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deck transformation
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(Definition)
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Let $p\co E\to X$ be a covering map. A deck transformation or covering transformation is a map $D\co E\to E$ such that $p\circ D=p$ , 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 $E$ . This group will be denoted by $\Au(p)$ and referred to as the group of deck transformations or as the automorphism group of $p$ . 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 $p\co E\to X$ and $p'\co E'\to X'$ is a pair of maps $(\tilde f,f)$ that make the following diagram commute
i.e. such that $p\circ\tilde f=f\circ p'$ .
Deck transformations should be perceived as the symmetries of $p$ (hence the notation $\Au(p)$ ), and therefore they should be expected to preserve any concept that is defined in terms of $P$ . Most of what follows is an instance of this meta-principle.
For this section we assume that the total space $E$ is connected and locally path connected. Notice that a deck transformation is a lifting of $p\co E\to X$ and therefore (according to the lifting theorem) it is uniquely determined by the image of a point. In
other words:
Proposition 1 Let $D_1,D_2\in \Au(p)$ . If there is $e\in E$ such that $D_1(e)=D_2(e)$ then $D_1=D_2$ . In particular if $D_1(e)=e$ for some $e\in E$ then $D_1={id}$ .
Another simple (or should I say double?) application of the lifting theorem gives
Proposition 2 Given $e,e' \in E$ with $p(e)=p(e')$ , there is a $D\in \Au(p)$ such that $D(e)=e'$ if and only if $p_*\left(\pi_1(E,e)\right)=p_*\left(\pi_1(E,e')\right)$ , where $p_*$ denotes $\pi_1(p)$ .
Proposition 3 Deck transformations commute with the monodromy action. That is if $*\in X$ , $e\in p^{-1}(*)$ , $\Gg\in \pi_1(X,*)$ and $D\in \Au(p)$ then $$D(x\cdot\Gg)=D(x)\cdot\Gg$$ where $\cdot$ denotes the monodromy action.
Proof. If $\tilde \Gg$ is a lifting of $\Gg$ starting at $e$ , then $D\circ\tilde\Gg$ is a lifting of $\Gg$ staring at $D(e)$ . 
We simplify notation by using $\pi_e$ to denote the fundamental group $\pi_1(E,e)$ for $e\in E$ .
Theorem 4 For all $e\in E$ $$\Au(p)\cong N\left(p_*(\pi_e)\right)/p_*(\pi_e)$$ where, $ N(p_*\pi_e)$ denotes the normalizer of $p_*\pi_e$ inside $\pi_1\left(X,p(e)\right)$ .
Proof. Denote $N(p_*\pi_e)$ by $N$ . Note that if $\Gg\in N$ then $p_*(\pi_{e\cdot\Gg})=p_*(\pi_e)$ . Indeed, recall that $p_*(\pi_e)$ is the stabilizer of $e$ under the momodromy action and therefore we have $$ p*(\pi_{e\cdot\Gg})= \operatorname{Stab}(e\cdot\Gg)=\Gg\operatorname{Stab}(e)\Gg^{-1}=\Gg p_* (\pi_e)\Gg^{-1}= p_* (\pi_e)$$ where, the last equality follows from the definition of normalizer. One can then define a map $$\Gf\co N\to \Au(p)$$ as follows: For $\Gg\in N$ let $\Gf(\Gg)$ be the deck transformation that maps $e$ to $e\cdot\Gg$ . Notice that Proposition 2 ensures the existence of such a deck transformation while Proposition 1 guarantees its uniqueness. Now
Corollary 5 If $p$ is regular covering then $$\Au(p)\cong \pi_1(X,*)/p_*\left(\pi_1(E,e)\right).$$
Corollary 6 If $p$ is the universal cover then $$\Au(p)\cong \pi_1(X,*)\,.$$
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Cross-references: universal cover, regular covering, first isomorphism theorem, theorem, obvious, loop, path, path connected, onto, homomorphism, proposition, equality, action, stabilizer, normalizer, fundamental group, monodromy action, application, 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, analogy, covering, Galois group, automorphism group, homeomorphisms, group, subgroup, inverses, operation, compositions, closed under, diagram, map, covering map
There are 96 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 16816 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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![$\displaystyle \xymatrix{ {E}\ar[dr]_{p}\ar[rr]^{D}&&{E}\ar[dl]^{p}\ &{X}& }$](http://images.planetmath.org:8080/cache/objects/4010/js/img1.png "$\displaystyle \xymatrix{ {E}\ar[dr]_{p}\ar[rr]^{D}&&{E}\ar[dl]^{p}\ &{X}& }$")
![$\displaystyle \xymatrix{ {E}\ar[d]_{p}\ar[r]^{D}&{E}\ar[d]^{p}\ {X}\ar[r]_{\text{id}}& {X} }$](http://images.planetmath.org:8080/cache/objects/4010/js/img2.png "$\displaystyle \xymatrix{ {E}\ar[d]_{p}\ar[r]^{D}&{E}\ar[d]^{p}\ {X}\ar[r]_{\text{id}}& {X} }$")
![$\displaystyle \xymatrix{ {E'}\ar[d]_{p'}\ar[r]^{\tilde f}&{E}\ar[d]^{p}\ {X'}\ar[r]_{f}& {X} }$](http://images.planetmath.org:8080/cache/objects/4010/js/img3.png "$\displaystyle \xymatrix{ {E'}\ar[d]_{p'}\ar[r]^{\tilde f}&{E}\ar[d]^{p}\ {X'}\ar[r]_{f}& {X} }$")