  We study one-dimensional, interacting, gapped fermionic systems described by
variants of the Peierls-Hubbard model and characterize their phases via a
topological invariant constructed out of their Green's functions. We
demonstrate that the existence of topologically protected, zero-energy states
at the boundaries of these systems can be tied to the values of their
topological invariant, just like when working with the conventional,
noninteracting topological insulators. We use a combination of analytical
methods and the numerical density matrix renormalization group method to
calculate the values of the topological invariant throughout the phase diagrams
of these systems, thus deducing when topologically protected boundary states
are present. We are also able to study topological states in spin systems
because, deep in the Mott insulating regime, these fermionic systems reduce to
spin chains. In this way, we associate the zero-energy states at the end of an
antiferromagnetic spin-one Heisenberg chain with the topological invariant 2.
