Visual Geometry and Topology

1 Polyhedra. Simplicial Complexes. Homologies.- 1.1 Polyhedra.- 1.2 Simplicial Homology Groups of Simplicial Complexes (Polyhedra).- 1.3 General Properties of Simplicial Homology Groups.- 2 Low-Dimensional Manifolds.- 2.1 Basic Concepts of Differential Geometry.- 2.2 Visual Properties of One-Dimensional Manifolds.- 2.3 Visual Properties of Two-Dimensional Manifolds.- 2.4 Cohomology Groups and Differential Forms.- 2.5 Visual Properties of Three-Dimensional Manifolds.- 3 Visual Symplectic Topology and Visual Hamiltonian Mechanics.- 3.1 Some Concepts of Hamiltonian Geometry.- 3.2 Qualitative Questions of Geometric Integration of Some Differential Equations. Classification of Typical Surgeries of Liouville Tori of Integrable Systems with Bott Integrals.- 3.3 Three-Dimensional Manifolds and Visual Geometry of Isoenergy Surfaces of Integrable Systems.- 4 Visual Images in Some Other Fields of Geometry and Its Applications.- 4.1 Visual Geometry of Soap Films. Minimal Surfaces.- 4.2 Fractal Geometry and Homeomorphisms.- 4.3 Visual Computer Geometry in the Number Theory.- Appendix 1 Visual Geometry of Some Natural and Nonholonomic Systems.- 1.1 On Projection of Liouville Tori in Systems with Separation of Variables.- 1.2 What Are Nonholonomic Constraints?.- 1.3 The Variety of Manifolds in the Suslov Problem.- Appendix 2 Visual Hyperbolic Geometry.- 2.1 Discrete Groups and Their Fundamental Region.- 2.2 Discrete Groups Generated by Reflections in the Plane.- 2.3 The Gram Matrix and the Coxeter Scheme.- 2.4 Reflection-Generated Discrete Groups in Space.- 2.5 A Model of the Lobachevskian Plane.- 2.6 Convex Polygons on the Lobachevskian Plane.- 2.7 Coxeter Polygons on the Lobachevskian Plane.- 2.8 Coxeter Polyhedra in the Lobachevskian Space.- 2.9 Discrete Groups of Motions of Lobachevskian Space and Groups of Integer-Valued Automorphisms of Hyperbolic Quadratic Forms.- 2.10 Reflection-Generated Discrete Groups in High-Dimensional Lobachevskian Spaces.- References.