Topological geometry




Topological geometry deals with incidence structures consisting of a point set P{displaystyle P}P and a family L{displaystyle {mathfrak {L}}}{mathfrak  {L}} of subsets of P{displaystyle P}P called lines or circles etc. such that both P{displaystyle P}P and L{displaystyle {mathfrak {L}}}{mathfrak  {L}} 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.




Contents






  • 1 Linear geometries


  • 2 History


  • 3 Topological projective planes


    • 3.1 Topological dimension


    • 3.2 2-dimensional planes


    • 3.3 Compact connected planes




  • 4 Compact projective spaces


  • 5 Stable planes


  • 6 Symmetric planes


  • 7 Circle geometries


    • 7.1 Möbius planes


    • 7.2 Homogeneous Möbius planes


    • 7.3 Laguerre planes


    • 7.4 Homogeneous Laguerre planes


    • 7.5 Minkowski planes


    • 7.6 Homogeneous Minkowski planes




  • 8 Notes


  • 9 References





Linear geometries


Linear geometries are incidence structures in which any two distinct points x{displaystyle x}x and y{displaystyle y}y are joined by a unique line xy{displaystyle xy}xy. Such geometries are called topological if xy{displaystyle xy}xy depends continuously on the pair (x,y){displaystyle (x,y)}(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 rôles of points and lines. A survey of linear topological geometries is given in Chapter 23 of the Handbook of incidence geometry.[1] 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.[2] Earlier, the topological properties of the real plane had been introduced via ordering relations on the affine lines, see, e.g., Hilbert,[3]Coxeter,[4] and O. Wyler.[5] The completeness of the ordering is equivalent to local compactness and implies that the affine lines are homeomorphic to R{displaystyle mathbb {R} }mathbb {R} 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 x2+y2=3{displaystyle x^{2}+y^{2}=3}{displaystyle 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 numbers, the quaternions or the octonion algebra.[6] 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 2m,1≤m≤4{displaystyle 2^{m},,1leq mleq 4}{displaystyle 2^{m},,1leq mleq 4}.



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 X{displaystyle X}X, the dimension dim⁡X{displaystyle dim X}{displaystyle dim X} can be characterized as follows:


If Sn{displaystyle mathbb {S} _{n}}{displaystyle mathbb {S} _{n}} denotes the n{displaystyle n}n-sphere, then dim⁡X≤n{displaystyle dim Xleq n}{displaystyle dim Xleq n} if, and only if, for every closed subspace A⊂X{displaystyle Asubset X}Asubset X each continuous map φ:A→Sn{displaystyle varphi :Ato mathbb {S} _{n}}{displaystyle varphi :Ato mathbb {S} _{n}} has a continuous extension ψ:X→Sn{displaystyle psi :Xto mathbb {S} _{n}}{displaystyle psi :Xto mathbb {S} _{n}}.


For details and other definitions of a dimension see [7] and the references given there, in particular Engelking[8] or Fedorchuk.[9]



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.[10] Equivalently, the point space is a surface. Early examples not isomorphic to the classical real plane E{displaystyle {mathcal {E}}}{displaystyle {mathcal {E}}} have been given by Hilbert[3][11] and Moulton.[12] 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{displaystyle 2}2-dimensional compact planes. The traditional way to distinguish E{displaystyle {mathcal {E}}}{displaystyle {mathcal {E}}} from the other 2{displaystyle 2}2-dimensional planes is by the validity of Desargues’s theorem or the theorem of Pappos (see, e.g., Pickert[13] for a discussion of these two configuration theorems). The latter is known to imply the former (Hessenberg[14]). 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,[3][15][13] hence it implies that the group of automorphisms is transitive on the set of quadrangles (4{displaystyle 4}4 points no 3{displaystyle 3}3 of which are collinear). In the present setting, a much weaker homogeneity condition characterizes E{displaystyle {mathcal {E}}}{displaystyle {mathcal {E}}}:


Theorem. If the automorphism group Σ{displaystyle Sigma }Sigma of a 2{displaystyle 2}2-dimensional compact plane P{displaystyle {mathcal {P}}}{{mathcal  P}} is transitive on the point set (or the line set), then Σ{displaystyle Sigma }Sigma has a compact subgroup Φ{displaystyle Phi }Phi which is even transitive on the set of flags (=incident point-line pairs), and P{displaystyle {mathcal {P}}}{{mathcal  P}} is classical.[10]


The automorphism group Σ=Aut⁡P{displaystyle Sigma =operatorname {Aut} {mathcal {P}}}{displaystyle Sigma =operatorname {Aut} {mathcal {P}}} of a 2{displaystyle 2}2-dimensional compact plane P{displaystyle {mathcal {P}}}{{mathcal  P}}, taken with the topology of uniform convergence on the point space, is a locally compact group of dimension at most 8{displaystyle 8}8, in fact even a Lie group. All 2{displaystyle 2}2-dimensional planes such that dim⁡Σ3{displaystyle dim Sigma geq 3}{displaystyle dim Sigma geq 3} can be described explicitly;[10] those with dim⁡Σ=4{displaystyle dim Sigma =4}{displaystyle dim Sigma =4} are exactly the Moulton planes, the classical plane E{displaystyle {mathcal {E}}}{displaystyle {mathcal {E}}} is the only 2{displaystyle 2}2-dimensional plane with dim⁡Σ>4{displaystyle dim Sigma {,>,}4}{displaystyle dim Sigma {,>,}4}; see also.[16]



Compact connected planes


The results on 2{displaystyle 2}2-dimensional planes have been extended to compact planes of dimension >2{displaystyle >2}{displaystyle >2}. This is possible due to the following basic theorem:


Topology of compact planes. If the dimension of the point space P{displaystyle P}P of a compact connected projective plane is finite, then dim⁡P=2m{displaystyle dim P=2^{m}}{displaystyle dim P=2^{m}} with m∈{1,2,3,4}{displaystyle min {1,2,3,4}}{displaystyle min {1,2,3,4}}. Moreover, each line is a homotopy sphere of dimension 2m−1{displaystyle 2^{m-1}}2^{m-1}, see [17] or.[18]


Special aspects of 4{displaystyle 4}4-dimensional planes are treated in,[19] more recent results can be found in.[20] The lines of a 4{displaystyle 4}4-dimensional compact plane are homeomorphic to the 2{displaystyle 2}2-sphere;[21] in the cases m>2{displaystyle m>2}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 B{displaystyle {mathcal {B}}}{{mathcal  B}} of a projective plane P{displaystyle {mathcal {P}}}{{mathcal  P}} is said to be a Baer subplane,[22] if each point of P{displaystyle {mathcal {P}}}{{mathcal  P}} is incident with a line of B{displaystyle {mathcal {B}}}{{mathcal  B}} and each line of P{displaystyle {mathcal {P}}}{{mathcal  P}} contains a point of B{displaystyle {mathcal {B}}}{{mathcal  B}}. A closed subplane B{displaystyle {mathcal {B}}}{{mathcal  B}} is a Baer subplane of a compact connected plane P{displaystyle {mathcal {P}}}{{mathcal  P}} if, and only if, the point space of B{displaystyle {mathcal {B}}}{{mathcal  B}} and a line of P{displaystyle {mathcal {P}}}{{mathcal  P}} have the same dimension. Hence the lines of an 8{displaystyle 8}8-dimensional plane P{displaystyle {mathcal {P}}}{mathcal {P}} are homeomorphic to a sphere S4{displaystyle mathbb {S} _{4}}{mathbb  {S}}_{4} if P{displaystyle {mathcal {P}}}{{mathcal  P}} has a closed Baer subplane.[23]


Homogeneous planes. If P{displaystyle {mathcal {P}}}{mathcal {P}} is a compact connected projective plane and if Σ=Aut⁡P{displaystyle Sigma =operatorname {Aut} {mathcal {P}}}{displaystyle Sigma =operatorname {Aut} {mathcal {P}}} is transitive on the point set of P{displaystyle {mathcal {P}}}{mathcal {P}}, then Σ{displaystyle Sigma }Sigma has a flag-transitive compact subgroup Φ{displaystyle Phi }Phi and P{displaystyle {mathcal {P}}}{mathcal {P}} is classical, see [24] or.[25] In fact, Φ{displaystyle Phi }Phi is an elliptic motion group.[26]


Let P{displaystyle {mathcal {P}}}{mathcal {P}} be a compact plane of dimension 2m,m=2,3,4{displaystyle 2^{m},;m{,=,}2,3,4}{displaystyle 2^{m},;m{,=,}2,3,4}, and write Σ=Aut⁡P{displaystyle Sigma =operatorname {Aut} {mathcal {P}}}{displaystyle Sigma =operatorname {Aut} {mathcal {P}}}. If dim⁡Σ>8,18,40{displaystyle dim Sigma >8,18,40}{displaystyle dim Sigma >8,18,40}, then P{displaystyle {mathcal {P}}}{{mathcal  P}} is classical,[27] and Aut⁡P{displaystyle operatorname {Aut} {mathcal {P}}}{displaystyle operatorname {Aut} {mathcal {P}}} is a simple Lie group of dimension 16,35,78{displaystyle 16,35,78}{displaystyle 16,35,78} respectively. All planes P{displaystyle {mathcal {P}}}{mathcal {P}} with dim⁡Σ=8,18,40{displaystyle dim Sigma =8,18,40}{displaystyle dim Sigma =8,18,40} are known explicitly.[28] The planes with dim⁡Σ=40{displaystyle dim Sigma =40}{displaystyle dim Sigma =40} are exactly the projective closures of the affine planes coordinatized by a so-called mutation (O,+,∘){displaystyle (mathbb {O} ,+,circ )}{displaystyle (mathbb {O} ,+,circ )} of the octonion algebra (O,+, ){displaystyle (mathbb {O} ,+, ,)}{displaystyle (mathbb {O} ,+, ,)}, where the new multiplication {displaystyle circ }circ is defined as follows: choose a real number t{displaystyle t}t with 1/2<t≠1{displaystyle 1/2<tneq 1}{displaystyle 1/2<tneq 1} and put a∘b=t⋅ab+(1−t)⋅ba{displaystyle acirc b=tcdot ab+(1-t)cdot ba}{displaystyle acirc b=tcdot ab+(1-t)cdot ba}. Vast families of planes with a group of large dimension have been discovered systematically starting from assumptions about their automorphism groups, see, e.g.,.[20][29][30][31][32] 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.;[33] see also [34] for more recent results in the case m=3{displaystyle m=3}m=3 and [30] for m=4{displaystyle m=4}m=4.



Compact projective spaces


Subplanes of projective spaces of geometrical dimension at least 3 are necessarily Desarguesian, see [35] §1 or [4] §16 or.[36] Therefore, all compact connected projective spaces can be coordinatized by the real or complex numbers or the quaternion field.[37]



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,[38] for the 2{displaystyle 2}2-dimensional case see also.[39]


Precisely, a stable plane S{displaystyle {mathcal {S}}}{displaystyle {mathcal {S}}} is a topological linear geometry (P,L){displaystyle (P,{mathfrak {L}})}{displaystyle (P,{mathfrak {L}})} such that


(1) P{displaystyle P}P is a locally compact space of positive finite dimension,


(2) each line L∈L{displaystyle L{,in ,}{mathfrak {L}}}{displaystyle L{,in ,}{mathfrak {L}}} is a closed subset of P{displaystyle P}P, and L{displaystyle {mathfrak {L}}}{mathfrak  {L}} is a Hausdorff space,


(3) the set {(K,L)∣K≠L,K∩L≠}{displaystyle {(K,L)mid Kneq L,;Kcap Lneq emptyset }}{displaystyle {(K,L)mid Kneq L,;Kcap Lneq emptyset }} is an open subspace O⊂L2{displaystyle {mathfrak {O}}subset {mathfrak {L}}^{2}}{displaystyle {mathfrak {O}}subset {mathfrak {L}}^{2}} ( stability),


(4) the map (K,L)↦K∩L:O→P{displaystyle (K,L){,mapsto ,}Kcap L:{mathfrak {O}}{,to ,}P}{displaystyle (K,L){,mapsto ,}Kcap L:{mathfrak {O}}{,to ,}P} is continuous.


Note that stability excludes geometries like the 3{displaystyle 3}3-dimensional affine space over R{displaystyle mathbb {R} }mathbb {R} or C{displaystyle mathbb {C} }mathbb {C} .


A stable plane S{displaystyle {mathcal {S}}}{displaystyle {mathcal {S}}} is a projective plane if, and only if, P{displaystyle P}P is compact.[40]


As in the case of projective planes, line pencils are compact and homotopy equivalent to a sphere of dimension 2m−1{displaystyle 2^{m{-}1}}{displaystyle 2^{m{-}1}}, and dim⁡P=2m{displaystyle dim P{,=,}2^{m}}{displaystyle dim P{,=,}2^{m}} with m∈{1,2,3,4}{displaystyle m{,in ,}{1,2,3,4}}{displaystyle m{,in ,}{1,2,3,4}}, see [17] or.[41] Moreover, the point space P{displaystyle P}P is locally contractible.[17][42]


Compact groups of (proper) stable planes
are rather small. Let Φd{displaystyle Phi _{d}}Phi_d denote a maximal compact subgroup of the automorphism group of the classical d{displaystyle d}d-dimensional projective plane Pd{displaystyle {mathcal {P}}_{d}}{displaystyle {mathcal {P}}_{d}}. Then the following theorem holds:
If a d{displaystyle d}d-dimensional stable plane S{displaystyle {mathcal {S}}}{displaystyle {mathcal {S}}} admits a compact group Γ{displaystyle Gamma }Gamma of automorphisms such that dim⁡Γ>dim⁡Φd−d{displaystyle dim Gamma {,>,}dim Phi _{d}{,-,}d}{displaystyle dim Gamma {,>,}dim Phi _{d}{,-,}d}, then S≅Pd{displaystyle {mathcal {S}}cong {mathcal {P}}_{d}}{displaystyle {mathcal {S}}cong {mathcal {P}}_{d}}, see.[43]


Flag-homogeneous stable planes. Let S=(P,L){displaystyle {mathcal {S}}=(P,{mathfrak {L}})}{displaystyle {mathcal {S}}=(P,{mathfrak {L}})} be a stable plane. If the automorphism group Aut⁡S{displaystyle operatorname {Aut} {mathcal {S}}}{displaystyle operatorname {Aut} {mathcal {S}}} is flag-transitive, then S{displaystyle {mathcal {S}}}{displaystyle {mathcal {S}}} is a classical projective or affine plane, or S{displaystyle {mathcal {S}}}{displaystyle {mathcal {S}}} is isomorphic to the interior of the absolute sphere of the hyperbolic polarity of a classical plane; see.[44][45][46]


In contrast to the projective case, there is an abundance of point-homogeneous stable planes, among them vast classes of translation planes, see [33] and.[47]



Symmetric planes


Affine translation planes have the following property:


{displaystyle *}* There exists a point transitive closed subgroup Δ{displaystyle Delta }Delta of the automorphism group which
contains a unique reflection at some and hence at each point.


More generally, a symmetric plane is a stable plane S=(P,L){displaystyle {mathcal {S}}{,=,}(P,{mathfrak {L}})}{displaystyle {mathcal {S}}{,=,}(P,{mathfrak {L}})} satisfying condition ({displaystyle *}*); see,[48] cf.[49] for a survey of these geometries. By [50] Corollary 5.5, the group Δ{displaystyle Delta }Delta is a Lie group and the point space P{displaystyle P}P is a manifold. It follows that S{displaystyle {mathcal {S}}}{displaystyle {mathcal {S}}} is a symmetric space. By means of the Lie theory of symmetric spaces, all symmetric planes with a point set of dimension 2{displaystyle 2}2 or 4{displaystyle 4}4 have been classified.[48][51]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 [52] are given by the plane sections of a quadratic surface S{displaystyle S}S in real projective 3{displaystyle 3}3-space; if S{displaystyle S}S is a sphere, the geometry is called a Möbius plane.[39] The plane sections of a ruled surface (one-sheeted hyperboloid) yield the classical Minkowski plane, cf.[53] for generalizations. If S{displaystyle S}S 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.[54]



Möbius planes


Möbius planes consist of a family C{displaystyle {mathfrak {C}}}{mathfrak  {C}} of circles, which are topological 1-spheres, on the 2{displaystyle 2}2-sphere S{displaystyle S}S such that for each point p{displaystyle p}p the derived structure (S∖{p},{C∖{p}∣p∈C∈C}){displaystyle (Ssetminus {p},{Csetminus {p}mid p{,in ,}C{,in ,}{mathfrak {C}}})}{displaystyle (Ssetminus {p},{Csetminus {p}mid p{,in ,}C{,in ,}{mathfrak {C}}})} is a topological affine plane.[55] In particular, any 3{displaystyle 3}3 distinct points are joined by a unique circle. The circle space C{displaystyle {mathfrak {C}}}{mathfrak  {C}} is then homeomorphic to real projective 3{displaystyle 3}3-space with one point deleted.[56] A large class of examples is given by the plane sections of an egg-like surface in real 3{displaystyle 3}3-space.



Homogeneous Möbius planes


If the automorphism group Σ{displaystyle Sigma }Sigma of a Möbius plane is transitive on the point set S{displaystyle S}S or on the set C{displaystyle {mathfrak {C}}}{mathfrak  {C}} of circles, or if dim⁡Σ4{displaystyle dim Sigma geq 4}{displaystyle dim Sigma geq 4}, then (S,C){displaystyle (S,{mathfrak {C}})}{displaystyle (S,{mathfrak {C}})} is classical and dim⁡Σ=6{displaystyle dim Sigma =6}{displaystyle dim Sigma =6}, see.[57][58]


In contrast to compact projective planes there are no topological Möbius planes with circles of dimension >1{displaystyle >1}{displaystyle >1}, in particular no compact Möbius planes with a 4{displaystyle 4}4-dimensional point space.[59] All 2-dimensional Möbius planes such that dim⁡Σ3{displaystyle dim Sigma geq 3}{displaystyle dim Sigma geq 3} can be described explicitly.[60][61]



Laguerre planes


The classical model of a Laguerre plane consists of a circular cylindrical surface C{displaystyle C}C in real 3{displaystyle 3}3-space R3{displaystyle mathbb {R} ^{3}}mathbb{R} ^{3} as point set and the compact plane sections of C{displaystyle C}C as circles. Pairs of points which are not joined by a circle are called parallel. Let P{displaystyle P}P denote a class of parallel points. Then C∖P{displaystyle Csmallsetminus P}{displaystyle Csmallsetminus P} is a plane R2{displaystyle mathbb {R} ^{2}}R^2, the circles can be represented in this plane by parabolas of the form y=ax2+bx+c{displaystyle y=ax^{2}+bx+c}{displaystyle y=ax^{2}+bx+c}.


In an analogous way, the classical 4{displaystyle 4}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 in the derived projective plane. By [62] or,[63] circles are homeomorphic to spheres of dimension 1{displaystyle 1}1 or 2{displaystyle 2}2. Hence the point space of a locally compact connected Laguerre plane is homeomorphic to the cylinder C{displaystyle C}C or it is a 4{displaystyle 4}4-dimensional manifold, cf.[64] A large class of 2{displaystyle 2}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 R2{displaystyle mathbb {R} ^{2}}R^2.


The automorphism group of a 2d{displaystyle 2d}2d-dimensional Laguerre plane (d=1,2{displaystyle d{,=,}1,2}{displaystyle 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{displaystyle 7d}{displaystyle 7d}. All automorphisms of a Laguerre plane which fix each parallel class form a normal subgroup, the kernel of the full automorphsm group. The 2{displaystyle 2}2-dimensional Laguerre planes with dim⁡Σ=5{displaystyle dim Sigma {,=,}5}{displaystyle dim Sigma {,=,}5} are exactly the ovoidal planes over proper skew parabolae.[65] The classical 2d{displaystyle 2d}2d-dimensional Laguerre planes are the only ones such that dim⁡Σ>5d{displaystyle dim Sigma {,>,}5d}{displaystyle dim Sigma {,>,}5d}, see,[66] cf. also.[67]



Homogeneous Laguerre planes


If the automorphism group Σ{displaystyle Sigma }Sigma of a 2d{displaystyle 2d}2d-dimensional Laguerre plane L{displaystyle {mathcal {L}}}{mathcal L} is transitive on the set of parallel classes, and if the kernel T◃Σ{displaystyle Ttriangleleft Sigma }{displaystyle Ttriangleleft Sigma } is transitive on the set of circles, then L{displaystyle {mathcal {L}}}{mathcal L} is classical, see [68][67] 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{displaystyle 2d}2d-dimensional Laguerre planes.



Minkowski planes


The classical model of a Minkowski plane has the torus S1×S1{displaystyle mathbb {S} _{1}times mathbb {S} _{1}}{displaystyle mathbb {S} _{1}times mathbb {S} _{1}} as point space, circles are the graphs of real fractional linear maps on S1=R∪{∞}{displaystyle mathbb {S} _{1}=mathbb {R} cup {infty }}{displaystyle mathbb {S} _{1}=mathbb {R} cup {infty }}. As with Laguerre planes, the point space of a locally compact connected Minkowski plane is 1{displaystyle 1}1- or 2{displaystyle 2}2-dimensional; the point space is then homeomorphic to a torus or to S2×S2{displaystyle mathbb {S} _{2}times mathbb {S} _{2}}{displaystyle mathbb {S} _{2}times mathbb {S} _{2}}, see.[69]



Homogeneous Minkowski planes


If the automorphism group Σ{displaystyle Sigma }Sigma of a Minkowski plane M{displaystyle {mathcal {M}}}{mathcal M} of dimension 2d{displaystyle 2d}2d is flag-transitive, then M{displaystyle {mathcal {M}}}{mathcal M} is classical.[70]


The automorphsm group of a 2d{displaystyle 2d}2d-dimensional Minkowski plane is a Lie group of dimension at most 6d{displaystyle 6d}{displaystyle 6d}. All 2{displaystyle 2}2-dimensional Minkowski planes such that dim⁡Σ4{displaystyle dim Sigma {,geq ,}4}{displaystyle dim Sigma {,geq ,}4} can be described explicitly.[71] The classical 2d{displaystyle 2d}2d-dimensional Minkowski plane is the only one with dim⁡Σ>4d{displaystyle dim Sigma >4d}{displaystyle dim Sigma >4d}, see.[72]



Notes





  1. ^ Grundhöfer & Löwen 1995


  2. ^ Skornyakov, L.A. (1954), "Topological projective planes", Trudy Moskov. Mat. Obschtsch., 3: 347–373.mw-parser-output cite.citation{font-style:inherit}.mw-parser-output q{quotes:"""""""'""'"}.mw-parser-output code.cs1-code{color:inherit;background:inherit;border:inherit;padding:inherit}.mw-parser-output .cs1-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Lock-green.svg/9px-Lock-green.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .cs1-lock-limited a,.mw-parser-output .cs1-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/d/d6/Lock-gray-alt-2.svg/9px-Lock-gray-alt-2.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .cs1-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/a/aa/Lock-red-alt-2.svg/9px-Lock-red-alt-2.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration{color:#555}.mw-parser-output .cs1-subscription span,.mw-parser-output .cs1-registration span{border-bottom:1px dotted;cursor:help}.mw-parser-output .cs1-hidden-error{display:none;font-size:100%}.mw-parser-output .cs1-visible-error{font-size:100%}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration,.mw-parser-output .cs1-format{font-size:95%}.mw-parser-output .cs1-kern-left,.mw-parser-output .cs1-kern-wl-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right,.mw-parser-output .cs1-kern-wl-right{padding-right:0.2em}


  3. ^ abc Hilbert 1899


  4. ^ ab Coxeter, H.S.M. (1993), The real projective plane, New York: Springer


  5. ^ Wyler, . (1952), "Order and topology in projective planes", Amer. J. Math., 74: 656–666


  6. ^ Conway, J.H.; Smith, D.A. (2003), On quaternions and octonions: their geometry, arithmetic, and symmetry, Natick, MA: A K Peters


  7. ^ Salzmann et al. 1995, §92


  8. ^ Engelking, R. (1978), Dimension theory, North-Holland Publ. Co.


  9. ^ Fedorchuk, V.V. (1990), "The fundamentals of dimension theory", Encycl. Math. Sci., Berlin: Springer, 17: 91–192


  10. ^ abc Salzmann 1967


  11. ^ Stroppel, M. (1998), "Bemerkungen zur ersten nicht desarguesschen ebenen Geometrie bei Hilbert", J. Geom., 63: 183–195


  12. ^ Moulton, F.R. (1902), "A simple non-Desarguesian plane geometry", Trans. Amer. Math. Soc., 3: 192–195


  13. ^ ab Pickert 1955


  14. ^ Hessenberg, G. (1905), "Beweis des Desarguesschen Satzes aus dem Pascalschen", Math. Ann. (in German), 61: 161–172


  15. ^ Hughes, D.R.; Piper, F.C. (1973), Projective planes, Berlin: Springer


  16. ^ Salzmann et al. 1995, Chapter 3


  17. ^ abc Löwen 1983


  18. ^ Salzmann et al. 1995, 54.11


  19. ^ Salzmann et al. 1995, Chapter 7


  20. ^ ab Betten, D. (1997), "On the classification of 4{displaystyle 4}4-dimensional flexible projective planes", Lect. Notes pure. appl. Math., 190: 9–33


  21. ^ Salzmann et al. 1995, 53.15


  22. ^ Salzmann, H. (2003), "Baer subplanes", Illinois J. Math., 47: 485–513


  23. ^ Salzmann et al. 1995, 55.6


  24. ^ Löwen, R. (1981), "Homogeneous compact projective planes", J. reine angew. Math., 321: 217–220


  25. ^ Salzmann et al. 1995, 63.8


  26. ^ Salzmann et al. 1995, 13.12


  27. ^ Salzmann et al. 1995, 72.8,84.28,85.16


  28. ^ Salzmann et al. 1995, 73.22,84.28,87.7


  29. ^ Hähl, H. (1986), "Achtdimensionale lokalkompakte Translationsebenen mit mindestens 17{displaystyle 17}17-dimensionaler Kollineationsgruppe", Geom. Dedicata (in German), 21: 299–340


  30. ^ ab Hähl, H. (2011), "Sixteen-dimensional locally compact translation planes with collineation groups of dimension at least 38{displaystyle 38}{displaystyle 38}", Adv. Geom., 11: 371–380


  31. ^ Hähl, H. (2000), "Sixteen-dimensional locally compact translation planes with large automorphism groups having no fixed points", Geom. Dedicata, 83: 105–117


  32. ^ Salzmann et al. 1995, §§73,74,82,86


  33. ^ ab Knarr 1995


  34. ^ Salzmann 2014


  35. ^ Hilbert 1899, §§22


  36. ^ Veblen, O.; Young, J.W. (1910), Projective Geometry Vol. I, Boston: Ginn Comp.


  37. ^ Kolmogoroff, A. (1932), "Zur Begründung der projektiven Geometrie", Ann. of Math. (in German), 33: 175–176


  38. ^ Salzmann et al. 1995, §§3,4


  39. ^ ab Polster & Steiner 2001


  40. ^ Salzmann et al. 1995, 3.11


  41. ^ Salzmann et al. 1995, 3.28,29


  42. ^ Grundhöfer & Löwen 1995, 3.7


  43. ^ Stroppel, M. (1994), "Compact groups of automorphisms of stable planes", Forum Math., 6: 339–359


  44. ^ Löwen, R. (1983), "Stable planes with isotropic points", Math. Z., 182: 49–61


  45. ^ Salzmann et al. 1995, 5.8


  46. ^ Salzmann 2014, 8.11,12


  47. ^ Salzmann et al. 1995, Chapters 7 and 8


  48. ^ ab Löwen, R. (1979), "Symmetric planes", Pacific J. Math., 84: 367–390


  49. ^ Grundhöfer & Löwen 1195, 5.26-31


  50. ^ Hofmann, K.H.; Kramer, L. (2015), "Transitive actions of locally compact groups on locally contractive spaces", J. Reine Angew. Math., 702: 227–243, 245/6


  51. ^ Löwen, R. (1979), "Classification of 4{displaystyle 4}4-dimensional symmetric planes", Math. Z., 167: 137–159


  52. ^ Steinke 1995


  53. ^ Polster & Steinke 2001, §4


  54. ^ Steinke, G. (1983), "Locally classical Benz planes are classical", Math. Z., 183: 217–220


  55. ^ Wölk, D. (1966), "Topologische Möbiusebenen", Math. Z. (in German), 93: 311–333


  56. ^ Löwen, R.; Steinke, G.F. (2014), "The circle space of a spherical circle plane", Bull. Belg. Math. Soc. Simon Stevin, 21: 351–364


  57. ^ Strambach, K. (1970), "Sphärische Kreisebenen", Math. Z. (in German), 113: 266–292


  58. ^ Steinke 1995, 3.2


  59. ^ Groh, H. (1973), "Möbius planes with locally euclidean circles are flat", Math. Ann., 201: 149–156


  60. ^ Strambach, K. (1972), "Sphärische Kreisebenen mit dreidimensionaler nichteinfacher Automorphismengruppe", Math. Z. (in German), 124: 289–314


  61. ^ Strambach, K. (1973), "Sphärische Kreisebenen mit einfacher Automorphismengruppe'", Geom. Dedicata (in German), 1: 182–220


  62. ^ Buchanan, T.; Hähl, H.; Löwen, R. (1980), "Topologische Ovale", Geom. Dedicata (in German), 9: 401–424


  63. ^ Salzmann et al. 1995, 55.14


  64. ^ Steibke 1995, 5.7


  65. ^ Steinke 1995, 5.5


  66. ^ Steinke 1995, 5.4,8


  67. ^ ab Steinke, G.F. (2002), "4{displaystyle 4}4-dimensional elation Laguerre planes admitting non-solvable automorphism groups", Monatsh. Math., 136: 327–354


  68. ^ Steinke, G.F. (1993), "4{displaystyle 4}4-dimensional point-transitive groups of automorphisms of 2{displaystyle 2}2- dimensional Laguerre planes", Result. Math., 24: 326–341


  69. ^ Steinke 1991, 4.6


  70. ^ Steinke, G.F. (1992), "4{displaystyle 4}4-dimensional Minkowski planes with large automorphism group", Forum Math., 4: 593–605


  71. ^ Polster & Steinke 2001, §4.4


  72. ^ Steinke 1995, 4.5 and 4.8




References




  • Grundhöfer, T.; Löwen, R. (1995), Buekenhout, F., ed., Handbook of incidence geometry: buildings and foundations, Amsterdam: North-Holland, pp. 1255–1324


  • Hilbert, D. (1899), The foundations of geometry, translation by E. J. Townsend, 1902, Chicago


  • Knarr, N. (1995), "Translation planes. Foundations and construction principles", Lecture Notes in Mathematics, Berlin: Springer, 1611


  • Löwen, R. (1983), "Topology and dimension of stable planes: On a conjecture of H. Freudenthal", J. reine angew. Math., 343: 108–122


  • Pickert, G. (1955), Projektive Ebenen (in German), Berlin: Springer


  • Polster, B.; Steinke, G.F. (2001), Geometries on surfaces, Cambridge UP


  • Salzmann, H. (1967), "Topological planes", Advances in Mathematics, 2: 1–60


  • Salzmann, H. (2014), Compact planes, mostly 8-dimensional. A retrospect, arXiv:1402.0304, Bibcode:2014arXiv1402.0304S


  • Salzmann, H.; Betten, D.; Grundhöfer, T.; Hähl, H.; Löwen, R.; Stroppel, M. (1995), Compact Projective Planes, W. de Gruyter


  • Steinke, G. (1995), "Topological circle geometries", Handbook of incidence geometry, Amsterdam: North-Holland: 1325–1354




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