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and think of the torus
S
1
¥
S
1
as the quotient space of
R
2
/~, where ~ is the equiva-
lence relation
(
)
(
)
xy
,
~
h
xy
,
.
mn
,
If p :
R
2
Æ
R
2
/~ is the quotient map, then (
R
2
,p) is the universal covering space of the
torus. With this interpretation, the maps h
m,n
are obviously the covering transforma-
tions. Note that h
m,n
= h
1,0
h
0,1
.
o
Using Corollary 7.4.2.18 and what we showed in Examples 7.4.2.19, 7.4.2.20, and
7.4.2.22 we now have alternate proofs of the facts stated in Theorem 7.4.1.10(1) and
(2) and Corollary 7.4.1.15, namely,
7.4.2.23. Corollary.
(1) p
1
(
S
1
) ª
Z
.
(2) p
1
(
P
n
) ª
Z
2
.
(3) p
1
(
S
1
¥
S
1
) ª
Z
≈
Z
.
All this talk about covering transformations and the last three examples leads to
another question. Suppose that we turn things around and start with a group of
homeomorphisms G of a space
Y
and define
YG Y
=
~,
(7.8)
where
()
yy
~
if
y
=
h
y
for
some h
Œ
G.
1
2
2
1
Definition.
The space
Y
/G in equation (7.8) is called the
quotient space
of
Y
modulo
the group
G.
If p :
Y
Æ
Y
/G is the quotient map, then is (
Y
,p,
Y
/G) a covering space with G the
group of covering transformations? The answer in general is no. At the very least the
homeomorphisms in G could not be allowed to have fixed points, but we need some-
thing stronger.
Definition.
A group of homeomorphism G of a space
X
is said to be
properly
discontinuous
if every point
x
in
X
has a neighborhood
U
so that all the sets h(
U
),
h ΠG, are disjoint.
Clearly, no homeomorphism in a properly discontinuous group of homeomor-
phism can have a fixed point. Furthermore, it is easy to see that the covering
transformations of a covering space form a properly discontinous group of homeo-
morphisms of the total space.
7.4.2.24. Theorem.
Let
Y
be a connected, locally path-connected topological space
and G a properly discontinuous group of homeomorphism of
Y
. If p :
Y
Æ
Y
/G is the
