  A many-particle Hamiltonian is proposed in order to explain the fractional
quantum Hall effect (FQHE) for fractional filling factors $\nu < 1$. The
solutions of the corresponding Hartree-Fock equations make it possible to
discuss the FQHE from the point of view of the single quasi-particle energy
spectrum. It is shown how the specific couplings in the many-particle
Hamiltonian depend on the magnetic field and the area density of electrons. The
degeneracies of the quasi-particle states are related to the fractional filling
factors $\nu$. It is suggested that the energy gaps obtained in the
quasi-particle energy spectrum are comparable with the experimentally measured
quantities. An explicit calculation for the FQH - conductance is given and its
character as a topological invariant is discussed.

Condensed Matter