Covering Space and Van Kampen theory methods of Fundamental Group

This paper examines two methods of computing fundamental group which are covering space method and the Van Kampen theory method. Van Kampen theory method is more analytical than the cover space method; the idea is used to solve a geometrical problem of global nature by first reducing it to homotopy theory problem which in turn reduces to an algebraic problem and solves as such. In 2001, H. Fausk et al [1] and Hu [2] showed the isomorphism between left and right adjoint theory and its application to homotopy categories, bearing in mind that homotopy group are higher dimension of fundamental group. In this paper detail explanations of fundamental group and homotopy group will be given. Various definitions and explanation of concepts, which are directly or indirectly related, will also be considered with illustration on how fundamental group can be calculated. This paper also reviews that in principle any space that can be broken up into pieces can have its fundamental group described by generators and relations via Van Kampens theorem