Topology in Molecular Biology

11 One-Forms and Deformed de Rham Complex 199

with coefficients in the representationρλω:π 1 (M)→C∗ of fundamental
group defined by the formula

ρλω([γ]) = exp

∫

γ

λω, [γ]∈π 1 (M),

We denote corresponding Betti numbers bybp(λ, ω),bp(λ, ω) = dimH∗ρλω
(M,C).
There is another interpretation ofHρ∗λω(M,C: the representationρλω :
π 1 (M)→C∗defines a local system of groupsC∗on the manifoldM. The coho-
mology ofMwith coefficients in this local system coincides withHρ∗λω(M,C).
Now we can assume thatωis a Morse 1-form, i.e. in a neighbourhood of
any pointω=df, wherefis a Morse function. In other wordsωgives a
multi-valued Morse function. The zeros ofωare isolated, and one can define
the index of each zero. The number of zeros ofωof indexpis denoted by
mp(ω).
Following Witten’s scheme Pazhitnov showed in [4] that for sufficiently
large real numbersλ
mp(ω)≥bp(λ, ω).

11.6 Solvmanifolds and Left-Invariant Forms

A solvmanifold (nilmanifold)Mis a compact homogeneous space of the form
G/Γ,whereGis a simply connected solvable (nilpotent) Lie group andΓis
a lattice inG[23].
Let us consider some examples of solvmanifolds (the first two of them are
nilmanifolds):

1. Ann-dimensional torusTn=Rn/Zn.


2. The Heisenberg manifoldM 3 =H 3 /Γ 3 , whereH 3 is the group of all
   matrices of the form ⎛

⎝

1 xz

01 y

001

⎞

⎠, x,y,z∈R,

and a latticeΓ 3 is a subgroup of matrices with integer entriesx, y, z∈Z.

e 1 =

⎛

⎝

010

000

000

⎞

⎠,e 2 =

⎛

⎝

000

001

000

⎞

⎠,e 3 =

⎛

⎝

001

000

000

⎞

⎠,

and the only one non-trivial structure relation: [e 1 ,e 2 ]=e 3. The left invariant
1-forms onH 3
e^1 =dx, e^2 =dy, e^3 =dz−xdy (11.17)

are dual toe 1 ,e 2 ,e 3 and

de^1 =0, de^2 =0, de^3 =d(dz−xdy)=−dx∧dy=−e^1 ∧e^2. (11.18)