Covering Projections

Abstract

Let exp : ℝ → S¹ denote the normalized exponential map defined by exp(t) = e^2πit, t ∈ ℝ. Recall (Chapter 2, Theorem 2.6.1, 2.6.2) that this map has the Path Lifting Property as well as the Homotopy Lifting Property. We now wish to generalize these results to a wider class of continuous maps $p:\tilde X \to X$, called covering projections. The theory of covering projections is of great importance not only in topology, but also in other branches of mathematics like complex analysis, differential geometry and Lie groups, etc. The concept of fundamental group, discussed in Chapter 2, is intimately related to covering projections. Among other things, we will see that the problem of “lifting” a continuous map f : A → X to a continuous map $\tilde f:A \to \tilde X$, where $p:\tilde X \to X$ is a covering projection, has a complete solution in terms of the fundamental groups of the spaces involved and the induced homomorphisms among them.