Topology in Molecular Biology

194 D.V. Millionschikov









ind(q 1 )=0

ind(q 2 )=1

ind(q 3 )=1

ind(q 4 )=2

f(q)=z

z 1

z 2

z 3

z 4

q 1

q 2

q 3

q (^4) m 2 (f)=dimH^2 (T^2 ,R)=1
m 1 (f)=dimH^1 (T^2 ,R)=2
m 0 (f)=dimH^0 (T^2 ,R)=1
Fig. 11.1.A height-functionf(q)=zforT^2
Now let us consider the case whenMnis not simply connected and the
two-formFis globally exact onMn(like in the Aharonov–Bohm experiment).
Two solutionsω 1 ,ω 2 of the equation dω=Fijdxi∧dxjthat correspond to
two different actionsS 1 (γ)andS 2 (γ) are determined up to a differential df
by their integrals

∫

γkωiover the basic cyclesγkofH^1 (M

n,Z). These integrals

can be interpreted as the fluxes of the continuation ofF(with possible singu-
larities) to some large manifoldM ̃n. Two variational systemsS 1 (γ)andS 2 (γ)
are quantum-mechanically equivalent if and only if all integrals

∫

γk(ω^1 −ω^2 )
over basic cyclesγkofH 1 (Mn,Z) are integer valued.
The formω 12 =ω 1 −ω 2 is a closed 1-form onMnand it determines a
representationρω 12 of the fundamental groupπ 1 (Mn):

ρω 12 :π 1 (Mn)→C∗,ρω 12 (γ) = exp

{

2 πi

∫

γ

ω 12

}

,γ∈π 1 (Mn).

LetMbe a finite-dimensional (or infinite-dimensional) manifold andS:
M→Ra function (functional) on it.
What are the relations between the set of the stationary points dS=0
(δS= 0) and the topology of the manifoldM?
IfSis a Morse function (generic situation), i.e. d^2 Sis non-degenerate at
critical points, then one can define the Morse index ind(P) of a critical point
PofSas the number of negative squares of the quadratic form d^2 S(P) (if it
is finite in the infinite-dimensional case) (Fig. 11.1) [22].
Under some natural hypotheses the following inequality can be established:

mp(S)≥bp(M) = dimHp(M).

11.4 Semi-Classical Motion of Electron and Critical Points of 1-Form

Points of 1-Form

The semi-classical model of electron motion is an important tool for investi-
gating conductivity in crystals under the action of a magnetic field [18, 19].
At the same time it is one of the most important examples of applications of
topological methods in the modern physics.