Graphics Reference
In-Depth Information
is one-to-one.
Proof.
See [Mass67] or [Jäni84]. Basically, if an element [f] maps to 0, then the map
p
°
f is homotopic to a constant in
X
and this homotopy lifts to a homotopy between f
and the constant map in
Y
.
A natural question is if
y
1
Πp
-1
(
x
0
), then what is the relation between the sub-
groups p
*
(p
1
(
Y
,
y
1
)) and p
*
(p
1
(
Y
,
y
0
)) in p
1
(
X
,
x
0
)? There is an easy answer.
7.4.2.10. Theorem.
Let (
Y
,p) be a covering space for a space
X
and let
x
0
Œ
X
. If
Y
is connected, then the subgroups p
*
(p
1
(
Y
,
y
0
)) in p
1
(
X
,
x
0
) as
y
0
ranges over the points
in p
-1
(
x
0
) generate a conjugacy class of subgroups in p
1
(
X
,
x
0
).
Proof.
See [Mass67]. The result follows easily from the following observations. Let
y
0
,
y
1
Πp
-1
(
x
0
). Let ˜ : [0,1] Æ
Y
be a curve with ˜ (0) =
y
0
and ˜ (1) =
y
1
. The curve a
= p
°
˜ : [0,1] Æ
X
is a loop at
x
0
. If [ ˜] Œp
1
(
Y
,
y
1
), then define ˜ : [0,1] Æ
Y
by
1
3
Œ
È
˘
˙
˜
()
=
˜
()
ma
t
3
t
,
t
0
,
,
Í
31
2
t
-
1
3
2
3
Ê
Ë
ˆ
¯
Œ
È
Í
˘
˙
˜
=
g
,
t
,
,
2
3
Œ
È
˘
˙
˜
(
)
=-
a
33
t
,
t
,
1
.
Í
Now, set g=p
°
˜ and m=p
°
˜ . It is easy to show that [ ˜] Œp
1
(
Y
,
y
0
) and [m] = [a]
-1
[g][a]
Œp
1
(
X
,
x
0
). See Figure 7.28.
Next, we would like to classify covering spaces. Let (
Y
,p) be a covering space for
a space
X
and let
x
0
Œ
X
and
y
0
Πp
-1
(
x
0
). First, we shall answer the question about
when maps from some arbitrary space
Z
into
X
lifts to a map into
Y
. Let
z
0
Œ
Z
. The
specific question is, given a map f : (
Z
,
z
0
) Æ (
X
,
x
0
), when does a lifting
˜
: (
Z
,
z
0
) Æ
(
Y
,
y
0
) exist? In terms of diagrams, we are given f and p and are looking for an
˜
that
will produce a commutative diagram
~
g
y
1
~
a
Y
y
0
P
a
g
X
Figure 7.28.
How loops in the total space project to
conjugate loops.
x
0
