EXISTENCE OF SIMPLICIAL COMPLEXS CONTAINING HOMOLOGY GROUPS ISOMORPHIC TO GIVEN FINITELY GENERATED ABELIAN GROUPS

Abstract

In Example 2.40 in [1], for each finite cyclic group, a CW Complex has been found with its p-th homologygroup isomorphic to itself (Moore Spaces). To calculate the homology groups of this CW complex, usage has been made of the homology of CW Complexes and the degree of a mapping from the sphere into itself. But it is not known whether Moore spaces are Simplicial Complexes or not. Our aim is to find a Simplicial Complex whose homology groups are isomorphic to finitely generated abelian groups, where we are to directly compute the 1st homology group of this Simplicial Complex. First, for each finite cyclic group, we construct a Simplicial Complex and use the similar method in [2] (§78) to compute its 1st homology group. This group is isomorphic to the given finite cyclic group. Later, we construct the other Simplicial Complex and compute its p-th homology group based on Mayer - Vietoris sequences in [3] (§25).