Graphics Reference
In-Depth Information
Figure 7.27.
Lifting a homotopy.
Y
~
h(z,t)
~
h
0
p
~
h
h
0
X
Z ¥ 0
h
h(z,t)
Z ¥ [0,1]
t
Z ¥ 1
h
1
The importance of Theorem 7.4.2.6 is not only that every path in the base space
lifts to path in the total space but that the lift is essentially unique, meaning that if
two lifted paths agree at one point, then they agree everywhere. The unique lifting
property generalizes to arbitrary connected spaces not just the interval [0,1]. Another
important lifting theorem is the following:
7.4.2.7. Theorem.
(The Homotopy Lifting Theorem) Let (
Y
,p) be a covering space
for a space
X
. Let h :
Z
¥ [0,1] Æ
X
be a continuous map. Define h
t
:
Z
Æ
X
by h
t
(y) =
h(y,t). If
˜
0
is a lifting of h
0
, then h lifts to a unique continuous map
˜
:
Z
¥ [0,1] Æ
Y
so that
˜
(y,0) =
˜
0
(y).
Proof.
See [Jäni84]. Figure 7.27 tries to indicate the relationship between the various
maps.
7.4.2.8. Corollary.
(The Monodromy Lemma) Let (
Y
,p) be a covering space for a
space
X
. Let g
0
, g
1
: [0,1] Æ
X
be two continuous curves that start at the same point
x
0
and end at the same point
x
1
, that is,
x
0
=g
0
(0) =g
1
(0) and
x
1
=g
0
(1) =g
1
(1). Assume
that g
0
and g
1
are homotopic by a homotopy h that fixes the endpoints, that is, h(t,0)
=
x
0
and h(t,1) =
x
1
, for all t Œ [0,1]. If, ˜
0
, ˜
1
: [0,1] Æ
Y
are liftings of g
0
and g
1
, respec-
tively, that start at the same point in
Y
, then ˜
0
and ˜
1
will end at the same point, that
is, ˜
0
(1) = ˜
1
(1).
Proof.
This is an easy consequence of Theorem 7.4.2.7.
Corollary 7.4.2.8 is an important uniqueness type theorem. It says that if one lifts
two homotopic paths that start and end at the same point, then the lifted paths will
also end at the
same
point if they start at the same point.
The next two results describe some relationships between the fundamental groups
of the total and base space of a covering space.
7.4.2.9. Theorem.
Let (
Y
,p) be a covering space for a space
X
. Let
x
0
Œ
X
and
y
0
Œ
p
-1
(
x
0
). Then the induced homomorphism
(
)
Æ
(
)
p
*
:
p
Yy
,
p
Xx
,
1
0
1
0
