Arrangement of Part I
In Part I, we treat the fundamental group and the closely related notion of covering space. The geometric idea for the construction of the fundamental group functor is homotopy of paths. Roughly speaking, a homotopy of a path is a deformation leaving the end points fixed. A composition of paths may be defined when the end point of one agrees with the initial point of the other. Familiar algebraic properties, like associativity, do not hold, but do hold up to homotopy. The result is a group structure on equivalences classes, called the fundamental group. This group is not just a topological invariant, but invariant under a larger class of maps, called homotopy equivalences. These topics are treated in Sections 2 and 3.