Graphics Reference
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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 }.
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