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In this talk, we will work almost exclusively with the Euler characteristic and some of the consequences of its topological invariance. We discuss by following that motivated the study of the characteristic. This gives a relation between topological invariant of the surface and a quantity derived from its combinatorial description. Secondly, we obtain an inequality relating the number of normal triangles and normal quadrilaterals, that depends on the maximum number of tetrahedrons that share a vertex. In this thesis, we disscuss this and related injectivity conditions and show that there are many rings.