We investigate small covers and quasitoric manifolds over the duals of neighborly simplicial polytopes with a small number of vertices in dimensions 4, 5, 6 and 7. In 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 .Link for slides.
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