Topological geometry

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

If ${\displaystyle \mathbb {S} _{n}}$ denotes the ${\displaystyle n}$-sphere, then ${\displaystyle \dim X\leq n}$ if, and only if, for every closed subspace ${\displaystyle A\subset X}$ each continuous map ${\displaystyle \varphi :A\to \mathbb {S} _{n}}$ has a continuous extension ${\displaystyle \psi :X\to \mathbb {S} _{n}}$.

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

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 ${\displaystyle {\mathcal {E}}}$ 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 ${\displaystyle 2}$-dimensional compact planes. The traditional way to distinguish ${\displaystyle {\mathcal {E}}}$ from the other ${\displaystyle 2}$-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 (${\displaystyle 4}$ points no ${\displaystyle 3}$ of which are collinear). In the present setting, a much weaker homogeneity condition characterizes ${\displaystyle {\mathcal {E}}}$:

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

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

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

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

Special aspects of 4-dimensional planes are treated in,[19] more recent results can be found in.[20] The lines of a ${\displaystyle 4}$-dimensional compact plane are homeomorphic to the ${\displaystyle 2}$-sphere;[21] in the cases ${\displaystyle m>2}$ the lines are not known to be manifolds, but in all examples which have been found so far the lines are spheres. A subplane ${\displaystyle {\mathcal {B}}}$ of a projective plane ${\displaystyle {\mathcal {P}}}$ is said to be a Baer subplane,[22] if each point of ${\displaystyle {\mathcal {P}}}$ is incident with a line of ${\displaystyle {\mathcal {B}}}$ and each line of ${\displaystyle {\mathcal {P}}}$ contains a point of ${\displaystyle {\mathcal {B}}}$. A closed subplane ${\displaystyle {\mathcal {B}}}$ is a Baer subplane of a compact connected plane ${\displaystyle {\mathcal {P}}}$ if, and only if, the point space of ${\displaystyle {\mathcal {B}}}$ and a line of ${\displaystyle {\mathcal {P}}}$ have the same dimension. Hence the lines of an 8-dimensional plane ${\displaystyle {\mathcal {P}}}$ are homeomorphic to a sphere ${\displaystyle \mathbb {S} _{4}}$ if ${\displaystyle {\mathcal {P}}}$ has a closed Baer subplane.[23]

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

Let ${\displaystyle {\mathcal {P}}}$ be a compact plane of dimension ${\displaystyle 2^{m},\;m=2,3,4}$, and write ${\displaystyle \Sigma =\operatorname {Aut} {\mathcal {P}}}$. If ${\displaystyle \dim \Sigma >8,18,40}$, then ${\displaystyle {\mathcal {P}}}$ is classical,[27] and ${\displaystyle \operatorname {Aut} {\mathcal {P}}}$ is a simple Lie group of dimension ${\displaystyle 16,35,78}$ respectively. All planes ${\displaystyle {\mathcal {P}}}$ with ${\displaystyle \dim \Sigma =8,18,40}$ are known explicitly.[28] The planes with ${\displaystyle \dim \Sigma =40}$ are exactly the projective closures of the affine planes coordinatized by a so-called mutation ${\displaystyle (\mathbb {O} ,+,\circ )}$ of the octonion algebra ${\displaystyle (\mathbb {O} ,+,\ \,)}$, where the new multiplication ${\displaystyle \circ }$ is defined as follows: choose a real number ${\displaystyle t}$ with ${\displaystyle 1/2<t\neq 1}$ and put ${\displaystyle a\circ b=t\cdot ab+(1-t)\cdot ba}$. 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 ${\displaystyle m=3}$ and [30] for ${\displaystyle m=4}$.

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

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

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

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

In an analogous way, the classical ${\displaystyle 4}$-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 ${\displaystyle 1}$ or ${\displaystyle 2}$. Hence the point space of a locally compact connected Laguerre plane is homeomorphic to the cylinder ${\displaystyle C}$ or it is a ${\displaystyle 4}$-dimensional manifold, cf.[64] A large class of ${\displaystyle 2}$-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 ${\displaystyle \mathbb {R} ^{2}}$.

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

If the automorphism group ${\displaystyle \Sigma }$ of a ${\displaystyle 2d}$-dimensional Laguerre plane ${\displaystyle {\mathcal {L}}}$ is transitive on the set of parallel classes, and if the kernel ${\displaystyle T\triangleleft \Sigma }$ is transitive on the set of circles, then ${\displaystyle {\mathcal {L}}}$ 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 ${\displaystyle 2d}$-dimensional Laguerre planes.

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

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

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