  We consider a spatially homogeneous and isotropic system of Dirac particles
coupled to classical gravity. The dust and radiation dominated closed
Friedmann-Robertson-Walker space-times are recovered as limiting cases. We find
a mechanism where quantum oscillations of the Dirac wave functions can prevent
the formation of the big bang or big crunch singularity. Thus before the big
crunch, the collapse of the universe is stopped by quantum effects and reversed
to an expansion, so that the universe opens up entering a new era of classical
behavior.
  Numerical examples of such space-times are given, and the dependence on
various parameters is discussed. Generically, one has a collapse after a finite
number of cycles. By fine-tuning the parameters we construct an example of a
space-time which is time-periodic, thus running through an infinite number of
contraction and expansion cycles.
