Topological geometry
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Topological geometry deals with incidence structures consisting of a point set P and a family \mathfrak of subsets of P called lines or circles etc. such that both P and \mathfrak carry a
topology In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformations, such ...
and all geometric operations like joining points by a line or intersecting lines are continuous. As in the case of
topological group In mathematics, topological groups are logically the combination of groups and topological spaces, i.e. they are groups and topological spaces at the same time, such that the continuity condition for the group operations connects these two str ...
s, 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 In mathematics, the Euclidean plane is a Euclidean space of dimension two. That is, a geometric setting in which two real quantities are required to determine the position of each point ( element of the plane), which includes affine notions of ...
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 structure In mathematics, an incidence structure is an abstract system consisting of two types of objects and a single relationship between these types of objects. Consider the points and lines of the Euclidean plane as the two types of objects and ignore al ...
s in which any two distinct points x and y are joined by a unique line xy. Such geometries are called ''topological'' if xy depends continuously on the pair (x,y) 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 plane In mathematics, a projective plane is a geometric structure that extends the concept of a plane. In the ordinary Euclidean plane, two lines typically intersect in a single point, but there are some pairs of lines (namely, parallel lines) that do ...
s.


History

A systematic study of these planes began in 1954 with a paper by Skornyakov. Earlier, the topological properties of the
real plane In mathematics, a plane is a Euclidean (flat), two-dimensional surface that extends indefinitely. A plane is the two-dimensional analogue of a point (zero dimensions), a line (one dimension) and three-dimensional space. Planes can arise as su ...
had been introduced via ordering relations on the affine lines, see, e.g.,
Hilbert David Hilbert (; ; 23 January 1862 – 14 February 1943) was a German mathematician, one of the most influential mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many a ...
,
Coxeter Harold Scott MacDonald "Donald" Coxeter, (9 February 1907 – 31 March 2003) was a British and later also Canadian geometer. He is regarded as one of the greatest geometers of the 20th century. Biography Coxeter was born in Kensington to ...
, and O. Wyler. The completeness of the ordering is equivalent to
local compactness In topology and related branches of mathematics, a topological space is called locally compact if, roughly speaking, each small portion of the space looks like a small portion of a compact space. More precisely, it is a topological space in which ev ...
and implies that the affine lines are
homeomorphic In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphi ...
to \R and that the point space is
connected Connected may refer to: Film and television * ''Connected'' (2008 film), a Hong Kong remake of the American movie ''Cellular'' * '' Connected: An Autoblogography About Love, Death & Technology'', a 2011 documentary film * ''Connected'' (2015 TV ...
. Note that the
rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ration ...
s 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 x^2 + y^2 = 3 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 number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...
s, the
quaternion In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by the Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. Hamilton defined a quatern ...
s or the
octonion In mathematics, the octonions are a normed division algebra over the real numbers, a kind of hypercomplex number system. The octonions are usually represented by the capital letter O, using boldface or blackboard bold \mathbb O. Octonions have e ...
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
manifold In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...
s of dimension 2^m,\, 1 \le m \le 4.


Topological dimension

The notion of the
dimension In physics and mathematics, the dimension of a Space (mathematics), mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any Point (geometry), point within it. Thus, a Line (geometry), lin ...
of a topological space plays a prominent rôle in the study of topological, in particular of compact connected planes. For a
normal space In topology and related branches of mathematics, a normal space is a topological space ''X'' that satisfies Axiom T4: every two disjoint closed sets of ''X'' have disjoint open neighborhoods. A normal Hausdorff space is also called a T4 space. Th ...
X, the dimension \dim X can be characterized as follows: If \mathbb_n denotes the n-sphere, then ''\dim X \le n if, and only if, for every closed subspace A \subset X each continuous map \varphi : A \to \mathbb_n has a continuous extension \psi : X \to \mathbb_n''. 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 A surface, as the term is most generally used, is the outermost or uppermost layer of a physical object or space. It is the portion or region of the object that can first be perceived by an observer using the senses of sight and touch, and is t ...
. Early examples not isomorphic to the classical real plane have been given by Hilbert and
Moulton Moulton may refer to: Places in the United Kingdom ;In England *Moulton, Cheshire *Moulton, Lincolnshire **Moulton Windmill * Moulton St Mary, Norfolk *Moulton, Northamptonshire **Moulton College, agricultural college **Moulton Park, industria ...
. 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 2-dimensional compact planes. The traditional way to distinguish from the other 2-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
automorphism In mathematics, an automorphism is an isomorphism from a mathematical object to itself. It is, in some sense, a symmetry of the object, and a way of mapping the object to itself while preserving all of its structure. The set of all automorphisms ...
s is transitive on the set of quadrangles (4 points no 3 of which are collinear). In the present setting, a much weaker homogeneity condition characterizes : Theorem. ''If the automorphism group \Sigma of a 2-dimensional compact plane is transitive on the point set (or the line set), then \Sigma has a compact subgroup \Phi which is even transitive on the set of flags'' (=incident point-line pairs), ''and is classical''. The automorphism group \Sigma = \operatorname of a 2-dimensional compact plane , taken with the topology of
uniform convergence In the mathematical field of analysis, uniform convergence is a mode of convergence of functions stronger than pointwise convergence. A sequence of functions (f_n) converges uniformly to a limiting function f on a set E if, given any arbitrarily s ...
on the point space, is a
locally compact group In mathematics, a locally compact group is a topological group ''G'' for which the underlying topology is locally compact and Hausdorff. Locally compact groups are important because many examples of groups that arise throughout mathematics are lo ...
of dimension at most 8, in fact even a
Lie group In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the additio ...
. All 2-dimensional planes such that \dim\Sigma \ge 3 can be described explicitly; those with \dim\Sigma = 4 are exactly the Moulton planes, the classical plane is the only 2-dimensional plane with \dim \Sigma > 4; see also.


Compact connected planes

The results on 2-dimensional planes have been extended to compact planes of dimension >2 . This is possible due to the following basic theorem: Topology of compact planes. ''If the dimension of the point space P of a compact connected projective plane is finite, then \dim P=2^m with m \in \. Moreover, each line is a
homotopy sphere In algebraic topology, a branch of mathematics, a ''homotopy sphere'' is an ''n''-manifold that is homotopy equivalent to the ''n''-sphere. It thus has the same homotopy groups and the same homology groups as the ''n''-sphere, and so every homotopy ...
of dimension 2^'', see or. Special aspects of 4-dimensional planes are treated in, more recent results can be found in. The lines of a 4-dimensional compact plane are homeomorphic to the 2-sphere; in the cases m>2 the lines are not known to be manifolds, but in all examples which have been found so far the lines are spheres. A subplane of a projective plane is said to be a
Baer Baer (or Bär, from german: bear, links=no) or Van Baer is a surname. Notable people with the surname include: Baer * Alan Baer, American tuba player * Arthur "Bugs" Baer (1886–1969), American journalist and humorist * Buddy Baer (1915–198 ...
subplane, if each point of is incident with a line of and each line of contains a point of . A closed subplane is a Baer subplane of a compact connected plane if, and only if, the point space of and a line of have the same dimension. Hence the lines of an 8-dimensional plane \mathcal P are homeomorphic to a sphere \mathbb_4 if has a closed Baer subplane. Homogeneous planes. ''If \mathcal P is a compact connected projective plane and if \Sigma = \operatorname is transitive on the point set of \mathcal P, then \Sigma has a flag-transitive compact subgroup \Phi and \mathcal P is classical'', see or. In fact, \Phi is an elliptic motion group. Let \mathcal P be a compact plane of dimension 2^m,\; m=2,3,4, and write \Sigma = \operatorname . If \dim\Sigma > 8,18,40, then is classical, and \operatorname is a
simple Lie group In mathematics, a simple Lie group is a connected non-abelian Lie group ''G'' which does not have nontrivial connected normal subgroups. The list of simple Lie groups can be used to read off the list of simple Lie algebras and Riemannian symme ...
of dimension 16,35,78 respectively. All planes \mathcal P with \dim\Sigma = 8,18,40 are known explicitly. The planes with \dim\Sigma = 40 are exactly the projective closures of the affine planes coordinatized by a so-called ''mutation'' (\mathbb,+,\circ) of the octonion algebra (\mathbb,+, \ \,), where the new multiplication \circ is defined as follows: choose a real number t with 1/2 < t \ne 1 and put a \circ b = t\cdot a b + (1-t)\cdot b a. 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 plane In mathematics, a translation plane is a projective plane which admits a certain group of symmetries (described below). Along with the Hughes planes and the Figueroa planes, translation planes are among the most well-studied of the known non-Desarg ...
s (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 m=3 and for m=4.


Compact projective spaces

Subplanes of
projective space In mathematics, the concept of a projective space originated from the visual effect of perspective, where parallel lines seem to meet ''at infinity''. A projective space may thus be viewed as the extension of a Euclidean space, or, more generally ...
s 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 In mathematics, hyperbolic geometry (also called Lobachevskian geometry or Bolyai– Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with: :For any given line ''R'' and point ''P'' ...
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 2-dimensional case see also. Precisely, a ''stable plane'' is a topological linear geometry (P,\mathfrak) such that # P is a locally compact space of positive finite dimension, # each line L\in\mathfrak is a closed subset of P, and \mathfrak is a Hausdorff space, # the set \ is an open subspace \mathfrak \subset \mathfrak^2 ( ''stability''), # the map (K,L) \mapsto K \cap L:\mathfrak \to P is continuous. Note that stability excludes geometries like the 3-dimensional affine space over \R or \Complex. A stable plane is a projective plane if, and only if, P is compact. As in the case of projective planes, line pencils are compact and homotopy equivalent to a sphere of dimension 2^, and \dim P = 2^m with m\in\, see or. Moreover, the point space P is locally contractible. Compact groups of (proper) stable planes are rather small. Let \Phi_d denote a maximal compact subgroup of the automorphism group of the classical d-dimensional projective plane _d. Then the following theorem holds:
''If a d-dimensional stable plane admits a compact group \Gamma of automorphisms such that \dim\Gamma > \dim\Phi_d-d, then \cong _d'', see. Flag-homogeneous stable planes. ''Let =(P,\mathfrak) be a stable plane. If the automorphism group \operatorname is flag-transitive, then is a classical projective or affine plane, or is isomorphic to the interior of the absolute sphere of the hyperbolic
polarity Polarity may refer to: Science *Electrical polarity, direction of electrical current *Polarity (mutual inductance), the relationship between components such as transformer windings * Polarity (projective geometry), in mathematics, a duality of ord ...
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: * There exists a point transitive closed subgroup \Delta of the automorphism group which contains a unique
reflection Reflection or reflexion may refer to: Science and technology * Reflection (physics), a common wave phenomenon ** Specular reflection, reflection from a smooth surface *** Mirror image, a reflection in a mirror or in water ** Signal reflection, in ...
at some and hence at each point. More generally, a ''symmetric plane'' is a stable plane = (P,\mathfrak) satisfying the aforementioned condition; see, cf. for a survey of these geometries. By Corollary 5.5, the group \Delta is a Lie group and the point space P is a manifold. It follows that is a
symmetric space In mathematics, a symmetric space is a Riemannian manifold (or more generally, a pseudo-Riemannian manifold) whose group of symmetries contains an inversion symmetry about every point. This can be studied with the tools of Riemannian geometry, l ...
. By means of the Lie theory of symmetric spaces, all symmetric planes with a point set of dimension 2 or 4 have been classified. ''They are either translation planes or they are determined by a
Hermitian form In mathematics, a sesquilinear form is a generalization of a bilinear form that, in turn, is a generalization of the concept of the dot product of Euclidean space. A bilinear form is linear in each of its arguments, but a sesquilinear form allows o ...
''. An easy example is the real hyperbolic plane.


Circle geometries

Classical models are given by the plane sections of a quadratic surface S in real projective 3-space; if S is a sphere, the geometry is called a
Möbius plane In mathematics, a Möbius plane (named after August Ferdinand Möbius) is one of the Benz planes: Möbius plane, Laguerre plane and Minkowski plane. The classical example is based on the geometry of lines and circles in the real affine plane. A s ...
. The plane sections of a ruled surface (one-sheeted hyperboloid) yield the classical
Minkowski plane In mathematics, a Minkowski plane (named after Hermann Minkowski) is one of the Benz planes (the others being Möbius plane and Laguerre plane). Classical real Minkowski plane Applying the pseudo-euclidean distance d(P_1,P_2) = (x'_1-x'_2) ...
, cf. for generalizations. If S is an elliptic cone without its vertex, the geometry is called a
Laguerre plane In mathematics, a Laguerre plane is one of the three types of Benz plane, which are the Möbius plane, Laguerre plane and Minkowski plane. Laguerre planes are named after the French mathematician Edmond Nicolas Laguerre. The classical Laguerre ...
. Collectively these planes are sometimes referred to as
Benz plane In mathematics, a Benz plane is a type of 2-dimensional geometrical structure, named after the German mathematician Walter Benz. The term was applied to a group of objects that arise from a common axiomatization of certain structures and split int ...
s. ''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 \mathfrak of circles, which are topological 1-spheres, on the 2-sphere S such that for each point p the ''derived'' structure (S\setminus\,\) is a topological affine plane. In particular, any 3 distinct points are joined by a unique circle. The circle space \mathfrak is then homeomorphic to real projective 3-space with one point deleted. A large class of examples is given by the plane sections of an egg-like surface in real 3-space.


Homogeneous Möbius planes

''If the automorphism group \Sigma of a Möbius plane is transitive on the point set S or on the set \mathfrak of circles, or if \dim\Sigma \ge 4, then (S,\mathfrak) is classical and \dim\Sigma = 6'', see. In contrast to compact projective planes there are no topological Möbius planes with circles of dimension >1 , in particular no compact Möbius planes with a 4-dimensional point space. All 2-dimensional Möbius planes such that \dim\Sigma \ge 3 can be described explicitly.


Laguerre planes

The classical model of a Laguerre plane consists of a circular cylindrical surface C in real 3-space \R^3 as point set and the compact plane sections of C as circles. Pairs of points which are not joined by a circle are called ''parallel''. Let P denote a class of parallel points. Then C \setminus P is a plane \R^2, the circles can be represented in this plane by parabolas of the form y = ax^2+bx+c. In an analogous way, the classical 4-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 An oval () is a closed curve in a plane which resembles the outline of an egg. The term is not very specific, but in some areas (projective geometry, technical drawing, etc.) it is given a more precise definition, which may include either one or ...
in the derived projective plane. By or, circles are homeomorphic to spheres of dimension 1 or 2. Hence the point space of a locally compact connected Laguerre plane is homeomorphic to the cylinder C or it is a 4-dimensional manifold, cf. A large class of 2-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 \R^2. The automorphism group of a 2d-dimensional Laguerre plane (d = 1, 2) 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 7d. All automorphisms of a Laguerre plane which fix each parallel class form a normal subgroup, the ''kernel'' of the full automorphism group. The 2-dimensional Laguerre planes with \dim\Sigma=5 are exactly the ovoidal planes over proper skew parabolae. The classical 2d-dimensional Laguerre planes are the only ones such that \dim\Sigma > 5d, see, cf. also.


Homogeneous Laguerre planes

''If the automorphism group \Sigma of a 2d-dimensional Laguerre plane is transitive on the set of parallel classes, and if the kernel T \triangleleft \Sigma is transitive on the set of circles, then 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 2d-dimensional Laguerre planes.


Minkowski planes

The classical model of a Minkowski plane has the
torus In geometry, a torus (plural tori, colloquially donut or doughnut) is a surface of revolution generated by revolving a circle in three-dimensional space about an axis that is coplanar with the circle. If the axis of revolution does not tou ...
\mathbb_1 \times \mathbb_1 as point space, circles are the graphs of real fractional linear maps on \mathbb_1 = \R \cup\. As with Laguerre planes, the point space of a locally compact connected Minkowski plane is 1- or 2-dimensional; the point space is then homeomorphic to a torus or to \mathbb_2 \times \mathbb_2, see.


Homogeneous Minkowski planes

''If the automorphism group \Sigma of a Minkowski plane of dimension 2d is flag-transitive, then is classical''. The automorphism group of a 2d-dimensional Minkowski plane is a Lie group of dimension at most 6d. All 2-dimensional Minkowski planes such that \dim\Sigma \ge 4 can be described explicitly. The classical 2d-dimensional Minkowski plane is the only one with \dim\Sigma > 4d, see.


Notes


References

* * * * * * * * * * * {{citation, first=G., last=Steinke, year=1995, title=Topological circle geometries, journal=Handbook of Incidence Geometry, pages=1325–1354, location=Amsterdam, publisher=North-Holland, doi=10.1016/B978-044488355-1/50026-8, isbn=9780444883551 Topology Incidence geometry