  The problem of the availability of trajectories for the Feynman path integral
is considered. Forbidden trajectories for single particle integrals are
featured in the case of quantum tunneling across barriers. In the case of
multiparticle systems of indistinguishable identical particles, some limits for
the availability of cyclotron braid trajectories are demonstrated, which leads
to the explanation of statistics and correlation in quantum Hall systems of
interacting 2D electrons. The homotopy-type restrictions for trajectories close
to general-relativity singularities are discussed with indication of quantum
properties of black holes manifesting themselves at quasar luminosity or at
neutron star merger collapses. The related supplementation to conventional
models of accretion disk luminosity in the close vicinity of the event horizon
of a super massive black hole is proposed and compared with observations.
