User:Jim.belk/Covering Spaces Draft

In mathematics, a covering space is a topological space C which "covers" another space X by a covering map p : C → X. Covering spaces are closely related to the fundamental group, and are thus a basic object of study in algebraic topology. Covering spaces (and more generally branched covers) are the natural domains of multivalued functions, making them an important tool in complex analysis, algebraic geometry, and the theory of Riemann surfaces. Covering spaces also play an important role in geometric topology, geometric group theory, differential geometry, and in the study of Lie groups and other topological groups.

As pointed out by Emil Artin, there is a strong similarity between the theory of covering spaces and Galois theory. This has been formalized in the notion of a Galois connection.

Formal definition
A cover of a topological space X is a space C together with a surjective map p : C → X having the following property: each point x &isin; X has a neighborhood U whose preimage is a disjoint union of open sets that map homeomorphically onto U. The space X is called the base space, the space C is the cover, and p is the covering map.

Sometimes authors require both X and C to be connected in the definition of a cover. In addition, most of the theory of covering spaces requires X and C to satisfy certain technical conditions: they must be path-connected, locally path-connected, and semi-locally simply connected. These requirements are necessary to exclude certain pathological examples, such as the Hawaiian earring.

Examples
The most basic example is the covering of the unit circle by the real line, via the map


 * $$p(\theta) = (\cos \theta, \sin \theta)\,$$

Each point of the circle is covered by infinitely many points on the line, one for each possible value of $$\theta$$.

Similarly, one can define a cover from $$C = (0,\infty) \times \mathbb{R}$$ to the punctured plane $$\mathbb{R}^2 - \{(0,0)\}$$ via the map


 * $$(r,\theta) \mapsto (r \cos \theta, r \sin \theta)\,$$

This covering map is essentially the complex exponential function, and the space C can be thought of as the natural domain for the complex logarithm.

The map p : C&times; → C&times; defined by p(z) = zn is a cover, where C&times; denotes the complex plane with the origin removed. Under this cover, each point z in C&times; has n different preimages, namely the set of possible nth roots of z.

The diagram below shows several different covers of a figure eight. The colors and arrows indicate the manner in which the covering spaces map to the base space.

Universal Covers
Every space X has a unique simply connected covering space, known as the universal cover. This cover has an important universal property: if p : U → X is the universal cover of X, and q : C → X is any other cover of X, then there exists a covering map r : U → C such that p = q o r. That is, the universal cover of X covers any other cover of X.

For example, the universal cover of the circle is the line, and the universal cover of the figure eight is an infinite tree. In fact, the universal cover of any graph is an infinite tree.

Universal covers are very important in the geometric study of manifolds. The universal cover of the torus is the Euclidean plane, while the universal cover of a higher-genus surface can be identified with the hyperbolic plane. The universal covers of 3-manifolds have eight possible geometries.

Universal covers are useful throughout mathematics, allowing you to replace any space X by one that is simply connected:
 * In the theory of analytic functions, universal covers of open sets in the complex plane are the natural domains for analytic continuation
 * Universal covers are quite useful in algebraic topology, because the universal cover of X has the same higher homotopy groups as X itself. This allows one to restrict the study of homotopy groups to simply connected spaces.
 * Every Lie group possesses a universal covering group with the same Lie algebra. The |representations of the universal covering group are the same as projective representations of the original group, a fact that has important applications in quantum mechanics.
 * In general relativity, the universal cover of a Lorentzian manifold is timelike simply connected, meaning that it has no closed timelike curves. Therefore, passing to the cover replaces a universe with time travel by a many-worlds universe that satisfies causality.
 * In group theory, the universal cover of a presentation complex for a group can be used to obtain the Cayley graph. By attaching more cells to the presentation complex, one can obtain an Eilenberg-MacLane space whose universal cover is contractible.

Properties of Covers
Local Homeomorphism: A covering map p : C → X is a always a local homeomorphism. If C and X are manifolds, they must have the same dimension, and p must map interior points to interior points and boundary points to boundary points. If C and X are graphs, each edge of C must map |homeomorphically to an edge of X, and each vertex of C must map to a vertex of X with the same degree.

Number of Sheets: For every x in X, the preimage of x is a discrete set of points in C. If X is connected, the cardinality of the preimage does not depend on the point x chosen. The number of points in the preimage is often called the number of sheets of the covering. For example, the map $$z \mapsto z^n$$ on the complex plane minus the origin is a cover with n sheets.

Path Lifting: If γ is a path in X from x0 to x1, and c0 is a point in C that maps to x0, then γ lifts to a unique path in C starting at c0. That is, there exists a unique path η : [0,1] → C such that η(0) = c0 and p o η = γ.

It is an important fact that a closed path in X may not lift to a closed path in the cover. The homotopy classes of loops in X that do lift to loops in C form a subgroup of the fundamental group of X. This is the basis for the Galois connection between subgroups of the fundamental group and covers.

Deck transformation group, regular covers
A deck transformation or automorphism of a cover p : C → X is a homeomorphism f : C → C such that p o f = p. The set of all deck transformations of p forms a group under composition, the deck transformation group Aut(p). Every deck transformation permutes the elements of each fiber. This defines a group action of the deck transformation group on each fiber. Note that by the unique lifting property, if f is not the identity and C is path connected, then f has no fixed points.

Now suppose p : C → X is a covering map and C (and therefore also X) is connected and locally path connected. The action of Aut(p) on each fiber is free. If this action is transitive on some fiber, then it is transitive on all fibers, and we call the cover regular. Every such regular cover is a principal G-bundle, where G = Aut(p) is considered as a discrete topological group.

Every universal cover p : D → X is regular, with deck transformation group being isomorphic to the fundamental group π(X).

The example p : C&times; → C&times; with p(z) = zn from above is a regular cover. The deck transformations are multiplications with n-th roots of unity and the deck transformation group is therefore isomorphic to the cyclic group Cn.

Monodromy action
Again suppose p : C → X is a covering map and C (and therefore also X) is connected and locally path connected. If x in X and c belongs to the fiber over x (i.e. p(c) = x), and γ:[0,1]→X is a path with γ(0)=γ(1)=x, then this path lifts to a unique path in C with starting point c. The end point of this lifted path need not be c, but it must lie in the fiber over x. It turns out that this end point only depends on the class of γ in the fundamental group π(X,x), and in this fashion we obtain a right group action of π(X,x) on the fiber over x. This is known as the monodromy action.

So there are two actions on the fiber over x: Aut(p) acts on the left and π(X,x) acts on the right. These two actions are compatible in the following sense:
 * f.(c.γ) = (f.c).γ

for all f in Aut(p), c in p-1(x) and γ in π(X,x).

If p is a universal cover, then the monodromy action is regular; if we identify Aut(p) with the opposite group of π(X,x), then the monodromy action coincides with the action of Aut(p) on the fiber over x.

Group structure redux
The deck transformation group and the monodromy action can be understood to relate the normal subgroups of the fundamental group $$\pi_1(X)$$ of X and the fundamental group $$\pi_1(C)$$ of the cover. Furthermore, these equate the conjugacy classes of subgroups of $$\pi_1(X)$$ and equivalence classes of coverings. As a result, one can conclude that X=C/Aut(p), that is, the manifold X is given as the quotient of the covering manifold under the action of the deck transformation group. These inter-relationships are explored below.

Let γ be a curve in X. Denote by $$\gamma_C$$ the lift of γ to C. Consider the set


 * $$\Gamma_p(c) = \{ \gamma : \gamma_C \mbox{ is a closed curve in } C

\mbox { passing through } c\in C \}$$

Note that $$\Gamma_p(c)$$ is a group, and that it is a subgroup of $$\pi_1(X,p(c))$$. Note also that it depends on c, and that different values of c in the same fiber yield different subgroups. Each such subgroups is conjugate to another by the monodromy action. To see this, pick two points $$c_1, c_2$$ in the same fiber: $$p(c_1)=p(c_2)=x$$ and let g be a curve in C connecting $$c_1$$ to $$c_2$$. Then p(g) is a closed curve in X. If $$\gamma_C$$ is a closed curve in C passing through $$c_1$$, then $$g\gamma_C g^{-1}$$ is a closed curve in C passing through $$c_2$$. Thus, we have shown


 * $$\Gamma_p(c_2) = g \Gamma_p(c_1) g^{-1}$$

and so we have the result that $$\Gamma_p(c_1)$$ and $$\Gamma_p(c_2)$$ are conjugate subgroups of $$\pi_1(X,x)$$. All of the conjugate subgroups may be obtained in this way.

It should be clear that two equivalent coverings lead to the same conjugacy class of subgroups of $$\pi_1(X,x)$$; there is a bijective correspondence between equivalence classes of coverings and conjugacy classes of subgroups of $$\pi_1 (X)$$.

Note that this implies that the fundamental group $$\pi_1(C)$$ is isomorphic to $$\Gamma_p$$. Let $$N(\Gamma_p)$$ be the normalizer of $$\Gamma_p$$ in $$\pi_1(X)$$. The deck transformation group Aut(p) is isomorphic to $$N(\Gamma_p)/\Gamma_p$$. If p is a universal covering, then $$\Gamma_p$$ is the trivial group, and Aut(p) is isomorphic to $$\pi_1(X)$$.

As a corollary, let us reverse this argument. Let Γ be a normal subgroup of $$\pi_1(X,x)$$. By the above arguments, this defines a (regular) covering $$p:C\rightarrow X$$. Let $$c_1$$ in C be in the fiber of x. Then for every other $$c_2$$ in the fiber of x, there is precisely one deck transformation that takes $$c_1$$ to $$c_2$$. This deck transformation corresponds to a curve g in C connecting $$c_1$$ to $$c_2$$.

Note that Aut(p) operates properly discontinuously on C, and so we have that X=C/Aut(p), that is, X is the manifold given by the quotient of the covering manifold by the deck transformation group.

Generalizations
A cover can be viewed as fiber bundle whose fibers are discrete, with regular covers being a special case of principle G-bundles. Even more general is the idea of a fibration, which can be thought of as a "homotopical" fiber bundle.

A branched covering is a covering map except at a discrete set of branch points. Branched coverings are very important in algebraic geometry and the study of Riemann surfaces.

In algebraic topology, Whitehead towers are associated to the higher homotopy groups in the same way that the universal cover is associated to the fundamental group.