## New families of highly neighborly centrally symmetric spheres

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- by Isabella Novik and Hailun Zheng PDF
- Trans. Amer. Math. Soc.
**375**(2022), 4445-4475 Request permission

## Abstract:

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, $\Delta ^d_n$, that is cs-$\lceil d/2\rceil$-neighborly. Here, several new cs constructions, related to $\Delta ^d_n$, 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 $\Omega (2^n)$ such pairwise non-isomorphic triangulations. It is also shown that for all $k>2$ and a sufficiently large $n$, there are $\Omega (2^n)$ 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 $\Delta ^d_n$, 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 $\Delta ^3_n$ are shellable and an affirmative answer to Murai–Nevo’s question about $2$-stacked shellable balls is given.## References

- C. Carathéodory,
*Über den Variabilitatsbereich det Fourierschen Konstanten von positiven harmonischen Furktionen.*Ren. Circ. Math. Palermo**32**(1911), 193–217. - David Gale,
*Neighborly and cyclic polytopes*, Proc. Sympos. Pure Math., Vol. VII, Amer. Math. Soc., Providence, R.I., 1963, pp. 225–232. MR**0152944** - J. Gouveia, A. Macchia, and A. Wiebe,
*General non-realizability certificates for spheres with linear programming.*arXiv:2109.15247, 2021. - Branko Grünbaum,
*Convex polytopes*, 2nd ed., Graduate Texts in Mathematics, vol. 221, Springer-Verlag, New York, 2003. Prepared and with a preface by Volker Kaibel, Victor Klee and Günter M. Ziegler. MR**1976856**, DOI 10.1007/978-1-4613-0019-9 - J. F. P. Hudson,
*Piecewise linear topology*, W. A. Benjamin, Inc., New York-Amsterdam, 1969. University of Chicago Lecture Notes prepared with the assistance of J. L. Shaneson and J. Lees. MR**0248844** - William Jockusch,
*An infinite family of nearly neighborly centrally symmetric $3$-spheres*, J. Combin. Theory Ser. A**72**(1995), no. 2, 318–321. MR**1357779**, DOI 10.1016/0097-3165(95)90070-5 - Gil Kalai,
*Many triangulated spheres*, Discrete Comput. Geom.**3**(1988), no. 1, 1–14. MR**918176**, DOI 10.1007/BF02187893 - Victor Klee,
*A combinatorial analogue of Poincaré’s duality theorem*, Canadian J. Math.**16**(1964), 517–531. MR**189039**, DOI 10.4153/CJM-1964-053-0 - Nathan Linial and Isabella Novik,
*How neighborly can a centrally symmetric polytope be?*, Discrete Comput. Geom.**36**(2006), no. 2, 273–281. MR**2252105**, DOI 10.1007/s00454-006-1235-1 - Frank Hagen Lutz,
*Triangulated manifolds with few vertices and vertex-transitive group actions*, Berichte aus der Mathematik. [Reports from Mathematics], Verlag Shaker, Aachen, 1999 (English, with German summary). Dissertation, Technischen Universität Berlin, Berlin, 1999. MR**1866007** - F. H. Lutz,
*Vertex-transitive triangulations.*http://page.math.tu-berlin.de/$\sim$lutz/stellar/vertex-transitive-triangulations.html, 1999–2011. - Antonio Macchia and Amy Wiebe,
*Slack ideals in Macaulay$2$*, Mathematical software—ICMS 2020, Lecture Notes in Comput. Sci., vol. 12097, Springer, Cham, [2020] ©2020, pp. 222–231. MR**4139490**, DOI 10.1007/978-3-030-52200-1_{2}2 - P. McMullen,
*The maximum numbers of faces of a convex polytope*, Mathematika**17**(1970), 179–184. MR**283691**, DOI 10.1112/S0025579300002850 - P. McMullen and G. C. Shephard,
*Diagrams for centrally symmetric polytopes*, Mathematika**15**(1968), 123–138. MR**238180**, DOI 10.1112/S0025579300002473 - P. McMullen and D. W. Walkup,
*A generalized lower-bound conjecture for simplicial polytopes*, Mathematika**18**(1971), 264–273. MR**298557**, DOI 10.1112/S0025579300005520 - T. S. Motzkin.
*Comonotone curves and polyhedra*, Bull. Amer. Math. Soc., 63:35, 1957. - Satoshi Murai and Eran Nevo,
*On the generalized lower bound conjecture for polytopes and spheres*, Acta Math.**210**(2013), no. 1, 185–202. MR**3037614**, DOI 10.1007/s11511-013-0093-y - Eran Nevo, Francisco Santos, and Stedman Wilson,
*Many triangulated odd-dimensional spheres*, Math. Ann.**364**(2016), no. 3-4, 737–762. MR**3466849**, DOI 10.1007/s00208-015-1232-x - Isabella Novik and Hailun Zheng,
*Highly neighborly centrally symmetric spheres*, Adv. Math.**370**(2020), 107238, 16. MR**4106648**, DOI 10.1016/j.aim.2020.107238 - Udo Pachner,
*P.L. homeomorphic manifolds are equivalent by elementary shellings*, European J. Combin.**12**(1991), no. 2, 129–145. MR**1095161**, DOI 10.1016/S0195-6698(13)80080-7 - Arnau Padrol,
*Many neighborly polytopes and oriented matroids*, Discrete Comput. Geom.**50**(2013), no. 4, 865–902. MR**3138140**, DOI 10.1007/s00454-013-9544-7 - J. Pfeifle,
*Positive Plücker tree certificates for non-realizability.*Experimental Mathematics (2022), DOI 10.1080/10586458.2021.1994487. - Ido Shemer,
*Neighborly polytopes*, Israel J. Math.**43**(1982), no. 4, 291–314. MR**693351**, DOI 10.1007/BF02761235 - Richard P. Stanley,
*The upper bound conjecture and Cohen-Macaulay rings*, Studies in Appl. Math.**54**(1975), no. 2, 135–142. MR**458437**, DOI 10.1002/sapm1975542135 - Richard P. Stanley,
*The number of faces of a simplicial convex polytope*, Adv. in Math.**35**(1980), no. 3, 236–238. MR**563925**, DOI 10.1016/0001-8708(80)90050-X - Günter M. Ziegler,
*Lectures on polytopes*, Graduate Texts in Mathematics, vol. 152, Springer-Verlag, New York, 1995. MR**1311028**, DOI 10.1007/978-1-4613-8431-1

## Additional Information

**Isabella Novik**- Affiliation: Department of Mathematics, University of Washington, Seattle, Washington 98195-4350
- MR Author ID: 645524
- ORCID: 0000-0003-3695-1337
- Email: novik@uw.edu
**Hailun Zheng**- Affiliation: Department of Mathematical Sciences, University of Copenhagen, Universitesparken 5, 2100 Copenhagen, Denmark
- MR Author ID: 1166475
- Email: hz@math.ku.dk
- Received by editor(s): May 13, 2020
- Received by editor(s) in revised form: January 1, 2022
- Published electronically: March 31, 2022
- Additional Notes: The first author was partially supported by NSF grants DMS-1664865 and DMS-1953815, and by Robert R. & Elaine F. Phelps Professorship in Mathematics.

The second author was partially supported by a postdoctoral fellowship from ERC grant 716424 - CASe. - © Copyright 2022 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**375**(2022), 4445-4475 - MSC (2000): Primary 52B05, 52B15, 57Q15
- DOI: https://doi.org/10.1090/tran/8631
- MathSciNet review: 4419065