In differential geometry, the integration along fibers of a k-form yields a
(
k
−
m
)
-form where m is the dimension of the fiber, via "integration".
Let
π
:
E
→
B
be a fiber bundle over a manifold with compact oriented fibers. If
α
is a k-form on E, then for tangent vectors wi's at b, let
(
π
∗
α
)
b
(
w
1
,
…
,
w
k
−
m
)
=
∫
π
−
1
(
b
)
β
where
β
is the induced top-form on the fiber
π
−
1
(
b
)
; i.e., an
m
-form given by: with
w
i
~
the lifts of
w
i
to E,
β
(
v
1
,
…
,
v
m
)
=
α
(
v
1
,
…
,
v
m
,
w
1
~
,
…
,
w
k
−
m
~
)
.
(To see
b
↦
(
π
∗
α
)
b
is smooth, work it out in coordinates; cf. an example below.)
Then
π
∗
is a linear map
Ω
k
(
E
)
→
Ω
k
−
m
(
B
)
. By Stokes' formula, if the fibers have no boundaries, the map descends to de Rham cohomology:
π
∗
:
H
k
(
E
;
R
)
→
H
k
−
m
(
B
;
R
)
.
This is also called the fiber integration.
Now, suppose
π
is a sphere bundle; i.e., the typical fiber is a sphere. Then there is an exact sequence
0
→
K
→
Ω
∗
(
E
)
→
π
∗
Ω
∗
(
B
)
→
0
, K the kernel, which leads to a long exact sequence, dropping the coefficient
R
and using
H
k
(
B
)
≃
H
k
+
m
(
K
)
:
⋯
→
H
k
(
B
)
→
δ
H
k
+
m
+
1
(
B
)
→
π
∗
H
k
+
m
+
1
(
E
)
→
π
∗
H
k
+
1
(
B
)
→
⋯
,
called the Gysin sequence.
Let
π
:
M
×
[
0
,
1
]
→
M
be an obvious projection. First assume
M
=
R
n
with coordinates
x
j
and consider a k-form:
α
=
f
d
x
i
1
∧
⋯
∧
d
x
i
k
+
g
d
t
∧
d
x
j
1
∧
⋯
∧
d
x
j
k
−
1
.
Then, at each point in M,
π
∗
(
α
)
=
π
∗
(
g
d
t
∧
d
x
j
1
∧
⋯
∧
d
x
j
k
−
1
)
=
(
∫
0
1
g
(
⋅
,
t
)
d
t
)
d
x
j
1
∧
⋯
∧
d
x
j
k
−
1
.
From this local calculation, the next formula follows easily: if
α
is any k-form on
M
×
I
,
π
∗
(
d
α
)
=
α
1
−
α
0
−
d
π
∗
(
α
)
where
α
i
is the restriction of
α
to
M
×
{
i
}
.
As an application of this formula, let
f
:
M
×
[
0
,
1
]
→
N
be a smooth map (thought of as a homotopy). Then the composition
h
=
π
∗
∘
f
∗
is a homotopy operator:
d
∘
h
+
h
∘
d
=
f
1
∗
−
f
0
∗
:
Ω
k
(
N
)
→
Ω
k
(
M
)
,
which implies
f
1
,
f
0
induces the same map on cohomology, the fact known as the homotopy invariance of de Rham cohomology. As a corollary, for example, let U be an open ball in Rn with center at the origin and let
f
t
:
U
→
U
,
x
↦
t
x
. Then
H
k
(
U
;
R
)
=
H
k
(
p
t
;
R
)
, the fact known as the Poincaré lemma.
Given a vector bundle π : E → B over a manifold, we say a differential form α on E has vertical-compact support if the restriction
α
|
π
−
1
(
b
)
has compact support for each b in B. We written
Ω
v
c
∗
(
E
)
for the vector space of differential forms on E with vertical-compact support. If E is oriented as a vector bundle, exactly as before, we can define the integration along the fiber:
π
∗
:
Ω
v
c
∗
(
E
)
→
Ω
∗
(
B
)
.
The following is known as the projection formula. We make
Ω
v
c
∗
(
E
)
a right
Ω
∗
(
B
)
-module by setting
α
⋅
β
=
α
∧
π
∗
β
.
Proof: 1. Since the assertion is local, we can assume π is trivial: i.e.,
π
:
E
=
B
×
R
n
→
B
is a projection. Let
t
j
be the coordinates on the fiber. If
α
=
g
d
t
1
∧
⋯
∧
d
t
n
∧
π
∗
η
, then, since
π
∗
is a ring homomorphism,
π
∗
(
α
∧
π
∗
β
)
=
(
∫
R
n
g
(
⋅
,
t
1
,
…
,
t
n
)
d
t
1
…
d
t
n
)
η
∧
β
=
π
∗
(
α
)
∧
β
.
Similarly, both sides are zero if α does not contain dt. The proof of 2. is similar.
◻
