Topological geometry
Topological geometry deals with incidence structures consisting of a point set formula_1 and a family formula_2 of subsets of formula_1 called lines or circles etc. such that both formula_1 and formula_2 carry a topology and all geometric operations like joining points by a line or intersecting lines are continuous. As in the case of topological groups, many deeper results require the point space to be (locally) compact and connected. This generalizes the observation that the line joining two distinct points in the Euclidean plane depends continuously on the pair of points and the intersection point of two lines  is a continuous function of these lines. Linear geometries. Linear geometries are incidence structures in which any two distinct points formula_6 and formula_7 are joined by a unique line formula_8. Such geometries are called "topological" if formula_8 depends continuously on the pair formula_10 with respect to given topologies on the point set and the line set. The "dual" of a linear geometry is obtained by interchanging the roles of points and lines. A survey of linear topological geometries is given in Chapter 23 of the "Handbook of incidence geometry". The most extensively investigated topological linear geometries are those which are  also dual topological linear geometries. Such geometries are known as topological projective planes. History. A systematic study of these planes began in 1954 with a paper by Skornyakov. Earlier, the topological properties of the real plane had been introduced via ordering relations on the affine lines, see, e.g., Hilbert, Coxeter, and O. Wyler. The completeness of the ordering is equivalent to local compactness and implies that the affine lines are homeomorphic to formula_11 and that the point space is connected. Note that the rational numbers do not suffice to describe our intuitive notions of plane geometry and that some extension  of the rational field is necessary. In fact, the equation formula_12 for a circle has no rational solution. Topological projective planes. The approach to the topological properties of projective planes via ordering relations is not possible, however, for the planes coordinatized by the complex numbers, the quaternions or the octonion algebra. The point spaces as well as the line spaces of these "classical" planes (over the real numbers, the complex numbers, the quaternions, and the octonions) are compact manifolds of dimension formula_13. Topological dimension. The notion of the dimension of a topological space plays a prominent rôle in the study  of topological, in particular of compact connected planes. For a normal space formula_14, the dimension formula_15 can be characterized as follows: If formula_16 denotes the formula_17-sphere, then "formula_18 if, and only if, for every closed subspace formula_19 each continuous map formula_20 has a continuous extension formula_21". For details and other definitions of a dimension see and the references given there, in particular Engelking or Fedorchuk. 2-dimensional planes. The lines of a compact topological plane with a 2-dimensional point space form a family of curves homeomorphic to a circle, and this fact characterizes these planes among the topological  projective planes. Equivalently, the point space is a surface. Early examples not isomorphic to the classical real plane formula_22 have been given by Hilbert and Moulton. The continuity properties of these examples have not been considered explicitly at that time, they may have been taken for granted. Hilbert’s construction can be modified to obtain uncountably many pairwise non-isomorphic formula_23-dimensional compact planes. The traditional way to distinguish formula_22 from the other formula_23-dimensional planes is by the validity of Desargues’s theorem or the theorem of Pappos (see, e.g., Pickert for a discussion of these two configuration theorems ). The latter is known to imply the former (Hessenberg). The theorem of Desargues expresses a kind of homogeneity of the plane. In general, it holds in a projective plane if, and only if, the plane can be coordinatized by a (not necessarily commutative) field, hence it implies that the group of automorphisms is transitive on the set of quadrangles (formula_26 points no formula_27 of which are collinear). In the present setting, a much weaker homogeneity condition characterizes formula_22: Theorem. "If the automorphism group formula_29 of a formula_23-dimensional compact plane formula_31 is transitive on the point set (or the line  set), then formula_29 has a compact subgroup formula_33 which is even transitive on the set of flags" (=incident point-line pairs), "and formula_31 is classical". The automorphism group formula_35 of a formula_23-dimensional compact plane formula_31, taken with the topology of uniform convergence on the point space, is a locally compact group of dimension at most formula_38, in fact even a Lie group. All formula_23-dimensional planes such that formula_40 can be described explicitly; those with formula_41 are exactly the Moulton planes, the classical plane formula_22 is the only formula_23-dimensional plane with formula_44; see also. Compact connected planes . The results on formula_23-dimensional planes have been extended to compact planes of dimension formula_46. This is possible due to the following basic theorem: Topology of compact planes. "If the dimension of the point space formula_1 of a compact connected projective plane is finite, then formula_48 with formula_49. Moreover, each line is a homotopy sphere of dimension formula_50", see or. Special aspects of 4-dimensional planes are treated in, more recent results can be found in. The lines of a formula_26-dimensional compact plane are homeomorphic to the formula_23-sphere; in the cases formula_53 the lines are not known to  be manifolds, but in all examples which have been found so far the lines are spheres. A subplane formula_54 of a projective plane formula_31 is said to be a Baer subplane, if each point of formula_31 is incident with a line of formula_54 and each line of formula_31 contains a point of formula_54. A closed subplane formula_54 is a Baer subplane of a compact connected plane formula_31 if, and only if, the point space of formula_54 and a line of formula_31 have the same dimension. Hence the lines of an 8-dimensional plane formula_64 are homeomorphic to a sphere formula_65  if formula_31 has a closed Baer subplane. Homogeneous planes. "If formula_64 is a compact connected projective plane and if formula_35 is transitive on the point set of formula_64, then formula_29 has a flag-transitive compact subgroup formula_33 and formula_64 is classical", see or. In fact, formula_33 is an elliptic motion group. Let formula_64 be a compact plane of dimension formula_75, and write formula_76. If formula_77, then formula_31 is classical, and formula_79 is a simple Lie group of dimension formula_80 respectively. All planes formula_64 with formula_82 are known explicitly. The planes with formula_83 are exactly the projective closures of the affine  planes coordinatized by a so-called "mutation" formula_84 of the octonion algebra formula_85, where the new multiplication formula_86 is defined as follows: choose a real number formula_87 with formula_88 and put formula_89. Vast families of planes with a group of large dimension have been discovered systematically starting from assumptions about their automorphism groups, see, e.g.. Many of them are projective closures of translation planes (affine planes admitting a sharply transitive group of automorphisms mapping each line to a parallel), cf.; see also for more recent results in the case formula_90 and for formula_91. Compact projective spaces. Subplanes of projective spaces  of "geometrical" dimension at least 3 are necessarily Desarguesian, see §1 or §16 or. Therefore, all compact connected projective spaces can be coordinatized by the real or complex numbers or the quaternion field. Stable planes. The classical non-euclidean hyperbolic plane can be represented by the intersections of the straight lines in the real plane with an open circular disk. More generally, open (convex) parts of the classical affine planes are typical stable planes. A survey of these geometries can be found in, for the formula_23-dimensional case see also. Precisely, a "stable plane" formula_93 is a topological linear geometry  formula_94 such that Note that stability excludes geometries like the formula_27-dimensional affine space over formula_11 or formula_104. A stable plane formula_93 is a projective plane if, and only if, formula_1 is compact. As in the case of projective planes, line pencils are compact and homotopy equivalent to a sphere of dimension formula_50, and formula_108 with formula_109, see or. Moreover, the point space formula_1 is locally contractible. Compact groups of (proper) stable planes are rather small. Let formula_111 denote a maximal compact subgroup of the automorphism group of the classical formula_112-dimensional projective plane formula_113. Then the following theorem holds : "If a formula_112-dimensional stable plane formula_93 admits a compact group formula_116 of automorphisms such that formula_117, then formula_118", see. Flag-homogeneous stable planes. "Let formula_119 be a stable plane. If the automorphism group formula_120 is flag-transitive, then formula_93 is a classical projective or affine plane, or formula_93 is isomorphic to the interior of the absolute sphere of the hyperbolic polarity of a classical plane"; see. In contrast to the projective case, there is an abundance of point-homogeneous stable planes, among them vast classes of translation planes, see and. Symmetric planes. Affine translation planes have the following property : More generally, a "symmetric plane" is a stable plane formula_124 satisfying the aforementioned condition; see, cf. for a survey of these geometries. By Corollary 5.5, the group formula_123 is a Lie group and the point space formula_1 is a manifold. It follows that formula_93 is a symmetric space. By means of the Lie theory of symmetric spaces, all symmetric planes with a point set of dimension formula_23 or formula_26 have been classified. "They are either translation planes or they are determined by a Hermitian form". An easy example is the real hyperbolic plane. Circle geometries. Classical models are given by  the plane sections of a quadratic surface formula_130 in real projective formula_27-space; if formula_130 is a sphere, the geometry is called a Möbius plane. The plane sections of a ruled surface (one-sheeted hyperboloid) yield the classical Minkowski plane, cf. for generalizations. If formula_130 is an elliptic cone without its vertex, the geometry is called a Laguerre plane. Collectively these planes are sometimes referred to as Benz planes. "A topological Benz plane is classical, if each point has a neighbourhood which is isomorphic to some open piece of the corresponding classical Benz plane". Möbius planes. Möbius planes consist of  a family formula_134 of circles, which are topological 1-spheres, on the formula_23-sphere formula_130 such that for each point formula_137 the "derived" structure formula_138 is a topological affine plane. In particular, any formula_27 distinct points are joined by a unique circle. The circle space formula_134 is then homeomorphic to real projective formula_27-space with one point deleted. A large class of examples is given by the plane sections of an egg-like surface in real formula_27-space. Homogeneous Möbius planes. "If the automorphism group formula_29 of a Möbius plane is transitive on the point set formula_130 or on the  set formula_134 of circles, or if formula_146, then formula_147 is classical and formula_148", see. In contrast to compact projective planes there are no topological Möbius planes with circles of dimension formula_149, in particular no compact Möbius planes with a formula_26-dimensional point space. All 2-dimensional Möbius planes such that formula_40 can be described explicitly. Laguerre planes. The classical model of a Laguerre plane consists of a circular cylindrical surface formula_152 in real formula_27-space formula_154 as point set and the compact plane sections of formula_152 as circles. Pairs of points which are not joined by a circle are called  parallel. Let formula_1 denote a class of parallel points. Then formula_157 is a plane formula_158, the circles can be represented in this plane by parabolas of the form formula_159. In an analogous way, the classical formula_26-dimensional Laguerre plane is related to the geometry of complex quadratic polynomials. In general, the axioms of a locally compact connected Laguerre plane require that the derived planes embed into compact projective planes of finite dimension. A circle not passing through the point of derivation induces an oval in the derived projective plane. By or, circles are homeomorphic to spheres of dimension formula_161 or  formula_23. Hence the point space of a locally compact connected Laguerre plane is homeomorphic to the cylinder formula_152 or it is a formula_26-dimensional manifold, cf. A large class of formula_23-dimensional examples, called ovoidal Laguerre planes, is given by the plane sections of a cylinder in real 3-space whose base is an oval in formula_158. The automorphism group of a formula_167-dimensional Laguerre plane (formula_168) is a Lie group with respect to the topology of uniform convergence on compact subsets of the point space; furthermore, this group has dimension at most formula_169. All automorphisms of a Laguerre plane  which fix each parallel class form a normal subgroup, the "kernel" of the full automorphism group. The formula_23-dimensional Laguerre planes with formula_171 are exactly the ovoidal planes over proper skew parabolae. The classical formula_167-dimensional Laguerre planes are the only ones such that formula_173, see, cf. also. Homogeneous Laguerre planes. "If the automorphism group formula_29 of a formula_167-dimensional Laguerre plane formula_176 is transitive on the set of parallel classes, and if the kernel formula_177 is transitive on the set of circles, then formula_176 is classical", see 2.1,2. However, transitivity of the automorphism group on the set of circles  does not suffice to characterize the classical model among the formula_167-dimensional Laguerre planes. Minkowski planes. The classical model of a Minkowski plane has the torus formula_180 as point space, circles are the graphs of real fractional linear maps on formula_181. As with Laguerre planes, the point space of a locally compact connected Minkowski plane is formula_161- or formula_23-dimensional; the point space is then homeomorphic to a torus or to formula_184, see. Homogeneous Minkowski planes. "If the automorphism group formula_29 of a Minkowski plane formula_186 of dimension formula_167 is flag-transitive, then formula_186 is classical". The automorphism group of  a formula_167-dimensional Minkowski plane is a Lie group of dimension at most formula_190. All formula_23-dimensional Minkowski planes such that formula_146 can be described explicitly. The classical formula_167-dimensional Minkowski plane is the only one with formula_194, see.