Small Covers and Quasitoric Manifolds over Neighborly Polytopes

Djordje Baralić*, Lazar Milenković

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review


We prove that the duals of neighborly simplicial n-polytopes with the number of vertices greater than 2⌈n2⌉+2+[n2]-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 Z2-cohomology rings. New examples of manifolds provide the first known examples of quasitoric manifolds in higher dimensions whose orbit polytopes have chromatic numbers χ(Pn) ≥ 3 n- 5.

Original languageEnglish
Article number87
JournalMediterranean Journal of Mathematics
Issue number2
StatePublished - Apr 2022


  • Small covers
  • neighborly polytopes
  • quasitoric manifolds
  • the classification problem
  • the lifting conjecture


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