25.1
The
Let V be a finite dimensional vector space over the reals. The tensor algebra of V is direct sum
Ten
(
V
)
=
R
⊕ V ⊕ V
⊗ 2 . . .
⊕ V
⊗ k . . .
V
⊗ l
It is made into an algebra by declaring that the product of a ∈ V
⊗ k is a ⊗ b ∈ V
⊗
( k + l
) .
and b ∈
It is characterized by the universal mapping property that any linear map V → A where A is an algebra over
R
extends to a unique map of algebras Ten
(
V
)
→ A.
The exterior algebra algebra is the quotient of exterior algebra by the relation v
⊗ v
= 0
.
The exterior algebra is denoted
� ∗ (
V
)
or
�(
V
)
. It is customary to denote the multiplication in the exterior algebra by
( a
, )
→ a ∧ b If v
1
. . . v
k is a basis for V then this relation is equivalent to the relations v
i v
i
∧ v
j
∧ v
i
= − v
j
∧ v
i
= 0 for i �= j
,
Thus
� ∗ (
V
)
has basis the products v i
1
∧ v i
2
. . . v i
k where the indices run over all strictly increasing sequences of numbers between 1 and n.
Since for each k there are
1 ≤ i
�
i n
�
<
i
2
≤ n
.
such sequences of length k we have
k
< . . .
<
i
k dim
(� ∗ (
V
))
= 2
n .
� ∗ (
V
) since the relation is homogenous the grading of the tensor algebra descends to a grading on the exterior algebra (hence the *).
We can apply this construction fiberwise to a vector bundle. The most impor­ tant example is the cotangent bundle of a manifold T
∗
X in which case we get the bundle of differential forms
� ∗ (
T
∗
X
)
or
� ∗ (
X
).
63

We will denote the space of smooth sections of
� ∗ dinates a typical element of
� ∗ (
X
)
looks like
(
X
)
by
� ∗ (
X
)
. In local coor­
ω
=
�
1 ≤ i i
< i
2
<...< i k
≤ n
ω i i i
2
...< i k d x i
1 ∧ d x i
2 ∧
.
.
.
d x i
k
.
Since the construction of
� ∗ (
X
)
was functorial in the cotangent bundle these bundles naturally pull back under diffeomorphism and if f : X → Y is any smooth map there is natural map
f
∗
:
� ∗ (
Y
)
→
� ∗ (
X
).
The most important thing about differential forms is the existence of a natural differential operator the exterior differential defined locally by the following rules
d f d
ω
=
= n
� i = 1
∂
f
∂
x i d x
�
i
1 ≤ i i
< i
2
<...< i k
≤ n d
ω i i i
2
...< i
k
∧ d x i
1 ∧ d x i
2 ∧
.
.
.
d x i
k
.
Notice that we can’t invariantly define a similar operator on the tensor algebra.
If we have a one form
θ
=
� f i d x
i i = 1 and try to define
D
θ
=
�
∂
f
i
∂
x j d x i = 1
j
⊗ d x
i then when if we have new coordinates y
1 .
.
.
y
n we have d x
i
= n
� j = 1
∂
x i
∂
y j d y
j and
θ
= n
� g m d y
m m = 1 where
∂
x i g
m
= f
i ∂
y m
64

D
θ
=
=
=
=
=
=
=
�
∂
f
i
∂
x j
d x
j i = 1
⊗ d x
i
∂
f
i
∂
x
j
∂
f
i
∂
y k
∂
f
i
∂
∂
∂
∂
x
y
y
x
i
l
k
j
∂
x
j
∂
y
m d y
∂
∂
x
y
i
l
m
⊗ d y
∂
x
j
∂
y m d y
m
l
⊗ d y
l
∂
y m
∂
f
i
∂
y m
∂
x
i
∂
y l
∂
x
i d y
∂
y l d y
m
m
⊗ d y
⊗ d y
l
l
�
∂
∂
y m
(
f
i
∂
x
i
∂
y l
)
− f
∂ 2 x
i
i ∂
y m ∂
y l
� d y
m
⊗ d y
l n
� m = 1
∂ g
l
∂
y
m d y
m
⊗ d y
l
− f
∂ 2 x
i
i ∂
y m ∂
y
l d y
m
⊗ d y
l .
Thus our definition depends on the choice of coordinates. Notice that when we pass to the exterior algebra this last expression vanishes that exterior derivative is well defined.
Theorem 25.1. d
2
= 0.
Proof. From the definition in local coordinates it suffices to check that d
2
= 0 on functions. d n
� 2 (
f
)
= i
,
j = 1
∂
x
∂ 2
i ∂ f
x j d x
i
∧ d x
j
= 0 since the f smooth so the matrix of second derivatives is symmetric.
Proposition 25.2. d
(
a ∧ b
)
= da ∧ b +
(
− 1
) deg
( a
)
∧ db
.
65

Proof. The bilinearity of the wedge product implies that it suffices to check the result when
a = f d x i
1 ∧ d x i
2 ∧
. . .
∧ d x i
k
.
Definition 25.3. A cochain complex is a graded vector space C = gether with a map d : C → C so that dC
i
⊂ C i + 1 and d
2
=
�
∞ i = 0
C
i to­
0. The cohomology groups of a cochain complex are defined to be
H i (
C
, d
)
= ker
(
d : C
i
→ C i + 1 )/
Ran
(
d : C i − 1
→ C i )