Digital Signal Processing Reference
In-Depth Information
ker
(
d
)
Definition 1
The quotient vector space
H
(
V
,
d
)
=
is called (co)homology
d
(
V
)
space of the differential space
(
V
,
d
)
.
2. Let
Z
be the group of integers. A
Z
-graded vector space is a vector space
V
with
a family
{
V
n
⊂
V
,
n
∈ Z}
of vector subspaces such that
V
=⊕
n
V
n
.
Elements of
V
n
are called homogeneous elements of degree
n
.
A linear endo-
morphism
f
is called homogeneous of degree
k
if
f
(
V
n
)
⊂
V
n
+
k
∀
n
.Weset
de
g(
k
Actually given a homogeneous endomorphism
f
it is always pos-
sible to modify the
f
)
=
)
∈
{
}
Z
-gradation of
V
such that
de
g(
f
1
,
−
1
.
3. A graded differential vector space is a
Z
-graded vector space
(
V
n
)
with differ-
(
,
)
such that
de
g(
)
∈ {
,
−
}
ential vector space structure
V
d
d
1
1
. The couple
(
{
V
n
,
∈ Z}
,
)
is called cochain complex.
The (co)homology space
H
n
d
(
V
,
d
)
of a graded differential vector space is
Z
-
graded by the homogeneous subspaces
ker
(
d
:
V
n
→
V
n
+
de
g(
d
)
)
H
n
(
V
,
d
)
=
.
(
V
n
−
de
g(
d
)
)
d
Usually a chain complex
(
V
,
d
)
with
de
g(
d
)
=
1 is called cochain complex; elements
of
ker
)
is called cohomology space.The terms chain complex, cycle, boundary and homology
space are used when
de
g(
(
d
)
are called cocycles and those of
d
(
V
)
are called coboundaries and
H
(
V
,
d
1.
Roughly speaking, homological algebra matches with constructing (co)chain
complexes and with understanding their meaning.
The aim of the following subsection is to construct a so-called KV cochain com-
plex of a locally flat manifold
d
)
=−
. Let us recall that
D
is a torsion free linear
connection whose curvature tensor vanishes identically.
Let
(
M
,
D
)
be the vector space of smooth vector fields in
M
and let
C
∞
(
X (
M
)
M
)
be the
ring of real valued smooth functions defined in
M
.Itisaleft
X
(
M
)
-module under
the action
X
(
f
)
=
df
(
X
).
is a left
C
∞
(
The vector space
X
(
M
)
M
)
-module as well.
We equip the vector space
X (
M
)
with the multiplication defined by
XY
=
D
X
Y
.
The multiplication we just defined satisfies the following requirements. For arbitrary
X
C
∞
(
,
Y
∈
X (
M
)
and
f
∈
M
)
one has
(
fX
)
Y
=
f
(
XY
),
