  In free fermion systems with given symmetry and dimension, the possible
topological phases are labeled by elements of only three types of Abelian
groups, Z_1, Z_2, or Z. For example non-interacting 1D fermionic
superconducting phases with S_z spin rotation and time-reversal symmetries are
classified by Z. We show that with weak interactions, this classification
reduces to Z_4. Using group cohomology, one can additionally show that there
are only four distinct phases for such 1D superconductors even with strong
interactions. Comparing their projective representations, we find all these
four symmetry protected topological phases can be realized with free fermions.
Further, we show that 1D fermionic superconducting phases with Z_n discrete S_z
spin rotation and time-reversal symmetries are classified by Z_4 when n=even
and Z_2 when n=odd; again, all these strongly interacting topological phases
can be realized by non-interacting fermions. Our approach can be applied to
systems with other symmetries to see which 1D topological phases can be
realized by free fermions.
