Polyhedra. Simplicial Complexes. Homologies

Abstract

Geometry and topology most often deal with geometrical figures, objects realized as a set of points in a Euclidean space (maybe of many dimensions). It is useful to view these objects not as rigid (solid) bodies, but as figures that admit continuous deformation preserving some qualitative properties of the object. Recall that the mapping of one object onto another is called continuous if it can be determined by means of continuous functions in a Cartesian coordinate system in space. The mapping of one figure onto another is called homeomorphism if it is continuous and one-to-one, i.e. establishes a one-to-one correspondence between points of both figures. In this case there exists the inverse mapping with the same properties. For example, a straight line segment and a continuous arc (without self-intersections) on a plane are homeomorphic (Fig. 1.1.1). Homeomorphic are also a square and a circle (Fig. 1.1.1), a cube and a tetrahedron (sometimes called a simplex) (Fig. 1.1.1), a plane and a sphere with one punctured (discarded) point (Fig. 1.1.1). In the latter case the homeomorphism can be realized using the so-called stereographic projection (Fig. 1.1.2). To this end one should put a standard sphere onto a plane, take the point of tangency for the south pole and the very top point for the north pole N.