Topology in Molecular Biology

11 One-Forms and Deformed de Rham Complex 201

LetG/Γ be a solvmanifold. One can identify its de Rham complex
Λ∗(G/Γ) with the subcomplex inΛ∗(G)

Λ∗Γinv(G)⊂Λ∗(G)

of left-invariant forms onGwith respect to the action of the latticeΓ.
The sub-complexΛ∗Γinv(G) contains in its turn the subcomplexΛ∗Γinv(G)
of left-invariant forms with respect to the action ofG.
Taking left-invariant vector fieldsX 1 ,...,Xp+1and a left-invariantp-form
ξ∈Λ∗Ginv(G) in the formula (11.12) we have:

dξ(X 1 ,...,Xp+1)=

∑

1 ≤i<j≤p+1

(−1)i+jξ([Xi,Xj],X 1 ,...,Xˆi,...,Xˆj,...,Xp+1).

(11.22)
The Lie algebra of left-invariant vector fields onGis naturally isomorphic to
the tangent Lie algebrag. Hence one can identify the spaceΛpGinv(G) with
the spaceΛp(g∗) of skew-symmetric polylinear functions ong.
The differential d defined by (11.22) provides us with the cochain complex
of the Lie algebrag:

R

d 0 =0

−−−−→g∗

d

−−−−→Λ^2 (g∗)

d

−−−−→Λ^3 (g∗)

d

−−−−→··· (11.23)

The dual of the Lie bracket [,]:Λ^2 (g)→ggives a linear mapping

δ:g∗→Λ^2 (g∗).

Consider a basise 1 ,...,enofgand its dual basise^1 ,...,en. Then we have
the following relation:

dek=−δek=−

∑

i<j

Cijkdei∧dej, (11.24)

where [ei,ej]=

∑

Cijkek. The differential d is completely determined by
(11.24) and the following property:

d(ξ 1 ∧ξ 2 )=dξ 1 ∧ξ 2 +(−1)degξ^1 ξ 1 ∧dξ 2 ,∀ξ 1 ,ξ 2 ∈Λ∗(g∗).

Cohomology of the complex (Λ∗(g∗),δ) is called thecohomology (with trivial
coefficients) of the Lie algebragand is denoted byH∗(g).
Let us consider the inclusion

ψ:Λ∗(g)→Λ∗(G/Γ).

LetG/Γbe a compact solvmanifold, whereGis a completely solvable Lie
group, thenψ:Λ∗(g)→Λ∗(G/Γ) induces the isomorphismψ∗:H∗(g)→
H∗(G/Γ,R) in cohomology (Hattori’s theorem [12], Nomizu’s theorem for
nilmanifolds [8]).