Embeddable Spherical Circle Planes (2009)
Type of ContentTheses / Dissertations
Degree NameMaster of Science
PublisherUniversity of Canterbury. Mathematics and Statistics
AuthorsLightfoot, Ashley Charlesshow all
Spherical circle planes are topological incidence geometries; one has a 2- sphere P and a collection of 1-spheres in P such that any three points in P determine exactly one of these 1-spheres (the ‘circles’ of the spherical circle plane). A determination of the homeomorphism type of the circle space (the collection of 1-spheres suitably topologised) of a topological 2-dimensional Möbius plane, which is a spherical circle plane with the additional property of the axiom of touching, was given by Karl Strambach in 1974. Embeddable spherical circle planes are a type of spherical circle plane that are not, in general, Möbius planes, constructed on a 2-sphere P in ℝ3 by taking the circles to be precisely the non-trivial plane intersections in ℝ3 with P. We show that the circle space of an embeddable spherical circle plane is homeomorphic to the 3-dimensional projective space minus one point. Furthermore, it is shown that the flag space of an embeddable spherical circle plane is homeomorphic to the flag space of the classical flat Möbius plane; a topological description of the latter is also given. An apparent gap is the literature is also filled: we prove the well-known conjecture that the flag space of a general spherical circle plane is a 4-dimensional manifold. Finally, we define the notion of isotopy equivalence between spherical circle planes and prove that embeddable spherical circle planes are isotopy equivalent to the classical flat Möbius plane.