Graphics Reference
In-Depth Information
Yy
,
0
˜
f
Ø
p
ææ
f
Zz
,
0
Xx
,
.
0
A necessary condition is clearly that f
*
(p
1
(
Z
,
z
0
)) Õ p
*
(p
1
(
Y
,
y
0
)). The amazing fact is
that this condition is also sufficient provided that some weak connectivity conditions
hold.
Definition.
A topological space
X
is said to be
locally path-connected
if every neigh-
borhood of a point contains a neighborhood that is path-connected.
Fortunately, the spaces of interest to us are locally path-connected. Manifolds are
trivially locally path-connected, but so are CW complexes (use induction on the
number of cells).
7.4.2.11. Theorem.
Let (
Y
,p) be a covering space for a space
X
. Let
Z
be a path-
connected and locally path-connected space. Let
x
0
Œ
X
,
y
0
Œ p
-1
(
x
0
), and
z
0
Œ
Z
. Then
a map f : (
Z
,
z
0
) Æ (
X
,
x
0
) lifts to a map
˜
: (
Z
,
z
0
) Æ (
Y
,
y
0
) if and only if f
*
(p
1
(
Z
,
z
0
)) Õ
p
*
(p
1
(
Y
,
y
0
)).
Proof.
See [Mass67] or [Jäni84]. The diagram below should help clarify what is
being said:
(
)
Yy
,
p
Yy
,
0
1
0
˜
f
Ø
p
Ø
p
*
f
(
(
)
)
Zz
,
ææ
Xx
,
p
p
Yy
,
0
0
*
1
0
f
(
)
ææ
(
(
)
)
Õ
(
)
*
p
Zz
,
f
p
Zz
,
p
Xx
,
.
1
0
*
1
0
1
0
We can deduce a number of important results from Theorem 7.4.2.11.
7.4.2.12. Theorem.
Let (
Y
1
,p
1
) and (
Y
2
,p
2
) be covering spaces for a space
X
, where
Y
1
and
Y
2
are path-connected and locally path-connected spaces. Let
x
0
Œ
X
and
y
i
Œ
p
i
-1
(
x
0
). The two covering spaces are isomorphic via a bundle isomorphism f : (
Y
1
,
y
1
)
Æ (
Y
2
,
y
2
) if and only if p
1*
(p
1
(
Y
1
,
y
1
)) = p
2*
(p
1
(
Y
2
,
y
2
)).
Proof.
See [Mass67]. The following diagram might again help:
(
)
(
)
p
Yy
,
p
Yy
,
f
Y
ææ
Y
111
122
1
2
p
p
p
p
1
*
2
*
1
2
X
Xx
(
(
)
)
(
)
(
(
)
)
p
p
Yy
,
Õ
p
,
p
p
Yy
,
.
1111
*
1
0
2122
*
To get the next theorem we need another technical definition.