The fundamental group and its applications
Terry C. Lawson · 2003
Abstract In this chapter we study, through the example of the fundamental group, the general method of algebraic topology. We give a means of associating to a geometric problem a (hopefully easier) algebraic problem to solve. Consider the problem of distinguishing between two surfaces. How can we tell, for example, that there is no homeomorphism between the sphere and the torus? According to algebraic topology, to solve this problem we should transform it into an algebraic problem that is readily solved. To each topological space X, associate to it some algebraic object, say a group g(X). Do this in such a way that homeomorphic spaces have isomorphic groups associated to them. Thus one way of telling that X is not homeomorphic to Y is by showing that g(X) is not isomorphic to g(Y). Of course, this works only when we can readily compute g(X) and g(Y) and decide whether or not they are isomorphic. This method is successful in distinguishing between surfaces.We now formalize the informal discussion above. In this section, we start by discussing briefly the concept of a group. Readers with a previous course in abstract algebra should just skim over the group theory material in this section to become familiar with our notation and viewpoint. We then apply these ideas to discuss how algebraic topology uses group theory to answer topological questions. In later sections and chapters, we will introduce more sophisticated results from group theory as it is needed.