The integral homology of orientable Seifert manifolds

For any orientable Seifert manifold M, the integral homology group H1(M)=H1(M;Z) is computed and explicit generators are found. This calculation gives a presentation for the p-torsion of H1(M) for any prime p. Since Seifert manifolds have dimension 3, H1(M) determines H∗(M;A) and H∗(M;A) as well, for any abelian group A. The complete details are given when A=Z, Z/ps. er to calculate the partition functions of the Dijkgraaf–Witten topological quantum field theories it is necessary to compute the linking form of the underlying 3-manifold. In the case of the orientable Seifert manifolds it is possible to compute the linking form. The calculation of the linking form involves finding a presentation of the torsion of the first integral homology of the orientable Seifert manifolds, which is the main result of this paper.