Singular homology theory

This thesis consists of 4 chapters. In the first chapter, mathematical definitions, lemmaand equations which will be used in the thesis and will ease the understanding take place. In thesecond chapter, we made the definitions of simplexes,simplicial complexes, oriented simplexes,boundary operator, chain groups,cycle group and boundary group, and hence homology groupswhich will be used in the thesis again.In the third chapter, with the formula of the fact that the groups of two spaces areisomorphic, we gave and examplified the theories and lemma of homology theory. In the lastchapter, we aplicate these homological properties ( properties of homology goups) on thealgebra.In general, in the thesis, to benefit from the Euler characteristic equation of surfaces, weconstructed the desired simplexes to make the surface of the house to homeomorphic. So, bycomputing the numbers of edges, vertices, faces. We enabled the objects to link to abeliangroups.Keywords: Simplexes,Simplicial Complexes, Oriented Simplexes, Boundary Operator, ChainGroups,Cycle Group and Boundary Group, Homology Group, Triangulation, HomolgicalAlgebra, Relative Homolgy, The Exact Homolgy Sequence of a Pair.