Authors: Baralić, Đorđe 
Milenković, Lazar
Affiliations: Mathematics 
Mathematical Institute of the Serbian Academy of Sciences and Arts 
Title: Small Covers and Quasitoric Manifolds over Neighborly Polytopes
Journal: Mediterranean Journal of Mathematics
Volume: 19
Issue: 2
First page: 87
Issue Date: 1-Apr-2022
Rank: ~M21
ISSN: 1660-5446
DOI: 10.1007/s00009-022-01989-5
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.
Keywords: neighborly polytopes | quasitoric manifolds | Small covers | the classification problem | the lifting conjecture
Publisher: Springer Link

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