Theory of Covering Spaces

Introduction. In this paper I study covering spaces in which the base space is an arbitrary topological space. No use is made of arcs and no assumptions of a global or local nature are required. In consequence, the fundamental group ir(X, Xo) which we obtain frequently reflects local properties which are missed by the usual arcwise group iri(X, Xo). We obtain a perfect Galois theory of covering spaces over an arbitrary topological space. One runs into difficulties in the non-locally connected case unless one introduces the concept of space [§l], a common generalization of topological and uniform spaces. The theory of covering spaces (as well as other branches of algebraic topology) is better done in the more general domain of spaces than in that of topological spaces. The Poincaré (or deck translation) group plays the role in the theory of covering spaces analogous to the role of the Galois group in Galois theory. However, in order to obtain a perfect Galois theory in the non-locally connected case one must define the Poincaré filtered group [§§6, 7, 8] of a covering space. In §10 we prove that every filtered group is isomorphic to the Poincaré filtered group of some regular covering space of a connected topological space [Theorem 2]. We use this result to translate topological questions on covering spaces into purely algebraic questions on filtered groups. In consequence we construct examples and counterexamples answering various questions about covering spaces [§11 ]. In §11 we also discuss other applications of the general theory of covering spaces—e.g., a theory of covering spaces with singularities [§2, Example 1; §11, Example l]; the defining of a "relative fundamental group" ir(X, A) [§2, Example 2; §11, Example 3]; the interpretation of the étale coverings in characteristic 0 of algebraic geometry as covering spaces [§11, Example 2]; and the resolution of the classical problem of the classification of the covering spaces of the topological space : {(0,0)} VJGraph [sin ( 1 /x), x > 0 ] [§11, Example 4]. The following table summarizes some of the similarities between Galois theory and the theory of covering spaces: