The Universal Coefficient Theorem for Homology and Cohomology: An Enigma of Computations

Main Article Content

Frank Kwarteng Nkrumah
Samuel Amoh Gyampoh
William Obeng-Denteh

Abstract

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.

Keywords:
Abelian group, homology, cohomology, exact sequences, tensor product, homomorphism, isomorphism, torsion product, extension, Kiinneth formula and cross product

Article Details

How to Cite
Nkrumah, F. K., Gyampoh, S., & Obeng-Denteh, W. (2019). The Universal Coefficient Theorem for Homology and Cohomology: An Enigma of Computations. Archives of Current Research International, 17(4), 1-5. https://doi.org/10.9734/acri/2019/v17i430116
Section
Short Research Article