  We investigate small covers and quasitoric over the duals of neighborly
simplicial polytopes with small number of vertices in dimensions $4$, $5$, $6$
and $7$. In the most of the considered cases we obtain the complete
classification of small covers. The lifting conjecture in all cases is verified
to be true. The problem of cohomological rigidity for small covers is also
studied and we have found a whole new series of weakly cohomologically rigid
simple polytopes. New examples of manifolds provide the first known examples of
quasitoric manifolds in higher dimensions whose orbit polytopes have chromatic
numbers $\chi (P^n)\geq 3n-5$.
