covering space

Let X and E be topological spacesMathworldPlanetmath and suppose there is a surjective continuous map p:EX which satisfies the following condition: for each xX, there is an open neighborhood U of x such that

  • p-1(U) is a disjoint union of open sets EiE, and

  • each Ei is mapped homeomorphically onto U via p.

Then E is called a covering space, p is called a covering map, the Ei’s are sheets of the covering of U and for each xX, p-1(x) is the fiber of p above x. The open set U is said to be evenly covered. If E is simply connected, it is called the universal covering space.

From this we can derive that p is a local homeomorphism, so that any local property E has is inherited by X (local connectedness, local path connectedness etc.). Covering spaces are foundational in the study of the fundamental groupPlanetmathPlanetmath of a topological space; in particular, there is a correspondence ( between connected coverings of X and subgroups of the group of deck transformationsMathworldPlanetmath of its universal covering space which is exactly analogous to the fundamental theorem of Galois theoryMathworldPlanetmath.

Covering maps are especially important in the study of Riemann surfaces; in this context, one sometimes discusses a generalized notion of covering map called a “ramified covering”; this allows one to replace a discrete set of the local homeomorphisms by maps that locally look like zzn in the complex planeMathworldPlanetmath near zero. Covering maps are also generalized in algebraic geometryMathworldPlanetmathPlanetmath; there the corresponding notion is that of étale morphism.

Note that this is a completely separate usage of the word “cover” than we encounter in “open cover”; confusion usually does not arise.

Title covering space
Canonical name CoveringSpace
Date of creation 2013-03-22 11:59:30
Last modified on 2013-03-22 11:59:30
Owner rmilson (146)
Last modified by rmilson (146)
Numerical id 21
Author rmilson (146)
Entry type Definition
Classification msc 55R05
Synonym covering map
Related topic Cover
Related topic Site
Related topic EtaleMorphism