covering space
Let and be topological spaces and suppose there is a surjective continuous map
which satisfies the following condition: for each , there is an open neighborhood of such that
-
•
is a disjoint union of open sets , and
-
•
each is mapped homeomorphically onto via .
Then is called a covering space, is called a covering map, the ’s are sheets of the covering of and for each , is the fiber of above . The open set is said to be evenly covered. If is simply connected, it is called the universal covering space.
From this we can derive that is a local homeomorphism, so that any local property
has is inherited by (local connectedness, local path connectedness etc.). Covering spaces
are foundational in the study of the fundamental group of a topological space; in particular, there is a correspondence (http://planetmath.org/ClassificationOfCoveringSpaces) between connected coverings of and subgroups of the group of deck transformations
of its universal covering space which is exactly analogous to the fundamental theorem of Galois theory
.
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 in the complex plane near zero. Covering maps are also generalized in algebraic geometry
; 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 |