The old and venerable subject of geometry has been changed radically by the famous book of
Hilbert [1899, 30] on the foundations of geometry. It was Hilbert's aim to give a simple axiomatic
characterization of the real (Euclidean) geometries. He expressed the necessary continuity
assumptions in terms of properties of an order. Indeed, the real projective plane is the only
desarguesian ordered projective plane where every monotone sequence of points has a limit, see the
elegant exposition in Coxeter [61].

However, the stipulation of an order excludes the geometries over the complex numbers (or over the quaternions or octonions) from the discussion. In order to include these geometries, the order properties have been replaced by topological assumptions (like local compactness and connectedness), see Kolmogoroff [32], Köthe [39], Skornjakov [54], Salzmann [55, 57], Freudenthal [57a,b]. This is the historical origin of topological geometry in the sense of this book.

Topological geometry studies incidence geometries endowed with topologies which are compatible with the geometric structure. The prototype of a topological geometry is a topological projective plane, that is, a projective plane such that the two geometric operations of joining distinct points and intersecting distinct lines are continuous (with respect to given topologies on the point set and on the line set). Only few results can be proved about topological planes in general. In order to obtain deeper results, and in order to stay closer to the classical geometries, we concentrate on compact, connected projective planes. Planes of this type exist in abundance; topologically they are very close to one of the four classical planes treated in Chapter 1, but they can deviate considerably from these classical planes in their incidence-geometric structure. Most theorems in this book have a homogeneity hypothesis requiring that the plane in question admits a collineation group which is large in some sense. Of course, there is a multitude of possibilities for the meaning of large. It is a major theme here to consider compact projective planes with collineation groups of large topological dimension. This approach connects group theory and geometry, in the spirit of F. Klein's Erlangen program. We shall indeed use various methods to describe a geometry in group-theoretic terms, see the remarks after (32.20). Usually, the groups appearing in our context turn out to be Lie groups.

In this book we consider mainly projective (or affine) planes. This restriction is made for conciseness. Let us comment briefly on some other types of incidence geometries (compare Buekenhout [95] for a panorama of incidence geometry). Topological projective spaces have been considered by Misfeld [68], Kühne-Löwen [92] and others, see also Groh [86a,b]. It is a general phenomenon that spatial geometries are automatically much more homogeneous than plane geometries; we just mention the validity of Desargues' theorem (and its consequences) in each projective space, and the more recent classification of all spherical buildings of rank at least 3 by Tits. This phenomenon effectuates a fundamental dichotomy between plane geometry and space geometry. Stable planes are a natural generalization of topological projective planes, leading to a rich theory, compare (31.26). Typical examples are obtained as open subgeometries of topological projective planes. The reader is referred to Grundhöfer-Löwen [95] and Steinke [95] for surveys on locally compact space geometries (including stable planes) and circle geometries, respectively.

Projective planes are the same thing as generalized triangles, and the generalized polygons are precisely the buildings of rank 2. A theory of topological generalized polygons and of topological buildings is presently developing, see Burns-Spatzier [87], Knarr [90] and Kramer [94] for fundamental results in this direction, compare also Grundhöfer-Löwen [95] Section 6. These geometries are of particular interest in differential geometry, see Thorbergsson [91, 92]. Another connection with differential geometry is provided by the study of symmetric planes, see Löwen [79a,b, 81b], Seidel [90a, 91], H. Löwe [94, 95], Grundhöfer-Löwen [95] 5.27ff.

Now we give a rough description of the contents of this book (see also the introduction of each chapter).

In Chapter 1, we consider in detail the classical projective planes over the real numbers, over the
complex numbers, over Hamilton's quaternions, and over Cayley's octonions; these classical division
algebras are denoted by **R**, **C**, **H**, **O**. The four classical planes are the prime examples
(and also the most homogeneous examples, as it turns out) of compact, connected projective planes.
We describe the full collineation groups of the classical planes, as well as various interesting
subgroups, like motion groups with respect to polarities. In the case of the octonion plane, this
comprises a complete and elementary description of some exceptional Lie groups (and of their actions
on the octonion plane), including proofs of their simplicity; e.g. the full collineation group has
type E_{6}, and the elliptic motion group is the compact group of type F_{4}.

Chapter 2 is a brief summary of notions and results concerning projective and affine planes, coordinates and collineations. It is meant as a reference for known facts which entirely belong to incidence geometry.

In Chapter 3 we study planes on the point set **R**^{2}, with lines which are homeomorphic to the real
line **R**, so that each line is a curve in **R**^{2}. This chapter is the most intuitive part of the
book. If the parallel axiom is satisfied, that is, if we have an affine plane, then we can form the
usual projective completion, which leads to a (topologically) 2-dimensional compact projective
plane. These planes have been studied by Salzmann in 1957-1967 with remarkable success.
He proved that
the full collineation group S of such a plane is a Lie group of dimension at most 8, that the
real projective plane P_{2}**R** is characterized by the condition dimS > 4, and that the
Moulton planes are the only planes of this type with dimS = 4. Furthermore, he explicitly
classified all 2-dimensional compact projective planes with dimS = 3. All these
classification results are proved in Chapter 3.

In Chapter 4 we begin a systematic study of topological projective planes in general. Most results
require compactness, and some results (like contractibility properties) are based on connectedness
assumptions. Note that, for every prime p, the plane over the p-adic numbers **Q**_{p} provides an
example of a compact, totally disconnected plane. We show that the four classical planes studied in
Chapter 1 are precisely the compact, connected Moufang planes; Moufang planes are defined by a very
strong homogeneity condition, which implies transitivity on triangles (and even on quadrangles).
Furthermore, we prove that the group of all continuous collineations of a compact projective plane
is always a locally compact group (with respect to the compact-open topology).

Chapter 5 deals with the algebraic topology of compact, connected projective planes of finite
topological dimension. As Löwen has shown, the point spaces of these planes have the very same
homology invariants as their classical counterparts considered in Chapter 1; moreover, the lines are
homotopy equivalent to spheres. We obtain that the topological dimension of a line in such a plane
is one of the numbers 1, 2 ,4, 8; the (point sets of the) corresponding planes have topological
dimensions 2, 4, 8, 16. The topological resemblance to classical planes has strong geometric
consequences, which are discussed in Section 55 and in Chapters 6-8. In fact, these results
determine a subdivision of the whole theory into four cases. In order to understand Chapters 6-8,
it suffices to be acquainted with the main *results * of Chapter 5; the *methods * of
proof in that chapter are not used in other chapters.

In Chapter 6 we consider compact, connected projective planes which are homogeneous in some sense. As indicated above, the idea of homogeneity plays a central rôle in this book. We prove that a compact, connected projective plane which admits an automorphism group transitive on points is isomorphic to one of the four classical planes treated in Chapter 1. This is a remarkable result; it says that for compact, connected projective planes, the Moufang condition is a consequence of transitivity on points. Furthermore, we consider groups of axial collineations and transitivity conditions for these groups, and we study planes which admit a classical motion group. Often, these homogeneity conditions are strong enough to allow an explicit classification of the planes in question. In Section 65 we employ the topological dimension of the automorphism group as a measure of homogeneity (this idea is fully developed in Chapters 3, 7, 8), and Section 66 is a short report on groups of projectivities in our context.

In Chapters 7 and 8 we determine all compact projective planes of dimension 4, 8 or 16 which admit
an automorphism group of sufficiently large topological dimension. This approach leads first to the
classical planes over **C**, **H**, **O**, and then the most homogeneous non-classical planes appear
in a systematic fashion. In contrast to Chapter 3, deeper methods are required, and proper
translation planes arise.

In Chapter 7 we study compact projective planes of topological dimension 4; these planes are the
topological relatives of the complex projective plane P_{2}**C**. We prove that the automorphism
group S of such a plane P is a (real) Lie group of dimension at most 16, and that the
complex projective plane is characterized by the condition dimS > 8. This result is one of the
highlights of the theory of 4-dimensional planes. If dimS ³ 7, then P is a translation plane (up to duality) or
a shift plane. All translation planes P with dimS ³ 7 and all shift planes P with
dimS ³ 6 have been classified explicitly; Chapter 7 contains a classification of the
translation planes with dimS ³ 8 and of the shift planes with dimS ³ 7. Finally,
we show in Section 75 that only the complex plane admits a complex analytic structure.

The theory of 4-dimensional compact planes is distinguished from the theory of higher-dimensional compact planes, regarding both the phenomena and the methods. For instance, the class of shift planes appears only in low dimensions. In higher dimensions, special tools connected with the recognition and handling of low-dimensional manifolds are not available.

Chapter 8 deals with compact projective planes of
topological dimension 8 or 16, that is, with the
relatives of the quaternion plane P_{2} **H** or of
the octonion plane P_{2} **O**. In some parts of this
chapter, the results are only surveyed,
with references to the literature. Again, classification
results on planes admitting automorphism groups of large
dimension constitute the main theme. It turns out that
such planes are often translation planes up to duality.
They carry a vector space structure, whence
special tools become available. Accordingly, the theory of translation
planes and the classification of the most homogeneous
ones form a theory on their own. Fundamental results of
this theory are developed in Section 81; in Section 82,
the classification of all 8-dimensional compact
translation planes P satisfying
dimAut P ³ 17 and of all 16-dimensional
compact translation planes P satisfying
dimAut P ³ 38 is presented.

In the following
sections, classification results of this kind are
extended to 8- and 16-dimensional compact planes
in general. For reasons of space, the results often
are not proved in their strongest form.
Salzmann [81a, 90] proved, on the basis of Hähl [78],
that the quaternion plane
P_{2} **H** is the only compact projective plane of
dimension 8 such that dimAut P > 18. In
Section 84, this result is proved under the stronger
assumption
dimAut P ³ 23. Similarly, the octonion plane
is known to be the only compact projective plane P
of dimension 16 such that dimAut P > 40,
see Salzmann [87], Hähl [88]. In Section 85,
we characterize the octonion plane by the stronger
condition that dimAut P ³ 57. The proofs of
these characterization results make use of the
corresponding characterization results for translation
planes.

In Section 86, we construct and characterize the compact Hughes planes of dimensions 8 and 16. They form two one-parameter families of planes with rather singular properties. Section 87 contains basic results indicating a viable route towards an extension of the classification results presented here. This should help the reader to go beyond the limitations of the exposition here; moreover, it may serve as a guide to future research. Among other things, these results explain the special rôle played in the classification by translation planes on the one hand and by Hughes planes on the other hand.

The final Chapter 9 is an appendix. Here we collect a number of results from topology, and we give a systematic outline of Lie theory, as required in this book. In this chapter we usually do not give proofs, but rather refer to the literature (with an attempt to give references also for folklore results). The topics covered in Chapter 9 include the topological characterization of Lie groups (Hilbert's fifth problem), and the structure and the classification of (simple) Lie groups; in fact, we require only results for groups of dimension at most 52. Furthermore we report on real linear representations of almost simple Lie groups, and we list all irreducible representations of these groups on real vector spaces of dimension at most 16. Finally, we deal with various classification results on (not necessarily compact) transformation groups.

This book gives a systematic account of many results which are scattered in the literature. Some results are presented in improved form, or with simplified proofs, others only in weakened versions. A few of the more recent results are only mentioned, because their proofs appear to be too complex to be included here. However, we hope to provide a convenient introduction to compact, connected projective planes, as well as a sound foundation for future research in this area.

Many colleagues and friends have offered helpful advice, or have read parts of the typescript and contributed improvements. In particular, the authors would like to thank Richard Bödi, Sven Boekholt, Michael Dowling, Karl Heinrich Hofmann, Norbert Knarr, Linus Kramer, Helmut Mäurer, Kai Niemann, Joachim Otte, Burkard Polster, Barbara Priwitzer, Eberhard Schröder, Jan Stevens, Peter Sperner, and Bernhild Stroppel. We are also indebted to the Oberwolfach Institute, which gave us the opportunity to hold several meetings devoted to the work on this book. Finally, we would like to thank de Gruyter Verlag and, in particular, Dr. Manfred Karbe, for support and professional advice.

Tübingen, September 1994

The authors

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