(p.684) B Quotients and universal covering spaces
(p.684) B Quotients and universal covering spaces
The purpose of the present appendix is to give a formal definition of some of the concepts used in Chapter 3. In particular, we define the notion of simple connectedness and of a universal covering space. However, we do not define what a quotient space is. The reader interested in a formal discussion of this notion is referred to, e.g., [15, Chapter 4]. Moreover, we assume the reader to be familiar with the concept of a manifold.
B.1 Simple connectedness
Let us begin by defining the concept of a simply connected manifold.
Let M be a manifold. A loop in M is a continuous function such that . A loop is said to be contractible if there is a continuous function such that , , and for all . The manifold M is said to be simply connected if every loop in M is contractible.
The spheres for and are simply connected manifolds. However, , and are not.
Due to Theorem 3.2, it would be desirable to only consider simply connected manifolds when trying to characterise topology via geometry. However, such a restriction would not be reasonable. On the other hand, it turns out that given any manifold, there is a canonically associated simply connected manifold, called the universal covering space:
Given a manifold M, there is a manifold , called the universal covering space of M, and a map , called the universal covering map (or universal covering projection), such that
• is simply connected,
• given , there is a connected open neighbourhood U of p such that , where A is some index set and and are disjoint if ,
• , restricted to , yields a homeomorphism from to U.
(p.685) Remark B.4
If is a Riemannian metric on M, then the above construction yields a natural Riemannian metric on , say ; in fact, . Moreover, if M is a closed manifold, then is geodesically complete.
Thinking of as the unit circle in the complex plane, there is a map defined by . Let us argue that the existence of this map demonstrates that is the universal covering space of and that is the universal covering projection. To start with, it is intuitively clear that is simply connected. Let and U be any connected open neighbourhood of p which does not coincide with the entire circle. Then is a union of disjoint sets, say , . Moreover, restricting to one of these sets, say , we obtain a homeomorphism from to U. As a consequence, is the universal covering space of and is the universal covering projection; cf. Figure 3.2 for an illustration.
Given a manifold, the universal covering space is unique up to diffeomorphism. Say now, for the sake of argument, that it is possible to classify simply connected manifolds. The question then arises how this knowledge can be used to obtain information concerning manifolds that are not simply connected. In other words, given a simply connected manifold M, how does one determine the manifolds that have M as a universal covering space? This question leads us to the concept of a deck transformation.
Let be the universal covering space of M and let be the covering projection. A deck transformation is then a diffeomorphism from to itself such that .
If there is a Riemannian metric on M, then the deck transformations are isometries of the corresponding metric on the universal covering space.
It turns out that the set of deck transformations, say Γ, form a group. Moreover, this group clearly acts in a natural way on the universal covering space. Furthermore, if g is a Riemannian metric on M, then Γ is a subgroup of the isometry group of the corresponding metric on the universal covering space. Finally, the group action has certain properties; it is free and properly discontinuous. The formal definition of these concepts is as follows; cf., e.g., [98, Definition 6, p. 188].
A group Γ of diffeomorphisms of a manifold M is said to act freely and properly discontinuously provided
1. each has a neighbourhood U such that if meets U for , then is the identity,
2. points not in the same orbit have neighbourhoods U and V such that for every , and V are disjoint.
For convenience, most of the time we simply say that the group acts nicely.
From our perspective, the main point of the definition is that if Γ acts freely and properly discontinuously on a simply connected manifold, then the quotient is also a manifold; cf., e.g., [98, Proposition 7, p. 188]. Moreover, M is the universal covering space of the quotient and the natural projection is the universal covering projection. As a consequence of the above considerations, there are two natural perspectives.
(p.686) Perspective 1
Let M be a manifold. Then there is a universal covering space and a group of diffeomorphisms Γ (acting freely and properly discontinuously on ) such that . Moreover, given a Riemannian metric on M, there is a corresponding metric on and Γ is a group of isometries.
Let be a simply connected manifold. If Γ is a group of diffeomorphisms of acting freely and properly discontinuously, then is a manifold. Moreover, the natural projection is a covering projection making into the universal covering space of M. If is endowed with a Riemannian metric and Γ is a subgroup of the isometry group, we, in addition, obtain a metric on M which is locally isometric to that of .
If we are in a position to classify the simply connected manifolds and the collection of free and properly discontinuous group actions on these manifolds, we are able to classify the non-simply connected manifolds as well.