  We study a model (i.e., the $t_2-V$ model involving {\em
next-nearest-neighbor hopping} and nearest-neighbor repulsion) in one-dimension
that generically depicts the dominant transport mechanism in cooperative strong
electron-phonon interaction systems. Using analytic and numerical approaches,
hard-core-bosons are shown to typically undergo a striking discontinuous
transition from a superfluid to a supersolid. Topological inequivalence of
rings with even and odd number of sites is manifested through observable
differences (in structure factor peaks) at the transition. Connections are also
identified between the $t_2-V$ model and other topologically interesting
models.
