The Fundamental Group

Abstract

For any topological space X and any point x₀ ∈ X, we will define a group, called the fundamental group of X, and denoted by π(X, x₀). (Actually, the choice of the point x₀ is usually of minor importance, and hence it is often omitted from the notation.) We define this group by a very simple and intuitive procedure involving the use of closed paths in X. From the definition, it will be clear that the group is a topological invarint of X; i.e., if two spaces are homeomorphic, thheir fundamental groups are isomorphic. This gives us the possibility of proving that two spaces are not homeomorphic by proving that their fundamental groups are nonisomorphic. For example, this method suffices to distinguish between the various compact surfaces and in many other cases.