Loops and Lifts

Abstract

Let us begin this chapter with a somewhat facetious metaphor: given a degree 3 map of Riemann Surfaces f : X → Y with branch locus B , imagine Y B is the world you live in and the map f gives the vertical projection from the heavens. Chapter 4 tells us that an astronomer1, pointing the telescope straight up into the sky, observes that the portion of heavens seen is always three copies of what surrounds her. A first guess is that the heavens are then exactly three copies of the Earth, but she suspects that there may be other possibilities, and devises the following experiment to test such theory. Standing at a particular point on Earth she finds a way to “mark” a , b , c the three points in the sky lying precisely above her head. Keeping her focus on the portion of sky identified by a , she starts walking around while looking into the telescope. If the heavens are indeed three copies of Earth, every time she comes back to the original point, her gaze should return to a . So if she finds one particular walk such that when she returns she is looking at b or c , this experiment will show that the global geometry of the heavens is in fact different from three copies of Earth. This chapter is devoted to turning this silly metaphor into actual mathematics. Our goal is to introduce the notion of coverings , i.e. pairs of spaces whose local geometry is identical; the global geometries are then controlled by the groups of loops of the two spaces. We begin by introducing the notion of homotopy of functions, corresponding to the idea of “wiggling” one function into another. We define the fundamental group of a pointed topological space as the group of homotopy equivalence classes of loops originating at the base point of the topological space. We show that, given a covering, the fundamental group of the source space is naturally identified with a subgroup of the fundamental group of the target space, and that in fact there is a perfect “dictionary” between coverings of a given (pointed) space and subgroups of its fundamental group, known as the Galois Correspondence of Covering Theory .