Algebraic Topology - Computer Graphics and Geometric Modeling - page 431

Definition. Cov( Y ,p) will denote the group of covering transformations of a cover-
ing space ( Y ,p).

7.4.2.17. Theorem. Let ( Y ,p) be a covering space for a path-connected and locally
path-connected space X . Let x 0 ΠX and y 0 Πp -1 ( x 0 ). Let G = p * (p 1 ( Y , y 0 )) and let N G
be the normalizer of G in p 1 ( X , x 0 ). Given an element [g] ΠN G , there is exactly one
covering transformation h [g] that maps y 0 into the end point ˜(1) of the lifting ˜ of g
that starts at y 0 . The map

( )

N

G Æ

Cov Y , p

[] Æ

g h

[] g

is a homomorphism with kernel G, that is,

N
G

G

( ) ª Cov Y , p .

Proof. See [Mass67] or [Jäni84].

7.4.2.18. Corollary. Let ( Y ,p) be the universal covering space for a path-connected
and locally path-connected space X and let x 0 Œ X . Then Cov( Y ,p) ªp 1 ( X , x 0 ). If
p 1 ( X , x 0 ) is finite and n =Ωp 1 ( X , x 0 )Ω, then ( Y ,p) is an n-fold covering.

7.4.2.19. Example. The covering transformations of the universal covering space
( R ,p) defined in Example 7.4.2.2 are the maps

h n : RR Æ

defined by

n () =+2p .

ht t n

The maps h n are obviously covering transformations. Note that h n = h n .

7.4.2.20. Example. The only covering transformation of the covering space S n over
P n is the antipodal map of S n .

7.4.2.21. Corollary. Let ( Y ,p) be the universal covering space for a path-connected
and locally path-connected space X and let x 0 Œ X . Then X ª Y /~, where ~ is the equiv-
alence relation defined by y ~ y ¢ if there is an h Œ Cov( Y ,p), such that y ¢=h( y ).

7.4.2.22. Example. Let m, n ΠZ . Define maps

2 2

h mn : RR Æ

,

by

( ) =+ ( ) hxyx my n , 2 p , + 2 p

mn ,