A principal G -bundle over S is defined by covering S by two charts, each homeomorphic to the open 2-ball such that their intersection is homeomorphic to the strip S × I. 2-balls are homotopically trivial and the strip is homotopically equivalent to the circle S. So a topological classification of the possible connections is reduced to classifying the transition functions. The transition function maps the strip to G, and the different ways of mapping a strip into G are given by the first homotopy group of G.