Graphics Reference
In-Depth Information
It is the bundles with discrete fibers that interest us in this section. In Section 8.10
we shall look at bundles whose fibers are vector spaces.
Definition.
A
covering space
for a space
X
is a locally trivial bundle with base space
X
with the property that every fiber is a discrete space. The covering is called an
n-fold covering
if every fiber consists of n points. The bundle automorphisms of a
covering space are called
covering transformations
.
Example 7.4.2.1 already described a 2-fold covering space. Here are some more
examples.
7.4.2.2. Example.
The map
1
p:
RS
Æ
()
=
(
)
pt
cos ,sin
t
t
defines a covering space (
R
,p) of
S
1
whose fibers
-
1
()
=+
{
}
pt t
2p
nn
Œ
Z
are a countable discrete set of points.
Consider the circle
S
1
7.4.2.3. Example.
as a subset of the complex plane
C
. The
map
1
1
p:
SS
Æ
()
=
n
p
zz
defines a bundle (
S
1
,p) over
S
1
that is an n-fold covering space for
S
1
.
7.4.2.4. Example.
The map
2
1 1 2 2
ƥà ¥
p:
RSSRR
()
=
(
(
) (
)
)
pst
,
cos ,sin
t
t
, cos ,sin
t
t
.
defines a is a covering space (
R
2
,p) of the torus
S
1
¥
S
1
.
That all the total spaces in our examples were manifolds should not be
surprising.
