TY - JOUR
T1 - Small Covers and Quasitoric Manifolds over Neighborly Polytopes
AU - Baralić, Djordje
AU - Milenković, Lazar
N1 - Publisher Copyright:
© 2022, The Author(s), under exclusive licence to Springer Nature Switzerland AG.
PY - 2022/4
Y1 - 2022/4
N2 - 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.
AB - 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.
KW - Small covers
KW - neighborly polytopes
KW - quasitoric manifolds
KW - the classification problem
KW - the lifting conjecture
UR - http://www.scopus.com/inward/record.url?scp=85126732690&partnerID=8YFLogxK
U2 - 10.1007/s00009-022-01989-5
DO - 10.1007/s00009-022-01989-5
M3 - מאמר
AN - SCOPUS:85126732690
VL - 19
JO - Mediterranean Journal of Mathematics
JF - Mediterranean Journal of Mathematics
SN - 1660-5446
IS - 2
M1 - 87
ER -