Graphics Reference
In-Depth Information
8.12.12. Lemma.
Let (
M
n
,s) be an oriented n-manifold. Let c
1
, c
2
: [0,1]
n
Æ
M
be
singular n-cubes that are extendable to a diffeomorphism on a neighborhood of [0,1]
n
.
Assume that the extensions of c
1
and c
2
are orientation-preserving diffeomorphisms
with respect to the standard orientation on
R
n
. If w is an n-form on
M
with the prop-
erty that
(
)
«
(
)
n
n
()
Õ
[]
[]
support
c
01
,
c
01
,
,
1
2
then
Ú
Ú
ww
=
.
c
c
1
2
Proof.
We have done all the hard work in Lemma 8.12.6. We basically want to apply
that lemma in the following way:
Ú
Ú
Ú
w
=
w
=
w
.
(
)
-
1
c
c
o
c
o
c
c
2
2
2
1
1
Certainly, h = c
2
-1
c
1
satisfies the hypothesis in the lemma. The only problem is that
h is not defined on all of [0,1]
n
, but this is where the hypothesis about the support of
w comes in. The proof of Lemma 8.12.6 remains valid in that case.
o
Definition.
Let
M
n
be an oriented manifold and let w be an n-form on
M
. If c is a
singular n-cube which is extendable to a diffeomorphism on a neighborhood of [0,1]
n
and if
(
)
n
()
Õ
[]
support
c
0,
,
then define
Ú
Ú
ww
=
.
(8.40)
c
Lemma 8.12.12 implies that
Ú
w
,
if it exists, is a well-defined value and does not depend on the choice of c.
Now let
M
n
be an oriented manifold. Let w be an n-form on
M
that has compact
support. Theorems 5.8.6 and 5.8.7 and the compactness of the support of w imply that
we can find a collection of coordinate neighborhoods (
U
i
,j
i
) of
M
satisfying:
(1) [0,1]
n
Ãj
i
(
U
i
)
(2) The maps c
i
=j
i
-1
|[0,1]
n
are orientation-preserving singular n-cubes for
M
with
respect to the standard orientation of
R
n
.
(3) If
V
i
= c
i
((0,1)
n
), then the collection {
V
i
} is an open cover of
M
.
(4) Only finitely many
V
i
intersect the support of w.
(5) There exists a partition of unity F subordinate to the cover {
V
i
}.