TITLE:
Cohomology of Homogeneous Spaces H1/H2
AUTHORS:
Yingling Liu, Yu Wang
KEYWORDS:
Homogeneous Space, Cochain Complex, de Rham Cohomology, Lie Algebra Cohomology, Invariant Forms
JOURNAL NAME:
Journal of Applied Mathematics and Physics,
Vol.14 No.1,
January
22,
2026
ABSTRACT: Let
G
be a compact connected Lie group with Lie algebra
g
,
H
1
a normal subgroup of
G
with Lie algebra
h
1
, and
H
2
⊆
H
1
a normal subgroup with Lie algebra
h
2
. This paper establishes an isomorphism between the de Rham cohomology of the quotient manifold
H
1
/
H
2
and the Lie algebra cohomology of the quotient Lie algebra
h
1
/
h
2
. By constructing explicit isomorphisms of cochain complexes, we prove
H
dR
•
(
H
1
/
H
2
)≅
H
•
(
h
1
/
h
2
,ℝ
)≅
(
(
Λ
•
(
h
1
/
h
2
)
)
*
)
g
,
where
(
(
Λ
•
(
h
1
/
h
2
)
)
*
)
g
denotes the space of
g
-invariant elements. This result transforms the geometric problem of computing the de Rham cohomology of the quotient
H
1
/
H
2
into an algebraic computation on the Lie algebra
h
1
/
h
2
.