III.93 Universal Covers

Let X be a TOPOLOGICAL SPACE [III.90]. A loop in X can be defined as a continuous function f from the closed interval [0, 1] to X such that f (0) = f (1). A continuous family of loops is a continuous function F from [0, 1]² to X such that F (t, 0) = F (t, 1) for every t; the idea is that for each t we can define a loop f_t by taking f_t (s) to be F (t, s), and if we do this then the loops f_t “vary continuously” with t. A loop f is contractible if it can be continuously shrunk to a point: more formally, there should be a continuous family of loops F (t,s) with F (0,s) = f (s) for every s and with all values of F (1, s) equal. If all loops are contractible, then X is said to be simply connected. For instance, a sphere is simply connected, but a torus is not because there are loops that “go around” the torus and therefore cannot be contracted (since any continuous deformation of a loop that goes around the torus goes around the same number of times).

Given any sufficiently nice path-connected space (that is, a space X such as a MANIFOLD [I.3 §6.9], with the property that any two points in X are linked by a continuous path), we can define a closely related simply connected space as follows. First, we pick an arbitrary “base point” x₀ in X. We then take the set of all continuous paths f from [0, 1] to X such that f (0) = x₀ (but we do not necessarily ask for f (1) to be x₀). Next, we regard two of these paths f and g as equivalent, or homotopic, if f (1) = g(1) and there is a continuous family of paths that begins with f and ends with g and always has the same beginning point and endpoint. That is, f and g are homotopic if there is a continuous function F from [0, 1]² to X such that F (t,0) = x₀ and F (t, 1) = f (1) = g (1) for every t, and F (0, s) = f (s) and F (1, s) = g (s) for every s. Finally, we define the universal cover of X to be the space of all homotopy classes of paths: that is, it is the QUOTIENT [I.3 §3.3] of the space of all continuous paths that start at xo by the EQUIVALENCE RELATION [I.2 §2.3] Of homotopy.

Let us see how this works in practice. As mentioned earlier, the torus is not simply connected, so what is its universal cover? To answer this question, it helps to think of the torus in a slightly artificial way: fix a point x₀and define the torus to be the set of all continuous paths that begin at x₀, with two of these paths regarded as equivalent if they have the same endpoint. If we do this, then for each path “all we care about” is where it ends, and the set of endpoints is clearly the torus itself. But this was not the definition of the universal cover. There we cared not just about the endpoint of a path but also about how we reached the endpoint. For instance, if the path happens to be a loop, in which case the endpoint is x₀itself, then we care about how many times that loop goes around the torus, and in what manner it goes around.

The torus can be defined as the quotient of ² by the equivalence relation where we define two points as equivalent if their difference belongs to ². Then any point in ² maps to a point in the torus (by the quotient map). Any continuous path on the torus then “lifts” uniquely to the plane in the following sense. Fix a point u₀in ² that maps to x₀in the torus. Then if you trace out any continuous path in the torus that starts at x₀, there will be exactly one way of tracing out a path in ² such that each point in that path maps to the appropriate point in the path in the torus.

Now suppose that we have two paths in the torus that start at x₀and end at the same point x_l. Then the “lifts” of those paths both start at u₀, but all we know about their endpoints is that they are equivalent: we do not know that they are the same. Indeed, if the first path is a contractible loop and the second is a loop that goes once around the torus, then their lifts will end at different points. It turns out (and if you try to visualize this then you will see that the result is very natural and plausible) that the “lifts” of two paths will end at the same point if and only if the original paths are homotopic. In other words, there is a one-to-one correspondence between homotopy classes of paths in the torus and points in ². This shows that ² is the universal cover of the torus. In a sense, the operation of passing from a space to its universal cover “unfolds” the quotienting operation that we use to get from the universal cover to the space.

A fruitful way to think of this example is to think about the natural GROUP ACTION [IV.9 §2] of ² on ². This associates with each element (m, n) of ² the translation (x,y) (x + m, y + n). We can then regard the torus as the quotient of ² by this action. That is, the elements of the torus are the orbits of the action (which are sets of the form {(x + m, y + n) : (m, n) ∈ ²}) with the quotient topology (which basically means that two translates of ² are close when you think they are close). The action of ² on ² is free and discrete, which means that each nonzero element of ² moves a small neighborhood of each point entirely off itself. It turns out that every sufficiently nice space X arises as the quotient of its universal cover by a similar group action: this group is the FUNDAMENTAL GROUP [IV.6 §2] Of X.

As its name suggests, the universal cover has a universal property. Roughly speaking, a cover of a space X is a space Y and a continuous surjection from Y to X such that the inverse image of a small neighborhood in X is a disjoint union of small neighborhoods in Y. If U is the universal cover of X and Y is any other cover of X, then U can be made into a cover of Y in a natural way. For instance, one can define a cover of the torus by an infinite cylinder by wrapping the cylinder around, and the cylinder can in turn be covered by the plane. Thus, all connected covers of X are quotients of the universal cover. What is more, each is the space of orbits for the action on the universal cover of a subgroup of the fundamental group of X. This observation sets up a correspondence between conjugacy classes of subgroups of the fundamental group of X and equivalence classes of covers. This “Galois correspondence” has many analogues elsewhere in mathematics, most classically in the theory of field extensions (see THE INSOLUBILITY OF THE QUINTIC [V.21]).

An example of the use of universal covers can be found in GEOMETRIC AND COMBINATORIAL GROUP THEORY [IV.10 §§7, 8].