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/p^s.In order 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.