  We study the $t-V-V'$ model in one dimension at half-filling. It is known
that for large enough $V$ fixed, as $V'$ is varied, the system goes from a
charge-density wave into a Luttinger liquid, then a bond-order, and then a
second charge density wave phase. We find that the Luttinger liquid state is
further split into two, separating parts with distinct values of the many-body
polarization Berry phase. Inside this phase, the variance of the polarization
is infinite in the thermodynamic limit, meaning that even if the polarization
differs, it would not be measurable. However, in the gapped phases on each side
of the Luttinger liquid, the polarization takes a different measurable value,
implying topologically distinction. The key difference is that the large-$V'$
phases are link-inversion symmetric, while the small-$V'$ one is site-inversion
symmetric. We show that large-$V'$ phase can be related to an $S=1$ spin chain,
and exhibits many features of the Haldane phase. The lowest lying states of the
entanglement spectrum display different degeneracies in the two cases, and we
also find string order in the large-$V'$ phase. We also study the system under
open boundary conditions, and suggest that the number of defects is related to
the topology.
