  We demonstrate an original development of path-integral quantization in the
case of a multiply connected configuration space of indistinguishable charged
particles on a 2D manifold and exposed to a strong perpendicular magnetic
field. The system occurs to be exceptionally homotopy-rich and the structure of
the homotopy essentially depends on the magnetic field strength resulting in
multi loop trajectories at specific conditions. We have proved, by a
generalization of the Bohr-Sommerfeld quantization rule, that the size of a
magnetic field flux quantum grows for multi loop orbits like
$(2k+1)\frac{h}{c}$ with the number of loops $k$. Utilizing this property for
electrons on the 2D substrate jellium we have derived upon the path integration
a complete FQHE hierarchy in excellent consistence with experiments. The
path-integral has been next developed to a sum over configurations, displaying
various patterns of trajectory homotopies (topological configurations), which
in the nonstationary case of quantum kinetics reproduces some unclear formerly
details in the longitudinal resistivity observed in experiments.
