Classical Tessellations and Three-Manifolds

Richly illustrated with 29 colour plates and over 200 line drawings Deals with euclidean,spherical and hyperbolic tessellations An unorthodox book, without explicit statements of theorems, but with geometrical descriptions and diagrams Includes an appendix on orbifolds

Inhalt
One.- S1-Bundles Over Surfaces.- 1.1 The spherical tangent bundle of the 2-sphere S2.- 1.2 The S1-bundles of oriented closed surfaces.- 1.3 The Euler number of ST(S2).- 1.4 The Euler number as a self-intersection number.- 1.5 The Hopf fibration.- 1.6 Description of non-orientable surfaces.- 1.7 S1-bundles over Nk.- 1.8 An illustrative example: IRP2 ? ?P2.- 1.9 The projective tangent S1-bundles.- Two.- Manifolds of Tessellations on the Euclidean Plane.- 2.1 The manifold of square-tilings.- 2.2 The isometries of the euclidean plane.- 2.3 Interpretation of the manifold of squaretilings.- 2.4 The subgroup ?.- 2.5 The quotient ?\E(2).- 2.6 The tessellations of the euclidean plane.- 2.7 The manifolds of euclidean tessellations.- 2.8 Involutions in the manifolds of euclidean tessellations.- 2.9 The fundamental groups of the manifolds of euclidean tessellations.- 2.10 Presentations of the fundamental groups of the manifolds M(?).- 2.11 The groups $$ \tilde \Gamma $$ as 3-dimensional crystallographic groups.- Appendix A.- Orbifolds.- Three.- Manifolds of Spherical Tessellations.- 3.1 The isometries of the 2-sphere.- 3.2 The fundamental group of SO(3).- 3.3 Review of quaternions.- 3.4 Right-helix turns.- 3.5 Left-helix turns.- 3.6 The universal cover of SO(4).- 3.7 The finite subgroups of SO(3).- 3.8 The finite subgroups of the quaternions.- 3.9 Description of the manifolds of tessellations.- 3.10 Prism manifolds.- 3.11 The octahedral space.- 3.12 The truncated-cube space.- 3.13 The dodecahedral space.- 3.14 Exercises on coverings.- 3.15 Involutions in the manifolds of spherical tessellations.- 3.16 The groups $$ \tilde \Gamma $$ as groups of tessellations of S3.- Four.- Seifert Manifolds.- 4.1 Definition.- 4.2 Invariants.- 4.3 Constructing the manifold from the invariants.- 4.4 Change of orientation and normalization.- 4.5 The manifolds of euclidean tessellations as Seifert manifolds.- 4.6 The manifolds of spherical tessellations as Seifert manifolds.- 4.7 Involutions on Seifert manifolds.- 4.8 Involutions on the manifolds of tessellations.- Five.- Manifolds of Hyperbolic Tessellations.- 5.1 The hyperbolic tessellations.- 5.2 The groups S?mn, 1/? + 1/m + 1/n < 1.- 5.3 The manifolds of hyperbolic tessellations.- 5.4 The S1-action.- 5.5 Computing b.- 5.6 Involutions.- Appendix B.- The Hyperbolic Plane.- B.5 Metric.- B.6 The complex projective line.- B.7 The stereographic projection.- B.8 Interpreting G*.- B.10 The parabolic group.- B.11 The elliptic group.- B.12 The hyperbolic group.- Source of the ornaments placed at the end of the chapters.- References.- Further reading.- Notes to Plate I.- Notes to Plate II.