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, Moebius, 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 Moebius, Laguerre, and Minkowski planes - and concludes by solving a topological version of the problem of Apollonius in Laguerre, Moebius, 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.
Introduction Circle Planes Introduction Definitions and Notation Models for Classical Circle Planes Derived Structures Antiregular Quadrangles Introduction Generalized Quadrangles Square Projections The Twisting Number Antiregular Quadrangles Characterization of Antiregular Quadrangles Laguerre Planes and Antiregular Quadrangles Introduction Laguerre Planes Constructed from Antiregular Quadrangles Antiregular Quadrangles Constructed from Laguerre Planes Constructing Topologies on the Lie Geometry Moebius Planes and Antiregular Quadrangles Introduction The Lie Geometry of a Moebius Plane The Lifted Lie Geometry of a Flat Moebius Plane Constructing Topologies on the Lifted Lie Geometry Characterizing Quadrangles Obtained from Flat Moebius Planes Minkowski Planes and Antiregular Quadrangles Introduction 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 Introduction Sisters of Laguerre Planes Sisters of Moebius Planes Sisters of Minkowski Planes The Problem of Apollonius Introduction The Problem of Apollonius in Laguerre Planes The Problem of Apollonius in Moebius Planes One Point and Two Circles Three Circles The Problem of Apollonius in Minkowski Planes Two Points and One Circle One Point and Two circles Three Circles Index Glossary References
Reviews for Topological Circle Planes and Topological Quadrangles
This book is a must read for anyone interested in incidence geometry and especially anybody interested in topological incidence geometry. -Mathematical Reviews, Issue 97b