Covering Spaces, An Introduction

Introduction

A covering space is actually two spaces, the total space (upstairs) and the base space (downstairs), and a continuous projection that carries the total space onto the base space. The projection is many to one, mapping different regions upstairs to the same region downstairs.

An important aspect of a covering space is its path lifting property. Given a path in the base space, establish even one point in its preimage, and the rest of the path's preimage in the total space is determined. As you move about downstairs, your preimage is forced to move with you upstairs.

It is possible to run around in circles downstairs, while your lift upstairs runs off to infinity. Again, this is possible because many different regions upstairs map to the same region downstairs. Each time you pass through the livingroom downstairs, your preimage could be passing through a different copy of the livingroom upstairs.

Throughout these pages, all spaces are assumed to be locally path connected. We need this for covering spaces to work properly.

Definition

A covering space consists of two spaces, S~ and S, and a continuous function p from S~ onto S. The ~ modifier is used to denote the total space, while the unadorned variable is the base space. The function p is sometimes called a projection, although this can be a bit misleading, since this has nothing to do with the components in a product space.

If U is a path connected open set in S, p covers U evenly if every path component in the preimage of U maps homeomorphically onto U. In other words, p carries many copies of U upstairs onto U downstairs.

In a covering space, every x in S has a path connected open set U containing x, that is evenly covered by p.

The preimage of a point x under p() is the fiber of x. This can also be spelled fibre of x, but wikipedia consistently refers to it as fiber, so I'm going to do the same.

Any homeomorphism from T onto S is a covering space. Since S is locally path connected, x lives in a path connected open set U, which has a single, open, homeomorphic preimage in T, whence U is evenly covered. The fiber of x is the preimage of x in T.

Another covering space is an arbitrary number of disjoint copies of S, each mapping to S. As above, the entire space S is evenly covered, but this time you have to select a preimage.

A more interesting covering space is the map from the unit circle onto itself, implemented by zⁿ in the complex plane. In other words, the circle is wrapped around itself n times. Now every x has an open arc, with n smaller arcs in the preimage. Select one of these arcs in the preimage, i.e. one of the path connected components, and the map zⁿ implements the homeomorphism. The fiber of x is now a set of size n.

Wrap the real line around the circle, as in E^it, and the fiber of x is infinite. We used this covering space to find the degree of a closed path in the circle.

Product Space

Consider the product of two covering spaces. Take S~ cross T~ for the total space, and S cross T for the base space. Run the two projections in parallel, on the two components. Since each projection is continuous from the total space into the base space, the composite function is continuous. Take x:y in S:T and embed it in U:V, such that U and V are evenly covered. since U and V are path connected, U:V is path connected. Any point in the preimage belongs to the cross product of two sets that are homeomorphic to U and V respectively. The product of these two sets is homeomorphic to U:V. Thus U:V is evenly covered, and the product of two covering spaces is a covering space.

Disjoint Union

The disjoint union of covering spaces is another covering space. This is left as an exercise for the reader. Basically, each component upstairs projects to its component downstairs, and they don't interfere with each other.

Components

Since all spaces are locally path connected, every path connected component can be covered with open sets, and is therefore open. The complement of such a component, i.e. the union of all the "other" path components, is also open. Thus each path component is open and closed, and is a proper component of the space. In the world of covering spaces, components and path components are synonymous, and connected and path connected are synonymous.