PlanetMath: classification of covering spaces

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classification of covering spaces

(Definition)

Let be a connected, locally path connected and semilocally simply connected space. Assume furthermore that has a basepoint .

A covering $p\colon\thinspace E\to X$ is called based if is endowed with a basepoint and . Two based coverings $p_i\colon\thinspace E_i\to X$ , are called equivalent if there is a basepoint preserving equivalence $T\colon\thinspace E_1\to E_2$ that covers the identity, i.e. is a homeomorphism and the following diagram commutes

$\displaystyle \begin{xy} *!C\xybox{ \xymatrix{ {(E_1,e_1)}\ar[dr]_{p}\ar[rr]^{T}&&{(E_2,e_2)}\ar[dl]^{p}\ &{(X,*)}.& } } \end{xy}$

Theorem 1 (Classification of connected coverings)

Equivalence classes of based coverings $p\colon\thinspace (E,e)\to (X,*)$ with connected total space are in bijective correspondence with subgroups of the fundamental group $\pi_1(X,*)$ . The bijection assigns to the based covering the subgroup $p_*\left(\pi_1(E,e)\right)$ .
Equivalence classes of coverings (not based) are in bijective correspondence with conjugacy class of subgroups of $\pi_1(X,*)$ .

Under the bijection of the above theorem normal coverings correspond to normal subgroups of $\pi_1(X,e)$ , and in particular the universal covering $\tilde \pi\colon\thinspace \tilde X\to X$ corresponds to the trivial subgroup while the trivial covering $\operatorname{id}\colon\thinspace X\to X$ corresponds to the whole group.

Normal coverings are sometimes called Galois coverings, and the group of deck transformations is someitmes called the Galois group of the covering space. The reason for this is that this theorem provides a direct analogy for the fundamental theorem of Galois theory. This theorem provides a correspondence between subgroups of the Galois group of a cover and covers that are its quotients, just as the fundamental theorem of Galois theory provides a correspondence between subextensions of a field extension and subgroups of its Galois group. Both fundamental theorems can be viewed as special cases of a more general theorem in the category of schemes; the correct tool is the study of étale morphisms and the étale fundamental group.

Proof. [Rough sketch of proof] We describe the based version. Clearly the set of equivalences of two based coverings form a torsor of the group of deck transformations $\operatorname{Aut}(p)$ . From our discussion of that group it follows then that equivalent (based) coverings give the same subgroup. Thus the map is well defined. To see that it is a bijection construct its inverse as follows: There is a universal covering $\tilde \pi\colon\thinspace \tilde X\to X$ and a subgroup $\pi$ of $\pi_1(X,*)$ acts on $\tilde X$ by the restriction of the monodromy action. The covering which corresponds to $\pi$ is then $\tilde X/\pi$ . $\qedsymbol$

"classification of covering spaces" is owned by Dr_Absentius. [ full author list (2) ]

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Cross-references: monodromy action, restriction, acts on, inverse, well defined, map, étale fundamental group, étale morphisms, schemes, category, field extension, quotients, Galois group of a cover, fundamental theorem of Galois theory, covering space, Galois group, deck transformations, group, trivial subgroup, universal, normal subgroups, normal coverings, conjugacy class, fundamental group, subgroups, bijective, equivalence classes, homeomorphism, identity, covers, equivalence, equivalent, covering, basepoint, semilocally simply connected, locally path connected, connected
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This is version 5 of classification of covering spaces, born on 2003-02-13, modified 2004-03-24.
Object id is 4033, canonical name is ClassificationOfCoveringSpaces.
Accessed 3495 times total.

Classification:

AMS MSC:	55R05 (Algebraic topology :: Fiber spaces and bundles :: Fiber spaces)
	55R15 (Algebraic topology :: Fiber spaces and bundles :: Classification)

Pending Errata and Addenda

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