No abstract

According to Soulie [1], computing the homology of a group is a fundamental question and can be a very difficult task. In his assertion, 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 proof the theorems, yet much is done on some properties of each of these theorems, which is enough for the calculation of (co) homology groups.

In this paper, we will first construct a Robertson – Walker like metric in (2 + 1) – dimensional space. The easiest way of doing this is to consider a 2-dimensional coordinate space as a space embedded in a 3-dimensional hypersurface. The curvature of each surface is determined using the spatial part of the Robertson – Walker like metric constructed. Our main goal is to find out if the Robertson – Walker like metric in (2 + 1) – dimensional space can be used as a prototype model to study Robertson – Walker in (3 + 1) dimensions since calculations involved in higher dimensions are tedious.

This study is an investigation of some of the relationships which exist between various generalizations of completely regular spaces. The primary aim of the study is to look at the separation axioms and delve more into one of the claims about completely regular space; “Every completely regular space is a regular space as well”.