Graphics Reference
In-Depth Information
The vector bundle f*x is called the
induced (vector) bundle
over
B
1
, the
(vector) bundle
over
B
1
induced by f
, or the
pullback (vector) bundle
. Define
˜
:
f
EE
1
Æ
by
˜
(
)
=
f
be e
,
.
1
The bundle map (
˜
,f) : f*xÆxis called the
canonical (vector bundle) map
from f*j
to x.
f*x is a well-defined n-plane bundle and (
˜
,f) is a vector bundle map.
8.9.3. Lemma.
Proof.
Easy.
We collect the main facts about induced bundles in the next theorem.
8.9.4. Theorem.
(1) If (
˜
,f) : hÆxis a vector bundle map between two n-plane bundles h and x,
then the induced vector bundle f*x is isomorphic to h.
(2) If j=(
E
,p,
B
) is a trivial vector bundle over
B
and if f :
B
1
Æ
B
is a map, then
f*j is a trivial vector bundle over
B
1
.
(3) Let x=(
E
,p,
B
) be a vector bundle. If
B
1
is a paracompact space and if f, g :
B
1
Æ
B
are homotopic maps, then the induced bundles f*x and g*x are iso-
morphic.
Proof.
To prove (1) show that the vector bundle map
(
˜
,
)
g
1
B
h
:
h
Æ
f
*
x
(
)
defined by
,
˜
(
)
()
=
() ()
˜
g
e
p
h
e
f
e
is the desired isomorphism. Fact (2) is easy. For fact (3) see [Huse66].
8.9.5. Corollary.
Every vector bundle over a contractible paracompact space
B
is
trivial.
Proof.
Let f be the identity map on
B
and let g :
B
Æ
B
be a constant map. If x is
any vector bundle over
B
, then f*x is easily seen to be isomorphic to x and g*x is trivial
by Theorem 8.9.4(2). Since f is homotopic to g, the Corollary now follows from
Theorem 8.9.4(3).
Next, we show how vector bundles over a space can be added.
