Difference between revisions of "Covering space"
Karl Jones (Talk | contribs) (Created page with "In mathematics, more specifically algebraic topology, a '''covering map''' (also '''covering projection''') is a continuous Function (mathema...") |
Karl Jones (Talk | contribs) (→Description) |
||
| Line 9: | Line 9: | ||
The definition implies that every covering map is a [[local homeomorphism]]. | The definition implies that every covering map is a [[local homeomorphism]]. | ||
| − | Covering spaces play an important role in [[homotopy theory]], [[harmonic analysis]], [[Riemannian geometry]] and [[differential topology]]. In Riemannian geometry for example, [[Ramification (mathematics)#In algebraic topology|ramification]] is a generalization of the notion of covering maps. Covering spaces are also deeply intertwined with the study of homotopy groups and, in particular, the [[fundamental group]]. An important application comes from the result that, if ''X'' is a "sufficiently good" [[topological space]], there is a [[bijection]] between the collection of all isomorphism classes of [[connected space|connected coverings]] of ''X'' and the subgroups of the [[fundamental group]] of ''X''. | + | Covering spaces play an important role in [[homotopy theory]], [[harmonic analysis]], [[Riemannian geometry]] and [[differential topology]]. In Riemannian geometry for example, [[Ramification (mathematics)#In algebraic topology|ramification]] is a generalization of the notion of covering maps. |
| + | |||
| + | Covering spaces are also deeply intertwined with the study of homotopy groups and, in particular, the [[fundamental group]]. | ||
| + | |||
| + | An important application comes from the result that, if ''X'' is a "sufficiently good" [[topological space]], there is a [[bijection]] between the collection of all isomorphism classes of [[connected space|connected coverings]] of ''X'' and the subgroups of the [[fundamental group]] of ''X''. | ||
== See also == | == See also == | ||
Revision as of 03:32, 14 September 2015
In mathematics, more specifically algebraic topology, a covering map (also covering projection) is a continuous function p from a topological space, C, to a topological space, X, such that each point in X has an open neighbourhood evenly covered by p (as shown in the image)
In this case, C is called a covering space and X the base space of the covering projection.
(TO DO: expand, organize, cross-reference, illustrate.)
Description
The definition implies that every covering map is a local homeomorphism.
Covering spaces play an important role in homotopy theory, harmonic analysis, Riemannian geometry and differential topology. In Riemannian geometry for example, ramification is a generalization of the notion of covering maps.
Covering spaces are also deeply intertwined with the study of homotopy groups and, in particular, the fundamental group.
An important application comes from the result that, if X is a "sufficiently good" topological space, there is a bijection between the collection of all isomorphism classes of connected coverings of X and the subgroups of the fundamental group of X.
See also
External links
- Covering space @ Wikipedia
