The Universal Coefficient Theorem for Homology and Cohomology an Enigma of Computations

Computing the homology of a group is a fundamental question and can be a very difficult task. A complete understanding of all the homology groups of mapping class groups of surfaces and 3-manifolds remains out of reach at present time. It is imperative that we give the universal coefficient theorem the supposed needed attention. In this article, we study some product topologies as well as the kiinneth formula for computing the (co) homology group of product spaces. The paper begins with study on the algebraic background with specific definitions and extends into four theorems considered as the Universal Coefficient Theorem. Though this article does not prove the theorems, yet much is done on some properties of each of these theorems, which is enough for the calculation of (co) homology groups.