On Algebraic and Topological Properties of Some Fundamental Groups

In this paper, we study the algebraic and topological properties of some topological spaces. We note that the fundamental group of a topological group is abelian and we study some spaces of the same homotopy type with the unit circle S1 . The basic group of the unit circle S1 is isomorphic to the additive group of integers. We say that a topological space is simply connected if it is path-connected and has a trivial fundamental group. We show that the fundamental group of a n-punctured plane is free and we characterize some surfaces as topologically distinct.