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

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