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![[parent]](http://images.planetmath.org:8080/images/uparrow.png)
deck transformation
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(Definition)
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Let
 be a covering map. A deck transformation or covering transformation is a map
 such that
 , that is, such that the following diagram commutes.
It is straightforward to check that the set of deck transformations is closed under compositions and the operation of taking inverses. Therefore the set of deck transformations is a subgroup of the group of homeomorphisms of  . This group will be denoted by
 and referred to as the group of deck transformations or as the automorphism group of  . It is worth noting that an alternative name for the group of deck transformations is the Galois group of the covering. This terminology arises from an analogy with the fundamental theorem of Galois theory
which gives the inclusion-reversing identification addressed in the classification of covering spaces.
In the more general context of fiber bundles deck transformations correspond to isomorphisms over the identity since the above diagram could be expanded to:
An isomorphism not necessarily over the identity is called an equivalence. In other words an equivalence between two covering maps
 and
 is a pair of maps
 that make the following diagram commute
i.e. such that
 .
Deck transformations should be perceived as the symmetries of  (hence the notation
 ), and therefore they should be expected to preserve any concept that is defined in terms of  . Most of what follows is an instance of this meta-principle.
For this section we assume that the total space  is connected and locally path connected. Notice that a deck transformation is a lifting of
 and therefore (according to the lifting theorem) it is uniquely determined by the image of a point. In other words:
Proposition 1 Let
 . If there is  such that
 then  . In particular if  for some  then
 id.
Another simple (or should I say double?) application of the lifting theorem gives
Proposition 2 Given
 with
 , there is a
 such that  if and only if
 , where  denotes  .
Proposition 3 Deck transformations commute with the monodromy action. That is if  ,
 ,
 and
 then
where  denotes the monodromy action.
Proof. If
 is a lifting of  starting at  , then
 is a lifting of  staring at  . 
We simplify notation by using  to denote the fundamental group
 for  .
Theorem 4 For all 
where,
 denotes the normalizer of  inside
 .
Proof. Denote
 by  . Note that if
 then
 . Indeed, recall that
 is the stabilizer of  under the momodromy action and therefore we have
where, the last equality follows from the definition of normalizer. One can then define a map
as follows: For
 let
 be the deck transformation that maps  to
 . Notice that Proposition 2 ensures the existence of such a deck transformation while Proposition 1 guarantees its uniqueness. Now
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"deck transformation" is owned by mathcam. [ full author list (3) | owner history (2) ]
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(view preamble)
Cross-references: universal cover, regular covering, first isomorphism theorem, obvious, loop, path, path connected, onto, homomorphism, proposition, equality, action, stabilizer, normalizer, fundamental group, monodromy action, simple, point, image, lifting theorem, lifting, locally path connected, connected, section, terms, preserve, symmetries, expanded, identity, isomorphisms, fiber bundles, classification of covering spaces, fundamental theorem of Galois theory, covering, Galois group, automorphism group, homeomorphisms, group, subgroup, inverses, operation, compositions, closed under, map, covering map
There are 76 references to this entry.
This is version 14 of deck transformation, born on 2003-02-10, modified 2004-06-04.
Object id is 4010, canonical name is DeckTransformation.
Accessed 12360 times total.
Classification:
| AMS MSC: | 55R05 (Algebraic topology :: Fiber spaces and bundles :: Fiber spaces) |
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Pending Errata and Addenda
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![$\displaystyle \begin{xy} *!C\xybox{ \xymatrix{ {E}\ar[dr]_{p}\ar[rr]^{D}&&{E}\ar[dl]^{p}\ &{X}& } } \end{xy}$](http://images.planetmath.org:8080/cache/objects/4010/l2h/img4.png "$\displaystyle \begin{xy} *!C\xybox{ \xymatrix{ {E}\ar[dr]_{p}\ar[rr]^{D}&&{E}\ar[dl]^{p}\ &{X}& } } \end{xy}$")
![$\displaystyle \begin{xy} *!C\xybox{ \xymatrix{ {E}\ar[d]_{p}\ar[r]^{D}&{E}\ar[d]^{p}\ {X}\ar[r]_{\text{id}}& {X} } } \end{xy}$](http://images.planetmath.org:8080/cache/objects/4010/l2h/img8.png "$\displaystyle \begin{xy} *!C\xybox{ \xymatrix{ {E}\ar[d]_{p}\ar[r]^{D}&{E}\ar[d]^{p}\ {X}\ar[r]_{\text{id}}& {X} } } \end{xy}$")
![$\displaystyle \begin{xy} *!C\xybox{ \xymatrix{ {E'}\ar[d]_{p'}\ar[r]^{\tilde f}&{E}\ar[d]^{p}\ {X'}\ar[r]_{f}& {X} } } \end{xy}$](http://images.planetmath.org:8080/cache/objects/4010/l2h/img12.png "$\displaystyle \begin{xy} *!C\xybox{ \xymatrix{ {E'}\ar[d]_{p'}\ar[r]^{\tilde f}&{E}\ar[d]^{p}\ {X'}\ar[r]_{f}& {X} } } \end{xy}$")
