizer Π induces on DF a group SU2F · P, where P is a closed non-compactone-parameter subgroup of F×, see Salzmann [108], p. 355, [100], (7.26), Hahl [54], 3.11and 3.14, and [114], 86.34. In fact, we have P =

e(1+iα)t

∣∣ t ∈ R⊂ C ⊂ H for some real

nonnegative constant α, cp. [54], Lemma (2.6).

Consider the derivation map Do : Σo → Aut(Ao)0. Since (o, L) is an antiflag, therestriction of Do to the group ∆ is an injection by Corollary (5.9). The action of ∆ on Lo

is equivalent to the action of Dp∆ on the spread So that defines the tangent translationplane Ao. Since the subplane DF is closed in PF, the point set P ′ is a closed embeddedsubmanifold of P according to Proposition (8.3). Since the subplane DF is desarguesian, wecan represent the affine point space A′ of DF by the translation group T = Aut(DF)[L,L]

∼=Rl, cp. Lemma (9.3). The lines through the origin o ∈ T are then linear subspaces thatbuild up the ordinary spread S′o on T of either the complex plane or the plane over thequaternions. Using this representation as a chart h for the point space of the affine plane


DLF , we get an isomorphism between the smooth affine planes DL

F and A′o that is equivariantwith respect to the actions of Aut(DL

F )o and Aut(A′o)0. The stabilizer Π consists of linearmappings on T, whence the restriction D′o : Π→ Aut(A′o) of the derivation map Do to thesubmanifold A′ is just the identity and the action of Πo on A′ = T is equivalent to theaction of D′0Π on ToP ′ = Th. With respect to a suitable basis of T we can write

D′oΠ =e(1+iα)tM

∣∣∣ M ∈ SU2F, t ∈ R.

The different types of Hughes planes can be distinguished by the way how D′oΠ acts onthe translation group T. This action is inequivalent for different values of α, and PαF is aMoufang plane if and only if α = 0 (Hahl [54], 4.3, 4.6, and [114], 86.36, 86.37).

We are going to introduce several groups of collineations acting on the projectivetangent translation plane Po at the point o which correspond to the groups Γ, ∆, and Π ofPF. Let Φ = Aut(Po)[P′o] be the subgroup of Aut(Po) which leaves the Baer subplane P′oof Po invariant. Futhermore, we denote by Ψ the induced group of Φ on P′o, i.e. we haveΨ = ΦP′o ∼= Φ

/ΦP′o

∼= Aut(P′o), and we put Υ = Ψ1ToS∧L∞ . Then Υ fixes the line L∞,

because ToS∧L∞ is an outer point on the inner line L∞, cp. Lemma (8.4) and Proposition(8.5). We assume now that the tangent plane Po is classical. Then we can represent Υ onTh as

Υ =etN

∣∣ N ∈ SU2F, t ∈ R

with respect to a suitable basis of Th. Since the action of Π on T is equivalent to the actionof Υ on Th, we conclude that α = 0, and so PF is a Moufang plane. Because the smoothstructure of a smooth plane is uniquely determined ([15]), we have proved the followingtheorem.

(11.2) Theorem. Let P be a smooth Hughes plane. If the tangent translation plane of

some inner point of P is classical, then P is isomorphic as a smooth plane to one of the

Moufang planes P2F with F ∈ H,O.

For 8-dimensional smooth Hughes planes it turns out that the tangent translationplanes at every inner point are always isomorphic to the classical affine plane over thequaternions.

(11.3) Theorem. Let PC = (P,L,F) be a smooth 8-dimensional Hughes plane and let o

be some inner point. Then the affine tangent translation plane Ao is isomorphic to A2H.

Proof. The stabilizer Γo,L of an inner antiflag (o, L) contains a closed subgroup Λ which isisomorphic to SL2C. Since Do : Λ→ Aut(Ao) is a closed embedding (this is true, because Λ

is semi-simple, see [32] or [36]), the plane Ao is a so-called SL2C-plane, see Hahl [51]. Sucha plane is either the classical affine plane (over H) or its collineation group has dimension


at most 16. The group Γo,L is isomorphic to GL2C ·T. In particular, we have dim Γo,L = 9.Since the restriction of Do to Γo,L is an injection, we get

dim Aut(Ao) ≥ dim DoΓo,L + dim Aut(Ao)[L∞,L∞] ≥ 9 + 8 = 17.

According to [51], (2.5), this implies that Ao∼= A2H. Since the collineation group of PC

acts transitively on the set P ′ of inner points, we get Ap∼= A2H for every p ∈ P ′.

Remark. In the proof of Theorem (11.3) we used Hahl’s characterization of SL2C-planes inthe class of 8-dimensional topological translation planes. It can be shown that in the case of16-dimensional topological translation planes there is a similar characterization of SL2H-planes. This allows us to generalize Theorem (11.3) to 16-dimensional Hughes planes.However, we omit this proof, because we are dealing in this chapter with 8-dimensionalplanes only. On the other hand, the non-classical 16-dimensional Hughes planes have anautomorphism group of dimension 36, which is smaller than gt(16) − 2 = 40 − 2 = 38.Hence, we do not have to consider these planes in order to prove the Main Theorem (ofthe Introduction).

(11.4) Corollary. Every 8-dimensional smooth Hughes plane is isomorphic as a smooth

projective plane to the desarguesian plane P2H.

We summarize the results of this chapter. Let P be an 8-dimensional compact pro-jective plane with automorphism group Γ. By a theorem of H. Salzmann [113], the planeP or its dual is a translation plane or P is a Hughes plane, if dim Γ ≥ 17. According toOtte’s Theorem [94], smooth projective translation planes are classical. Thus our previoustheorem yields the following corollary.

(11.5) Corollary. Let P be an 8-dimensional smooth projective plane with automorphism

Γ. Then P is either isomorphic to the quaternion plane P2H and Γ ∼= PSL3H · H× (and

hence dim Γ = 35), or dim Γ ≤ 16 holds.


CHAPTER 12


A 16-dimensional compact topological projective plane P is isomorphic to the Moufangplane P2O defined over the octonions if its collineation group has dimension greater than40, see Salzmann, [109] –[112]. Moreover, there are non-classical (translation) planes witha collineation group of dimension 40. Hence, the critical value gt(16) is 40. In this chapterwe will prove the following theorem.

(12.1) Theorem. Let P be a 16-dimensional smooth projective plane with collineation

group Γ. Then P is either isomorphic (as a smooth projective plane) to the Moufang plane

P2O, or dim ∆ ≤ 38 holds. In particular, we have gs(16) ≤ 38.

Throughout this chapter let P = (P,L,F) be a smooth 16-dimensional projectiveplane with Γ as its group of continuous (and hence smooth) collineations. We assume that∆ is a closed connected subgroup of the collineation group Γ with 39 ≤ dim ∆ ≤ 40. Weput F∆ :=

x ∈ P ∪ L

∣∣ ∀δ ∈ ∆ : xδ = x

as the set consisting of all fixed elements of ∆.Since we will often use Lemma (6.8), we will apply it without further reference.

Tangent translation planes of P. In the following two sections we suppose that ∆

has some fixed flag (p, L). We are going to determine the tangent translation plane Ap =(TpP, Sp) of P at the fixed point p. In order to do that, we consider the derivation map

Dp : ∆→ Aut(Ap)o,SL: δ 7→ Dpδ,

where 0 is the origin of the point set TpP . Note that Dp∆ fixes the subspace SL = TpL,because ∆ fixes L, see the remark before Lemma (5.4). The map Dp is a continuous grouphomomorphism with kernel kerDp = ∆[p,p] (see Theorem (5.3) and Corollary (5.9)). PutΣ := Aut(Pp). Since a smooth projective translation plane is classical by Otte’s Theorem,we may assume that dim ∆[p,p] ≤ 15. Because Pp is a compact 16-dimensional translationplane by Theorem (2.5), we obtain from Lemma (6.9) the inequality

dim Σ = dim Σ0 + 16 ≥ dim Dp∆ + 16 = dim∆− dim ∆[p,p] + 16 ≥ dim ∆ + 1 = 40.


The compact translation planes with a collineation group of dimension at least 40 arecompletely classified. The classification is due to H.Hahl, see [57], [58], and compare [114],82.26(5), 82.27, and 82.21. The following theorem collects the relevant information whichwe will need later on.

(12.2) Theorem. Let T be a topological 16-dimensional compact projective translation

plane with collineation group A. Assume that dim A ≥ 40. Then the following assertions

hold.

(i) Either T ∼= P2O and dim A = 78, or T is isomorphic to some plane defined over

the mutations Oα (see [46], XI.14 or [114], 82.27) and dim A = 40.

(ii) Let L∞ be a translation axis of T and let R, S be two lines such that their

intersection 0 = R ∧ S is not incident with L∞. Let B denote the connected

component of the stabilizer AR,S,L∞ . Then B contains a largest compact subgroup

M, and B = M · P · B[o,L∞] for some closed one-parameter subgroup P of B and

B[o,L∞]∼= R>0. If T ∼= P2O then M ∼= Spin8R, and we have M ∼= G2(−14) in the

case of dim A = 40.

The dimension of the stabilizer of two lines. Let K 6= L be a line of P through thepoint p. We denote by Φ the connected component of the stabilizer Aut(P)K,L and by Ψ

a Levi subgroup of Φ. We are going to consider the derivation map Dp : Φ → Ω, whereΩ is the connected component of the stabilizer Aut(Ap)SK ,SL

and Ap is the affine tangenttranslation plane of P at the point p. In this section we want to determine upper boundsfor the dimensions of Φ. The next lemma will be the key result for this task.

(12.3) Lemma. If dim Ψ ≥ 14, then the derivation map Dp : Φ→ Ω is a closed mapping.

We prove Lemma (12.3) in several steps. We start with a lemma on subgroups of Liegroups. An analytic subgroup H of a Lie group G is a subgroup of G which has a Lie groupstructure such that the inclusion map ι : H → G is a Lie group homomorphism, cp. [90],§2, Chap. 9; note that in [90] analytic subgroups are called virtual Lie subgroups. A virtualLie subgroup of a Lie group must not be a closed subgroup. In contrast, a Lie subgroupH of G is a Lie group with respect to the induced smooth structure of G. A subgroup ofG is a Lie subgroup if and only if it is closed in G. We recall a few abbreviations. Thetorus rank of a Lie group G, i.e. the dimension of a maximal torus subgroup, is written byrkG. The centralizer of H in G is denoted by CGH.

(12.4) Lemma. Let H be an analytic subgroup of a Lie group G, and let T be a torus

subgroup of H of rank r. If r ≥ rkG−1, the centralizer CHT is closed in G. In particular,

CHT is a Lie subgroup of G.


Proof. A subgroup U of G is closed if and only if every closed one-parameter subgroup ofU is closed in G, see Hochschild [62], XVI, Theorem 2.4. Assume that there is a closedone-parameter subgroup P of CHT which is not closed in G. Then P ∩ T = 1l, becauseP is not compact, otherwise it would be closed in G. Moreover, the closure P of P inG is compact and connected, [62], XVI, 2.3. Since P is commutative, the closure P iscommutative as well. The dimension of P is greater than dimP = 1, because a subgroupof maximal dimension of a connected Lie group coincides with the whole group. Theproduct B = P · T is a direct product, i.e. we have B = P × T , for P commutes with T

by hypothesis and P is disjoint to T . Since P is closed in CHT , we get P ∩ CHT = P ,whence P ∩ T = P ∩ T = 1l. Thus we conclude that B = P × T = P × T is a compactabelian Lie subgroup with dimB > dimB. In particular, we have B ∼= Tr+k, where k ≥ 2.This, however, contradicts the fact that the torus rank of G is at most r + 1. This provesthe Lemma.

(12.5) Lemma. Let Λ ≤ Aut(P) be a closed subgroup that fixes two distinct lines G and

H. Let Υ be a Levi subgroup of Λ, and let√

Λ denote the solvable radical of Λ. Setting

z = G ∧H, the inequality

dim DzΛ = dim Υ + dim√

DzΛ ≤ dim Υ− rkΥ + 6

holds.

Proof. The Levi decomposition of Λ is Λ = Υ ·√

Λ, where dim(Υ ∩√

Λ) = 0, see [91],Chapt. 1, 4.1. By Nagami [89], (2.1), this yields dim Λ = dim Υ + dim

√Λ. The derivation

map Dz is a Lie isomorphism between the Lie groups Υ and DzΥ, and Υ is compact, seeProposition (6.1). Hence, we obtain the first equation. By Corollary (7.2) and Corollary(5.9), the maximal dimension of a closed solvable subgroup of Aut(Az)SG,SH

is 6. Amaximal solvable subgroup of DzΛ is contained in some group

√DzΛ · T, where T is a

maximal torus subgroup of Υ. This proves the inequality.

Proof of Lemma (12.3). We use the Levi decomposition Φ = Ψ·√

Φ of Φ. By Proposition(6.1), the Levi subgroup Ψ is compact and the restriction of the derivation map to Ψ isinjective. Thus, in order to prove Lemma (12.3), it is sufficient to verify that the radicalΠ :=

√DpΦ = Dp

√Φ is closed in Ω. If dim Ψ ≥ 14, every closed solvable subgroup of

Ψ of maximal dimension is at least 2-dimensional according to the classification of quasi-simple Lie groups, see the tables of J. Tits [123]. Hence, we conclude by Lemma (12.5) thatdim Π ≤ 6 − 2 = 4. A quasi-simple compact Lie group Ψ of dimension at least 14 cannotact on a manifold of dimension n less than 5, for otherwise dimG ≤ 1

2n(n+ 1) ≤ 10 holdsby Montgomery-Zippin [88], Chap. VI, Theorem 2. Thus, the Levi subgroup Ψ commuteswith the radical

√Φ. Of course, the same is true for DpΨ and Π.


According to Theorem (6.7), the group Ψ is locally isomorphic to some subgroup ofSpin8R. Consider the two-sheeted covering map π : Spin8R → SO8R. A subgroup Θ ofSpin8R is mapped via π onto a subgroup π(Θ), and Θ is an at most two-sheeted coveringgroup of π(Θ). Recalling that SO5R cannot act on a compact connected projective planeby M.Luneburg [87], II, Korollar 1 or [114], 55.40, we may apply Lemma (6.19) in orderto obtain that DpΨ ∼= Ψ is isomorphic to one of the groups

Spin8R, Spin7R, SU4C, G2(−14). (∗)

The first three groups Spin8R, Spin7R and SU4C have torus rank at least 3. By M. Lune-burg [87], II, Satz 2 (see also [114], 55.37), the torus rank of Ω is at most 4. Thus theassumptions of Lemma (12.4) are satisfied for G = Ω, H = DpΦ and some 3-dimensionaltorus subgroup T of DpΨ. Hence we conclude that the centralizer CDpΦT is closed in Ω.By what we have shown above, the radical Π of DpΦ is a (closed) subgroup of CDpΦT andhence of Ω, too. This proves Lemma (12.3) in the case, where Ψ is isomorphic to one ofthe groups Spin8R, Spin7R or SU4C.

Now let us consider the case Ψ ∼= G2(−14). According to Theorem (12.2), a maximalcompact subgroup M of Ω is isomorphic to either Spin8R or G2(−14). We will show that thecentralizer of DpΨ in M is trivial. This is obvious if M ∼= G2(−14). Hence, we may assumethat M ∼= Spin8R, and consequently, it is sufficient to verify that the centralizer of DpΨ inGL8R has no nontrivial compact subgroup. Using the second table of Lemma (6.19), weget CGL8RΨ ∼= R2. This implies that CDp∆(DpΨ) is closed in Ω. Since Π =

√Dp∆ is a

(closed) subgroup of CDp∆(DpΨ), the group Π is closed in CΩDpΨ. Thus, we infer that Π

is also closed in Ω. This finishes the proof of Lemma (12.3).

(12.6) Stabilizer Theorem. Let P be a smooth 16-dimensional projective plane. Let Φ

be a connected closed subgroup of the collineation group of P which fixes two distinct lines

K and L. Let Ψ be a Levi subgroup of Φ. Then exactly one of the following statements is

true:

(i) P is isomorphic (as a smooth projective plane) to the octonion plane P2O,

(ii) Ψ ∼= Spin8R and dim Φ ≤ 38,

(iii) dim Ψ ≤ 14 and dim Φ ≤ 31.

Proof. Let us assume that P is not isomorphic to the octonion plane. Then P is neither atranslation plane nor a dual one according to Otte’s Theorem. Thus we have dim Φ[p,p] <

16, where p = K ∧ L. Consequently, we may assume that, say, dim Φ[p,K] < 8 holds.We have mentioned already that the group Ψ is compact and locally isomorphic to somesubgroup of Spin8R. Assume first that dim Ψ = dim Spin8R = 28. Then Ψ ∼= Spin8R,because the group (P)SO8R cannot act on the tangent plane Ap, see M. Luneburg [87], II,


Korollar 1 or [114], 55.40. An irreducible real representation of a group locally isomorphicto Spin8R has dimension at least 8. Thus, the group Ψ acts trivially on the elation groupΦ[p,K], since we have dim Φ[p,K] < 8. The center of Spin8R is isomorphic to Z2 × Z2.These involutions cannot act as Baer involutions on Ap, since Spin8R neither acts triviallyon a (8-dimensional) Baer subplane nor acts nontrivially (as the group SO8R) on a Baersubplane. Hence, the center of Ψ contains a central involution ω with center p. ByCorollary (5.24), this involution ω is not an elation and we may apply Lemma (6.14) inorder to get Φ[p,K] = 1l. Thus we obtain

dim Φ[p,p] ≤ Φ[p,K] + 8 ≤ 8

by Salzmann [112], (F). According to Theorem (12.2), the group DpΨ is a maximal compactsubgroup of Ω, and Lemma (6.9) together with Theorem (A.2) of the Appendix yields

dim Φ = dim DpΦ + dim kerDp = dim DpΦ + dim Φ[p,p] ≤ (dim Ψ + 2) + 8 = 38.

Now, assume that 14 ≤ dim Ψ < dim Spin8R = 28 holds. Then, by Lemma (12.3), themap Dp : Φ → Ω is closed. Let K be a maximal compact subgroup of Φ that containsthe (compact) Levi subgroup Ψ. According to the list (∗) and the discussion precedingthis theorem, the group K is isomorphic to one the groups Spin7R, SU4C× T

/〈−1l〉, U4C,

SU4C, or G2(−14). Let us first consider the last group. Then dim K = 14, and

dim Φ ≤ dim K + 2 + dim Φ[p,p] ≤ 14 + 2 + 15 = 31

follows. The groups Spin7R and SU4C both have a unique 8-dimensional irreducible realrepresentation which maps their central involution α onto −1l. Since K is compact and fixesthe lines K and L through p, the derivation map Dp maps K bijectively into GL8R×GL8Rif we identify the point space of Ap with the product TpK ×TpL. If DpK is trivial on oneof the spread elements TpK or TpL, then DpK is a homology group with axis, say, TpK(note: since DpK is compact, it cannot contain an elation with center TpK). According toT. Buchanan and H. Hahl [26], a homology group of a locally compact connected translationplane is a closed subgroup of the multiplicative group of the quaternions. Thus, thederivative Dp maps the central involution α onto −1l of GL16R. In particular, this provesthat Dpα is a homology with center o and so α is a central collineation with center p. Sincewe have dim Φ[p,K] < 8, the group K must act trivially on the elation group Φ[p,K]. NowLemma (6.14) applies and we get Φ[p,K] = 1l. This provides the inequality

dim Φ ≤ dim K + 2 + dim Φ[p,p] ≤ 21 + 2 + 8 = 31.

Turning to the last case, we assume that dim Ψ ≤ 13. If dimΨ ≤ 10, then dim DpΦ =dim Ψ + dim Π ≤ 10 + 6 = 16, where Π =

√DpΦ, and hence we get

dim Φ = dim DpΦ + dim Φ[p,p] ≤ 16 + 15 = 31.


Thus, it remains to consider the case 11 ≤ dim Ψ ≤ 13. Then Ψ is not quasi-simple andwe conclude by using the classification of quasi-simple Lie groups that the torus rank ofΨ is at least 3. This yields dim Π ≤ 6 − 3 = 3, and as before we derive the inequalitydim Φ ≤ (13 + 3) + 15 = 31. This proves the Theorem.

Remark. The dimension of the stabilizer of two lines of P2O has dimension 46. Using re-construction methods, it is possible to show that a 16-dimensional (not necessarily smooth)compact projective plane P admitting a 38-dimensional collineation group which containsa subgroup isomorphic to Spin8R and which fixes two distinct lines is the classical Mo-ufang plane over the Cayley numbers. Hence, for smooth projective planes part (ii) of thepreceding theorem can be sharpended as follows:

(ii) Ψ ∼= Spin8R and dim Φ ≤ 37.

The next proposition shows that we may restrict our attention to the case, where theset F∆ of fixed elements of ∆ is given by F∆ = p, L for some flag (p, L), and where aLevi subgroup Ψ of ∆ is isomorphic to G2(−14).

(12.7) Proposition. If F∆ 6= p, L, where (p, L) is a flag, or if a Levi subgroup Ψ of ∆

is not isomorphic to the compact exceptional Lie group G2(−14), then P ∼= P2O.

Proof. If ∆ has more than four fixed elements, then up to duality we may assume that∆ fixes three points o, u, v. Select a fourth point p ∈ P not on one of the lines o ∨ u,o ∨ v, and u ∨ v. Then the set o, u, v, p forms a (possibly degenerated) quadrangle, andwe get dim ∆ = dim p∆ + dim ∆p ≤ 16 + dim ∆p. If o, u, v, p is non-degenerate, then ∆p

stabilizes a quadrangle. From Salzmann [106], we infer that dim ∆p ≤ 14, whence dim ∆ ≤16 + 14 = 30 follows. If, on the other hand, the quadrangle o, u, v, p is degenerate, thenwe may choose a fifth point q 6= o, p on the line o∨ p such that u, v, p, q is non-degenerate.Since ∆p fixes the line o ∨ p, we get dim ∆p = dim q∆p + dim ∆p,q ≤ 8 + 14 = 22, and inthis case, dim ∆ ≤ 38 follows. Hence we may assume that ∆ fixes at most four elements.If ∆ fixes two points or two lines, then P ∼= P2O by Theorem (12.6). If ∆ fixes an anti-flag,we have P ∼= P2O according to Salzmann [109], (2.2). If ∆ fixes exactly one element, thenP ∼= P2O due to Salzmann [111], (C). Finally, if ∆ has no fixed elements at all, then ∆ hasto be a semi-simple group (see Salzmann [110], (2.1). Note that the involution of a centraltorus subgroup of ∆ is a reflection by [105], (1.8), and the center of this reflection is fixedby the whole group ∆.) This implies again that P ∼= P2O (Salzmann [111], (A)). Thus wehave proved the first part of the proposition.Now let Ψ 6∼= G2(−14). According to the first part we may assume that F∆ = p, L forsome flag (p, L). Then P or its dual plane is a translation plane by M. Luneburg, [87],


V, Satz, and Otte’s Theorem [94] on translation planes gives P ∼= P2O. This proves theproposition.

We recall that P is a 16-dimensional smooth projective plane, that the group ∆ is aclosed connected subgroup of Aut(P) with dim ∆ ≥ 39 which fixes exactly one flag (p, L),and that a Levi subgroup Ψ of ∆ is isomorphic to the compact exceptional Lie groupG2(−14). We proceed by verifying a series of small lemmas. We will always omit the proofsof the dual statements.

(A) We have dim ∆ = 39, dim ∆K = 31 for every line K ∈ Lp \ L, and Ψ is a maximal

compact subgroup of ∆. Moreover, the group ∆ acts transitively on Lp \ L and on

L \ p.

Since Ψ ∼= G2(−14), we obtain from the Stabilizer Theorem (12.6) that dim ∆K ≤ 31 holds.This gives

39 ≤ dim ∆ = dim ∆K + dimK∆ ≤ 31 + 8 = 39,

whence we have dim ∆ = 39 and dim ∆K = 31. The second statement follows immediatelyfrom the proof of Theorem (12.6). Let us turn to the last assertion. Since Lp \ L ∼= R8

is connected and because every orbit of ∆ on Lp \ L is 8-dimensional and hence is openin Lp \ L, we conclude that ∆ acts transitively on Lp \ L.

Using the tables of J. Tits [123], we get the following list.

(B) A compact semi-simple Lie group Υ of torus rank at most two is locally isomorphic to

one of the groups SOkR, SU3C, or G2(−14), where 3 ≤ k ≤ 5.

(C) A proper closed semi-simple subgroup Υ of G2(−14) has dimension at most 8.

If Υ is a proper closed subgroup of G2(−14) then dim Υ ≤ 9 holds according to Theorem 2of Chapt.VI of [88]. This excludes the groups SO5R and G2(−14) from the list in (B) andhence assertion (C) is proved.

(D) For every line K ∈ Lp \ L a Levi subgroup of the stabilizer ∆1K is isomorphic to

G2(−14). In particular, any two Levi subgroups of the stabilizers ∆K are conjugate in ∆.

Fix some stabilizer ∆K and let Υ be a Levi subgroup of ∆1K . Assume that Υ 6∼= G2(−14).

Using (C), we obtain dimΥ ≤ 8, and we infer from Lemma (6.9) and Lemma (12.5) that

dim ∆K = dim Dp∆K + dim(∆K)[p,p] ≤ (dim Υ− rkΥ + 6) + 15 ≤ 14 + 15 = 29

holds which is a contradiction to (A). Hence assertion (D) is proved.


(E) dim ∆[p,p] = 15, and dually, dim ∆[L,L] = 15.

The derivation map Dp : ∆K → Ω is closed by Lemma (12.3) and (D). Using the StabilizerTheorem (12.6), we infer from (A) and (D) that

31 = dim Dp∆K + dim kerDp ≤ (dim Ψ + 2) + dim ∆[p,p] = 16 + dim ∆[p,p]

holds for every line K ∈ Lp \ L. This shows that dim ∆[p,p] = 15.

Furthermore, the inequality above is in fact an equality, whence assertion

(F) dim Dp∆K = dim Ψ + 2

is true for every line K ∈ Lp \ L.

(G) We have dim q∆K ≥ 7 and dimK∆q ≥ 7 for every point q ∈ L \ p and every line

K ∈ Lp \ L.

Fix some point q ∈ L \ p and some line K ∈ Lp \ L. Then, from (A), Lemma (12.5)and the Stabilizer Theorem (12.6), we obtain the inequality

31 = dim ∆K = dim ∆K,q + dim q∆K ≤ dim Dp∆K,q + dim kerDp ∆K,q+ dim q∆K

≤ dim Dp∆K + dim(∆[p,p])q + dim q∆K

≤ dim Ψ + 2 + dim(∆[p,p])q + dim q∆K

≤ 14 + 2 + 8 + dim q∆K

= 24 + dim q∆K ,

which gives dim q∆K ≥ 7.

(H) dim ∆[p,L] = 8.

Since ∆ acts transitively on L \ p by (A), we know that dim ∆[q,L] is independent of thepoint q ∈ L\p. According to Salzmann [112], §0, (G), this implies that ∆[p,L] is transitiveon K \ p for every line K ∈ Lp \ L. In particular, we have dim ∆[p,L] = dimK = 8.

(I) dim ∆[q,L] = 7 and, dually, dim ∆[p,K] = 7 holds for every point q ∈ L \ p and every

line K ∈ Lp \ L.

According to Salzmann [112], (F), we have the inequality dim ∆[L,L] ≤ dim ∆[q,L] + 8 forany point q ∈ L. Thus, we infer from dim ∆[L,L] = 15 that dim ∆[q,L] ≥ 7. If dim ∆[q,L] = 8for q 6= p, then dim ∆[L,L] = 16 follows, a contradiction to (E).

(12.8) Proposition. There exists a non-trivial homology ω ∈ ∆[p].


Proof. We have dim ∆K = 31 for any line K ∈ Lp \ L by (A). Assertion (D) says thatevery Levi subgroup Ψ of ∆1

K is isomorphic to G2(−14). The group Ψ is a maximal compactsubgroup of ∆ (and hence of ∆1

K as well), and dim Dp∆K = dim Ψ+2 holds by (F). Hencewe may apply Theorem (A.2) of the Appendix in order to obtain that H = (Dp∆K)1[o,L∞]

is isomorphic to R>0. Since H fixes every tangent space TpG for G ∈ Lp, every collineationof the inverse image Θ = D−1

p H fixes every line through the point p. Conversely, Dp mapsevery central collineation of ∆1

[p] into H. This shows that ∆1[p] = Θ. By Proposition (7.3),

the group ∆1[p] can be written as a semi-direct product ∆1

[p] = ∆1[p,A] n ∆1

[p,p] for someline A 6∈ Lp. In particular, the group ∆[p,A] is not trivial, because of ∆[p,p] = kerDp andH 6= 1l. This proves the Proposition.

Fix K ∈ Lp \ L and choose a line M ∈ L \ Lp such that M∆ 6= M. Such aline exists because of F∆ = p, L. Due to Proposition (12.8) we can apply a theorem ofH.Hahl [52], Corollary 1.3, which yields M∆ = M∆[p,p] . Using (E) we get

dimM∆ = dimM∆[p,p] = dim ∆[p,p] = 15

anddim ∆M = dim ∆− dimM∆ = 39− 15 = 24.

The stabilizer ∆M fixes the antiflag (p,M). Thus we have (∆M )[p,p] = 1l and fromLemma (6.9) we conclude that

dim Dp∆K,M = dim ∆K,M − dim(∆K,M )[p,p] = dim ∆K − dimM∆K − 0 ≥ 24− 8 = 16.

Since ker Dp ∆K,M= (∆K,M )[p,p] = 1l, the restriction Dp ∆K,M

is an injection.

(12.9) Lemma. Every Levi subgroup Υ of ∆1K,M is isomorphic to G2(−14).

Proof. Since Ψ ∼= G2(−14), the group Υ is a subgroup of G2(−14). If Υ 6∼= G2(−14), thendim Υ ≤ 8 by (C) and

dim Dp∆K,M ≤ dim Υ + 6− rkΥ ≤ 8 + 6 = 14

follows, which is a contradiction. Thus we have Υ ∼= G2(−14).

According to Lemma (12.9) and the remarks preceding this lemma, we can argue as inProposition (12.8) in order to get the following corollary. Note that a homology of ∆K,M

with center q = M ∧ L must have K as its axis.


(12.10) Corollary. There exists a non-trivial homology ω∗ ∈ ∆[q,K].

(12.11) Corollary. The stabilizers ∆1q and ∆1

K can be decomposed in the following way:

∆1q∼= ∆1

q,K n ∆1[q,L] and ∆1

K∼= ∆1

q,K n ∆1[p,K].

Proof. Proposition (12.8) enables to apply Corollary 1.3 of [52] which yields K∆1q = K∆1

[q,q] .In particular, this gives dim ∆q

/∆q,K

= dim ∆[q,q]. Since ∆1[q,q] = ∆1

[q,L] is a normalsubgroup of ∆1

q that intersects ∆q,K trivially, the first decomposition is proved, The seconddecomposition is proved dually.

Now we have collected all the information we need in order to proof Theorem (12.1).Our proof is inspired by the proof of Satz 4 in Chapter VI of [87].

Proof of Theorem (12.1). By Lemma (12.9), we may assume that the Levi subgroupΨ of ∆ is contained in ∆1

K,M . Let Z be the connected component of the centralizer C∆Ψ.Since Ψ fixes both q and K, the subgroups ∆1

q, ∆1K and ∆1

q,K are normalized by Ψ. Using[87], VI, Lemma, we thus get

dim Zq ≡ dim ∆q(mod 7), dim ZK ≡ dim ∆K(mod 7), dim Zq,K ≡ dim ∆q,K(mod 7).

The fixed set FΨ of Ψ is a 2-dimensional subplane of P and the centralizer Z acts almosteffectively on FΨ, see [87], VII, Satz 5(iv), and p, q,K,L are contained in FΨ. Hence wehave

1 ≥ dim qZ = dim Z− dim Zq, and 1 ≥ dimKZ = dim Z− dim ZK ,

because Z fixes p as well as L, and q ∈ L ∩ FΨ, K ∈ Lp ∩ FΨ. If the orbit qZK is one-dimensional, we use Corollary (12.11) and (I) to get the contradiction

7 = dim∆[p,K] = dim ∆K − dim ∆q,K ≡ dim ZK − dim Zq,K = dim qZK = 1(mod 7).

Hence we may assume that the orbit qZK (and, dually, the orbit KZq ) is zero-dimensional.Then Z1

K fixes q and, dually, Z1q fixes K. The stabilizer of a quadrangle of a 2-dimensional

compact projective plane is trivial, [101], 4.1, whence we have dim Zq ≤ 3 and dim ZK ≤ 3.From [87], VI, Lemma, we have dim Z ≡ dim ∆(mod 7), which implies that dim Z = 4. Inparticular, we have Z 6= Zq ∪ ZK . Choose a collineation ζ ∈ Z \ (Zq ∪ ZK). Select somepoint u ∈ K ∩ FΨ distinct from p. From Z1

Kζ = (Z1K)ζ we infer that

(ZK,Kζ )1 ≤ Z1K ∩ Z1

Kζ = Z1K ∩ (Z1

K)ζ = Z1q,K ∩ (Z1

q,K)ζ ≤ Zq,qζ ,K,Kζ ,

whence the stabilizer Zu,K,Kζ fixes the quadrangleq, qζ , (u ∨ q) ∧Kζ , (u ∨ qζ) ∧Kζ

.

However, dim ZK,Kζ ≥ 2 holds, because of K,Kζ ∈ Lp ∩ FΨ and dim Z = 4. Conse-quently, we have dim Zu,K,Kζ ≥ 1, which is a contradiction to Salzmann [101], 4.1. Thisfinishes the proof of the Theorem.

Appendix 109

Appendix on Affine Translation Planes

Let A = (A,L) be a locally compact topological affine translation plane of dimensionn = 2l = 2k. We choose some point 0 in A as well as three distinct lines X,Y,E ∈ L0.Fixing a ”unit point” e in E \ 0, the affine translation plane A is coordinatized bysome quasifield Q whose additive group is isomorphic to Rl. In this setting we haveA = Q × Q ≈ R2l. The origin 0 has coordinates (0, 0) and X = Q × 0, Y = 0 × Q,E = diag(Q × Q). The connected component A of the collineation group Aut(A) of A isa semi-direct product A = A0 n T, where T ∼= R2l is the group of translations and A0 isthe stabilizer of the origin 0. Moreover, the stabilizer AX,Y can be expressed in terms oflinear mappings of the real vector space Q, namely

AX,Y =(B,C) : Q2 → Q2 : (x, y) 7→ (Bx,Cy)

∣∣ B,C ∈ GL(Q).

Since we have E = diag(Q × Q) the stabilizer of the three lines X, Y and E can beexpressed as

AX,Y,E =(B,B) : Q2 → Q2 : (x, y) 7→ (Bx,By)

∣∣ B ∈ GL(Q).

The following theorems of H.Hahl (see [50], 2.1) are crucial in the theory of locally compacttopological translation planes.

(A.1) Theorem. The stabilizer B = AX,Y,E has a largest compact subgroup M with

M = (A,A) ∈ A | detA = 1, and dim B/M ≤ 1 holds. Moreover, the group M is a closed

subgroup of SOlR.

(A.2) Theorem. The stabilizer B′ = AX,Y has a largest compact subgroup M′ with

M′ = (A,B) ∈ A | detA = detB = 1, and dim B′/M′ ≤ 2 holds. Moreover, the group M′

is a closed subgroup of SOlR×SOlR. If dim B′/M′ = 2, then there is a closed one-parameter

subgroup P of B′ such that B′ can be wriiten as B′ = M′ · P · A[0,L∞].

The next proposition shows that the torus rank of the stabilizer B (or, equivalently,of M) of a 2k-dimensional locally compact affine translation plane A cannot reach theexponent k.

(A.3) Proposition. We have rkB < k.

110 Appendix

Proof. By M.Luneburg [87], II, Satz 2 (see also [114], 55.37), the torus rank r of theautomorphism group Γ of the projective closure P of A cannot exceed the exponent kof the dimension 2k of A. Moreover, the group Γ contains three different commutingreflections if r = k ≥ 2. The group B, considered as a group of collineations of P fixingthe line L∞ at infinity, fixes the degenerate quadrangle defined by the lines X, Y , E, andL∞. Thus the reflections in question must have the common center 0 = X ∧ Y whichcontradicts a result of H. Salzmann [107], §1(2). Hence we have r < k if k ≥ 2. It remainsto study the case k = 1. Then, the plane P is isomorphic to the real projective plane (seeSalzmann [100], 7.24), and the group B is a subgroup of the additive group of the realnumbers. In particular, the group B has no torus subgroup, whence rkB = 0 follows.

Representation of affine planes by spreads. Every affine translation plane A can becoordinatized by some (not necessarily topological) quasifield Q. Every quasifield containsa subfield F called its kernel, and Q can be regarded as a vector space over F. If A is alocally compact connected topological affine plane, then the kernel F is isomorphic to R, Cor H, see [26]. Moreover, the quasifield Q has dimension 2k (0 ≤ k ≤ 3) over F. Since theset A of points can be identified with the R-vector space Q × Q ∼= Rn, the lines throughthe origin o are represented by linear subspaces of dimension l = 1

2n. This representationof an affine plane is due to Andre [1]. The family S of all linear subspaces through theorigin covers the point set A and any two such subspaces are complementary in A. Sucha family S is called a spread on A. A spread S ⊂ Gl(Rn) defines a locally compact affinetranslation plane if and only if S is compact in the Grassmann topology of Gl(Rn), seeLowen [84]. There is a slightly different point of view due to Bruck and Bose [25], wherethe vector space Q × Q is replaced by the projective space PnF. Within this model it iseasier to describe the projective completion of A. We will not discuss the constructionof Bruck and Bose in greater detail, since we are mainly interested in affine translationplanes. For more information on locally compact translation planes we recommend therecent exposition of N.Knarr, [71].

Representation of spreads by Grassmann coordinates. Let S be a (compact) spreadof Gl(Rn). Choose two distinct subspaces X and Y in S. Then we may represent eachelement S of S \ X,Y by a regular linear map λS : X → Y , namely as the graph(x, λS(x)) | x ∈ X. Define λX : X → Y as the zero map, and put S′ := S \ Y . Themapping λ : S 7→ λS : S′ → L(X,Y ) ∼= Rl×l is a homeomorphism from S to λ(S), seeBourbaki, [17], VI, §3.5. If S is the generating spread of a smooth affine plane, then S

is even a smooth submanifold of Gl(Rn), cp. Otte, Korollar 5.15. With respect to theordinary smooth structure on Rl×l, the map λ is a diffeomorphism between S and λ(S).

Glossary 111

Glossary

S = (P,L,F) A smooth stable plane with set P of points, set L of lines, and set F of flagsP = (P,L,F) A smooth projective planeO ⊂ L× L The open (domain) of the intersection map ∧ of a smooth stable planeLp The line pencil through the point p ∈ Pp ∨ q The line joining the points p and qK ∧ L The intersection of the two lines K and Ln = 2l = 2k The dimension of a compact connected projective plane, 1 ≤ k ≤ 4

Ap A tangent translation plane of P at the point p ∈ P .Pp The projective completion of Ap with L∞ as the line at infinitySp The tangent spread defined by the line pencil Lp

SL The line of Sp which is tangent to L ∈ Lp, i.e. SL = TpL

Γ The group of all continuous (smooth) collineations of a smooth geometryΓ1 The connected component of the identity of Γ

∆ A closed connected subgroup of Γ

Ψ A Levi subgroup of ∆

Σ A quasi-simple subgroup of ∆√∆ The solvable radical of ∆

K A maximal compact subgroup of ∆

Γ[c] The subgroup of central collineations of Γ with center cΓ[A] The subgroup of central collineations of Γ with axis AΓ[c,c] The subgroup of elations of Γ with center cΓ[A,A] The subgroup of elations of Γ with axis A

rkG The dimension of a maximal torus subgroup of a Lie group G

A The affine collineation group of the tangent plane Ap

A0 The stabilizer of the origin of A

T The translation group of Ap

SA0 The mappings of A0 that have determinant 1

Z(G) The center of the group G〈M〉 A group generated by Mker The kernel of a group homomorphism

112 Glossary

GnN The semidirect product of G and N (N is the normal subgroup)CGH The centralizer of H in GK× The multiplicative group of the (skew)field K1l The identity mapping or the trivial group

xG The orbit of x under the group GGx The stabilizer of x in GG[A] The pointwise stabilizer of A ⊆ X in GgA

The restriction of g to the subset AGA

The group induced by G on A∼= isomorphic≈ homeomorphic

ρ, ηp,L, ηL,P ProjectivitiesπP : F → P The canonical projection from F to the set P of pointsπL : F → L The canonical projection from F to the set L of linesϑ The characteristic map of a smooth stable plane

(TX,X, τX) The tangent bundle of the smooth manifold XDf The derivative of the function f

ξ = (E,B, π) A locally trivial bundle with total space E, base space B, and projection πE(ξ) The total space of the bundle ξE(ξ)0 The total space of the vector bundle ξ with the zero section removedPξ The projective bundle associated to the vector bundle ξ

R The real numbersC The complex numbersH The the quaternionsO The Cayley numbers (octonions)Sn The unit sphere in Rn+1

GLnF The general linear group over the field FGL(V ) The general linear group of the vector space VOn(F, h) The orthogonal group over the field F with respect to the form h

OnR The standard orthogonal group over the real numbersO(V ) The standard orthogonal group of the real vector space VT = R

/Z The torus group

Ω The universal covering group of SL2R

Vk(Rn) The Stiefel manifold of k-frames in Rn

Gk(Rn) The Grassmann manifold of k-dimensional subspaces of Rn

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