  We develop a non-singular bouncing cosmology using a non-trivial coupling of
general relativity to fermionic fields. The usual Big Bang singularity is
avoided thanks to a negative energy density contribution from the fermions. Our
theory is ghost-free since the fermionic operator that generates the bounce is
equivalent to torsion, which has no kinetic terms. The physical system consists
of standard general relativity plus a topological sector for gravity, and
fermionic matter described by Dirac fields with a non-minimal coupling. We show
that a scale invariant power-spectrum generated in the contracting phase can be
recovered by suitable choices of fermion number density and bare mass, thus
providing a possible alternative to the inflationary scenario.
