Covering Spaces

Abstract

In order to bring to a satisfactory conclusion the theory of the last chapter, it is necessary to show that the space X _∞, together with the given action on it by the infinite cyclic group, is uniquely defined by the oriented link L under consideration. Here it will be seen that X _∞ is a certain covering space of the exterior of L, and the theory of coverings will show it to be well defined. That is the present motivation, but it should be understood that the theory of covering spaces is an important part of many areas of mathematics (particularly Riemann surfaces and geometric structures on manifolds). It is intimately related to the study of the (appropriately named) fundamental group of a fairly general type of topological space. Thus the following discussion will be in the language of general topological spaces.