Topological Circle Planes and Topological Quadrangles
This research note presents a complete treatment of the connection between topological circle planes and topological generalized quadrangles. The author uses this connection to provide a better understanding of the relationships between different types of circle planes and to solve a topological version of the problem of Apollonius.
Topological Circle Planes and Topological Quadrangles begins with a foundation in classical circle planes and the real symmetric generalized quadrangle and the connection between them. This provides a solid base from which the author offers a more generalized exploration of the topological case. He also compares this treatment to the finite case.
Subsequent chapters examine Laguerre, Möbius, and Minkowski planes and their respective relationships to antiregular quadrangles. The author addresses the Lie geometry of each and discuss the relationships of circle planes-the "sisters" of Möbius, Laguerre, and Minkowski planes - and concludes by solving a topological version of the problem of Apollonius in Laguerre, Möbius, and Minkowski planes.
The treatment offered in this volume offers complete coverage of the topic. The first part of the text is accessible to anyone with a background in analytic geometry, while the second part requires basic knowledge in general and algebraic topology. Researchers interested in geometry-particularly in topological geometry-will find this volume intriguing and informative. Most of the results presented are new and can be applied to various problems in the field of topological circle planes.
Table of Contents
Definitions and Notation
Models for Classical Circle Planes
The Twisting Number
Characterization of Antiregular Quadrangles
Laguerre Planes and Antiregular Quadrangles
Laguerre Planes Constructed from Antiregular Quadrangles
Antiregular Quadrangles Constructed from Laguerre Planes
Constructing Topologies on the Lie Geometry
Möbius Planes and Antiregular Quadrangles
The Lie Geometry of a Möbius Plane
The Lifted Lie Geometry of a Flat Möbius Plane
Constructing Topologies on the Lifted Lie Geometry
Characterizing Quadrangles Obtained from Flat Möbius Planes
Minkowski Planes and Antiregular Quadrangles
The Point Space and Parallel Classes
The Circle Space
The Other Spaces
The Derivation of a Minkowski Plane
The Lie Geometry of a Minkowski Plane
The Lifted Lie Geometry of a Minkowski Plane
The Topology on the Lifted Lie Geometry
Characterizing Quadrangles Obtained from Minkowski Planes
Relationship of Circle Planes
Sisters of Laguerre Planes
Sisters of Möbius Planes
Sisters of Minkowski Planes
The Problem of Apollonius
The Problem of Apollonius in Laguerre Planes
The Problem of Apollonius in Möbius Planes
One Point and Two Circles
The Problem of Apollonius in Minkowski Planes
Two Points and One Circle
One Point and Two circles
"This book is a must read for anyone interested in incidence geometry and especially anybody interested in topological incidence geometry."
-Mathematical Reviews, Issue 97b