Some Homological Groups Related To Simplicial Complexes

Abstract

The aim of the paper is to study some of the homological groups in general and related these groups with simplicial complexes. Characterization these groups revealed the successful method to study the simplicial complex which has the following two properties: (a) each q-simplex determines (q +1) faces of dimension q-1, (b) the faces of a simplex determine the simplex and a semi-simplicial complex K is a collection of elements {f} called simplexes together with two functions. The main examples of homological groups are rchain group, r-cycle group and r-boundary group. When we calculating the Euler characteristic of surface, we need to building a multi-surface equivalent to the original surface, therefore in this paper we achieved that the homological groups are a type of improvement for the Euler characteristic. If there is no simplex of order two (2-simplexs) in K, then B1(K) and H1(K) are equal to Z1(K). Also if K is a simplex complex, then r-chain (Cr(K) is a group. We obtained that if three points and three lines such that is triangulation of the rings and there is no simplex of order two (2-simplexs) in K, in this case the boundary homological group equal zero and H1(K)=Z1(K).