Covering Spaces

Covering Spaces

Covering Space

A covering space is a continuous map such that each point has an open neighborhood such that

• is a disjoint union of open sets in : ,
• for each , the restriction is a homeomorphism.

Such a neighborhood is called evenly covered by , and the set of all such open neighbourhoods is called a good cover of .

e.g.

• Let’s consider the space covered by .
  
  A good cover for can be , where: is a neighborhood of the ‘a’ loop. is a neighborhood of the ‘b’ loop. is a small neighborhood of the wedge point.
  
  The image above shows a portion of a covering space of . The preimages of the open sets, for example , consist of disjoint copies of open sets where each copy is homeomorphic to .
• The Empty Set: For any space , the map is a covering space. Any open cover of is a good cover since the preimage of any set is the empty set.
• The Identity Map: is a trivial covering space. Any open cover is a good cover.
• The Real Line over the Circle: The exponential map defined by is a covering map, wrapping the real line around the circle infinitely many times. Locally, it looks like is a bunch of evenly spaced intervals mapping homeomorphically onto an arc of the circle. A concrete application of this can be seen in the proof of the fundamental group of .
• The Complex Plane without the Origin: The map given by is a covering map. This is essentially the exponential map viewed through polar coordinates.
• The n-fold Cover of the Circle: For a fixed integer , the map defined by is a covering space. This map wraps the circle around itself times. In general, one can think of this as “the sphere over real projective space”, that is, which identifies antipodal points is a covering space.
  

Universal Cover

A universal cover of a topological space is a covering space with a covering map such that is simply connected.

e.g.

• The map is a universal cover, since is simply-connected.
• For , the map is a universal cover because is simply-connected.
• The universal cover of is an infinite 4-valent tree, which is the Cayley graph of the free group on two generators.
  

Proposition

Suppose is the universal cover of , is the universal cover of , then is up to homeomorphism.

e.g. The universal cover of a torus is .

The Homotopy Lifting Property

Homotopy Lifting Property

Let be a covering space. Let be a homotopy, and let be a lift of the initial map (meaning ). Then, there exists a unique lift of the entire homotopy, , such that and .
The property is summarized by the following commutative diagram:

Proof Sketch The goal is to find a unique map that makes the full diagram commute. For each point , we can lift the path to a path in starting at . We can then find a neighborhood of and lift the homotopy on the “tube” to a map . Because these local lifts are unique, they must agree on the overlaps of neighborhoods, such as on . This allows us to “glue” these local lifts together to construct the desired global lift .

A key result connecting the topology of a covering space to its base space involves their fundamental groups. The homotopy lifting property leads to this fundamental proposition.

Proposition

For any covering space , the homomorphism induced on the fundamental groups, , is injective.

Proof Sketch If a loop class in maps to the identity in , it means its projection is null-homotopic in . By the homotopy lifting property, this null-homotopy can be lifted to a homotopy in that starts at and ends at a lift of the constant path at . Since lifts are unique, the endpoint of this homotopy must be the constant path at , proving that is null-homotopic. Thus, the kernel of is trivial.

Remark

This injectivity allows us to view as a subgroup of . The image, , consists of all loop classes in whose representative loops lift to loops in based at .

Proposition

For any covering space , the homomorphism induced on the higher homotopy groups, for is an isomorphism.

Sheets, Fibers, and Group Index

The geometric picture of a covering space having multiple “sheets” over the base space can be made precise and connected to the algebraic structure of the fundamental groups. In fact, a covering space is also an example of a fiber bundle where the fibers are discrete sets.

Lemma

For a covering space , if the base space is connected, then the cardinality of the fiber, , is constant for all . This constant value is called the number of sheets of the cover.

Proof The function mapping a point to the number of points in its fiber, , is locally constant due to the nature of evenly covered neighborhoods. Since is connected, any locally constant function on it must be globally constant.

e.g.

• The map from to is an -sheeted cover.
• The standard covering map is a 2-sheeted cover.
  

The main theorem of this section provides a beautiful correspondence between the geometry of the sheets and the algebra of group theory.

Proposition

If and are path-connected, the number of sheets of the covering is equal to the index of the subgroup within the group .

Proof Sketch We construct a bijection from the set of left cosets to the fiber .

1. First, define a map that takes a loop class and maps it to the endpoint of its lift which starts at .
2. This map is constant on the left cosets of . For any , the lift of terminates at the same point as the lift of , so . Therefore, induces a map .
3. Surjectivity: Any point in the fiber can be reached by a path from (since is path-connected). The projection of this path gives a loop class in that maps to .
4. Injectivity: If two cosets map to the same point in the fiber, it means the lifts of their representative loops end at the same point. This implies that the loop lifts to a loop in , meaning . Thus, the cosets must have been the same.

e.g. Let’s analyze the 3-sheeted cover of .

• The fundamental group of the base space is the free group on two generators, .
• The covering space is a graph with fundamental group , with generators we can call .
• The induced map sends these generators to loops in . For instance, they might correspond to , , and .
• The image subgroup is .
• A group-theoretic calculation shows that the index of this subgroup, , is 3. This matches the number of sheets of the cover, confirming the theorem.