ABSTRACT
We shall now study one of the most basic concepts in algebraic topology viz., the covering spaces. They are closely related to the study of the fundamental groups of spaces on the one hand and to the study of the action of the discontinuous groups on the other. Having met the notion of fundamental groups, it is time to study the theory of covering spaces and their relation with fundamental groups. We shall also study a little bit about the ‘discontinuous groups’, vis-a-vis covering spaces and fundamental group. Classically however, these concepts occurred in the reverse order. The study of discontinuous groups goes back to the time of Gauss and occurred in the theory of elliptic functions and then in the theory of modular forms. During the time of Riemann, the notion of covering space started taking shape in what is today known as the theory of Riemann surfaces. The fundamental group appeared for the first time in the third installment of the celebrated papers ANALYSIS SITUS of Poincare´, around the turn of this century. Nowadays, these three notions have taken deep root in almost all branches of mathematics. They have been found useful and, in any case, make a very delightful subject of study.
