DE RHAM COHOMOLOGY
JOSHUA P. EGGER
Abstract.
De Rahm Cohomology is a powerful tool which allows one to extract purely topo
logical information about a manifold, essentially by doing algebra on its cotangent bundle.
A
particularly useful method of computing de Rham cohomology groups was discovered by Austrian
mathematicians Mayer and Vietoris, and involves partitioning a manifold into subspaces, ideally
ones whose cohomology groups are already known. In this paper, we briefly cover the prerequi
site homological algebra, introduce de Rham cohomology, then proceed to prove the invariance
of cohomology under smooth homotopy which allows for a simple proof of the Poincaré lemma.
We then develop the Mayer Vietoris sequence, perform a few computations, including a simple
homological proof of the Brouwer fixedpoint theorem, then conclude with an introduction to co
homology with compact support, followed by a discussion and proof of a version of the Poincaré
Duality Theorem, which links the dual notions of homology and cohomology.
Contents
1.
Preliminaries
1
2.
Chain Complexes and De Rahm Cohomology
2
3.
Invariance of de Rham Cohomology under Smooth Homotopy
3
4.
The MayerVietoris Sequence
4
5.
Applications
6
6.
Poincaré Duality
8
References
14
1.
Preliminaries
We will list several basic properties of, but will mostly assume familiarity with the basic properties
of manifolds, exterior algebra, and differential forms.
Recall that a
n
dimensional manifold
M
consists of a set
M
and a
differential structure
which allows us to do calculus on the manifold.
This structure consists of an atlas
A
= (
U
α
, ϕ
α
)
α
∈
I
, where
I
is some indexing set, and each
ϕ
α
:
U
α
→
ϕ
α
(
U
α
)
∈
R
n
, which is called a
chart
, maps the set
U
α
⊆
M
diffeomorphically to the
parameter domain
ϕ
α
(
U
α
)
in Euclidean
n
space. Given a manifold
M
with cotangent space
T
*
M
(the dual space to the tangent space
TM
), a differential
k
form is an element of the
k
fold exterior
power
Λ
k
T
*
M
, a graded algebra.
If we have coordinates
x
1
, ..., x
n
for the chart
(
U
α
, ϕ
α
)
and
x
∈
U
α
, the space
Λ
k
T
*
M
, which we shall henceforth refer to as
Ω
k
(
M
)
, is spanned by the elements
dx
i
1
∧ · · · ∧
dx
i
k
, where the
i
k
are increasing.
Recall that a differential
k
form
ω
∈
Ω
k
(
M
)
is said to be
closed
if
dω
= 0
, and
exact
if there
exists a
(
k

1)
form
α
∈
Ω
k

1
(
M
)
such that
dα
=
ω
. Since
d
(
dω
) =
d
2
ω
= 0
for any
ω
∈
Ω
k
(
M
)
, it
is clear that every exact form is closed. The converse, however, is not true. Not all closed forms are
exact, the standard example being the
1
form
ω
=
dθ
=

ydx
+
xdy
x
2
+
y
2
defined on
R
2
\{
0
}
, where
θ
is the
polar angle. The form
ω
=
dθ
has no global primitive on
R
2
\ {
0
}
, yet it does have a local primitive
in any convex neighborhood.
This is typical of closed forms which fail to be exact: every closed
form on a manifold can be made exact under certain circumstances. In this case, the nonconvexity
of the punctured plane
R
2
\ {
0
}
prevents
dθ
from being exact. This notion is made precise by the
1
2
J. EGGER
Poincaré Lemma, which states that all closed forms on a contractible manifold are exact.
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