The Fundamental Group

Abstract

The aim of this section is to make the homotopy equivalence classes of paths that start and end at a fixed point in a space into a group. We will see later how this group tells us about the covering spaces of X, as well as determining the first homology and cohomology groups (when X is an open set in the plane). In this chapter it will be convenient again to have all paths defined on the same interval. So a path in a topological space X will be a continuous map γ: [0,1]→X. We say that γ is a path from the point x = γ(0) to the point x′ = γ(1). In this chapter a homotopy of paths will always fix the endpoints x and x′, i.e., H(0, s) = x and H(1, s) = x′ for all 0 ≤ s ≤ 1.