coverings

Coverings in topology describe a relationship between spaces via a covering map. A covering map p: E -> B is a continuous surjection such that for every b in B there exists an open neighborhood U of b with p^{-1}(U) a disjoint union of open sets, each of which is mapped homeomorphically onto U by p. E is called the covering space and B the base space.

A key example is the map p: S^1 -> S^1 given by p(z) = z^n, which is an n-fold covering. The universal cover of the circle is p: R -> S^1, t -> e^{it}. More generally, many spaces admit coverings that locally look like a simple product U x F, with p projecting to U.

Coverings have lifting properties: path lifting and unique path lifting. Given a path in B and a starting point in E over the path’s initial point, there exists a unique lift in E beginning at that point. Homotopies can often be lifted as well, leading to connections with fundamental groups.

When B is path-connected, locally path-connected, and semilocally simply connected, it has a universal cover ~B with a simply connected total space and a covering map p: ~B -> B. The fiber p^{-1}(b0) can be identified with the fundamental group π1(B, b0), and this group acts as the group of deck transformations on ~B.

Connected coverings of a space correspond to conjugacy classes of subgroups of π1(B, b0). A covering is regular (Galois) precisely when the corresponding subgroup is normal, in which case the deck transformation group is isomorphic to π1(B, b0)/H.