New families of highly neighborly centrally symmetric spheres
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Jockusch [J. Combin. Theory Ser. A 72 (1995), pp. 318-321] constructed an infinite family of centrally symmetric (cs, for short) triangulations of 3-spheres that are cs-2-neighborly. Recently, Novik and Zheng [Adv. Math. 370 (2020), 16 pp.] extended Jockusch's construction: for all d and n > d, they constructed a cs triangulation of a d-sphere with 2n vertices, Δdn, that is cs-⌈d/2⌉-neighborly. Here, several new cs constructions, related to Δdn, are provided. It is shown that for all k > 2 and a sufficiently large n, there is another cs triangulation of a (2k − 1)-sphere with 2n vertices that is cs-k-neighborly, while for k = 2 there are Ω(2n) such pairwise non-isomorphic triangulations. It is also shown that for all k > 2 and a sufficiently large n, there are Ω(2n) pairwise non-isomorphic cs triangulations of a (2k − 1)-sphere with 2n vertices that are cs-(k − 1)-neighborly. The constructions are based on studying facets of Δdn, and, in particular, on some necessary and some sufficient conditions similar in spirit to Gale's evenness condition. Along the way, it is proved that Jockusch's spheres Δ3n are shellable and an affirmative answer to Murai-Nevo's question about 2-stacked shellable balls is given.
Original language | English |
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Journal | Transactions of the American Mathematical Society |
Volume | 375 |
Issue number | 6 |
Pages (from-to) | 4445-4475 |
ISSN | 0002-9947 |
DOIs | |
Publication status | Published - 2022 |
Bibliographical note
Publisher Copyright:
© 2022 American Mathematical Society
Links
- https://arxiv.org/pdf/2005.01155.pdf
Accepted author manuscript
- https://www.ams.org/journals/tran/2022-375-06/S0002-9947-2022-08631-1/home.html
Final published version
ID: 310503089