Small Covers and Quasitoric Manifolds over Neighborly Polytopes

We prove that the duals of neighborly simplicial n -polytopes with the number of vertices greater than 2 n 2 + 2 + n 2 - 3 cannot appear as the orbit spaces of a small cover for all n ∈ N . We investigate small covers and quasitoric manifolds over the duals of neighborly simplicial polytopes with 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 C-rigidity for small covers is also studied and we have found a whole new series of ‘exceptional’ polytopes, which are polytopes such that small covers over them are classified up to a homeomorphism by their graded Z 2 -cohomology rings. New examples of manifolds provide the first known examples of quasitoric manifolds in higher dimensions whose orbit polytopes have chromatic numbers χ ( P n ) ≥ 3 n - 5 .