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Hailun Zheng

Publications and source records attributed to Hailun Zheng.

29 records · Page 2Linked to original sources
arXiv2020

Highly neighborly centrally symmetric spheres

In 1995, Jockusch constructed an infinite family of centrally symmetric $3$-dimensional simplicial spheres that are cs-$2$-neighborly. Here we generalize his construction and show that for all $d\geq 3$ and $n\geq d+1$, there exists a centrally symmetric $d$-dimensional simplicial sphere with $2n$ vertices that is cs-$\lceil d/2\rceil$-neighborly. This result combined with work of Adin and Stanley completely resolves the upper bound problem for centrally symmetric simplicial spheres.

math.CO↗
arXiv2020

Ear Decomposition and Balanced Neighborly Simplicial Manifolds

We find the first non-octahedral balanced 2-neighborly 3-sphere and the balanced 2-neighborly triangulation of the lens space $L(3,1)$. Each construction has 16 vertices. We show that there exists a balanced 3-neighborly non-spherical 5-manifold with 18 vertices. We also show that the rank-selected subcomplexes of a balanced simplicial sphere do not necessarily have an ear decomposition.

math.CO↗
arXiv2020

The upper bound theorem for flag homology 5-manifolds

We prove that among all flag homology $5$-manifolds with $n$ vertices, the join of $3$ circles of as equal length as possible is the unique maximizer of all the face numbers. The same upper bounds on the face numbers hold for $5$-dimensional flag Eulerian normal pseudomanifolds.

math.CO↗
arXiv2019

Centrally symmetric and balanced triangulations of $\mathbb{S}^2\times \mathbb{S}^{d-3}$ with few vertices

A small triangulation of the sphere product can be found in lower dimensions by computer search and is known in few other cases: Klee and Novik constructed a centrally symmetric triangulation of $\mathbb{S}^i\times \mathbb{S}^{d-i-1}$ with $2d+2$ vertices for all $d\geq 3$ and $1\leq i\leq d-2$; they also proposed a balanced triangulation of $\mathbb{S}^1\times \mathbb{S}^{d-2}$ with $3d$ or $3d+2$ vertices. In this paper, we provide another centrally symmetric $(2d+2)$-vertex triangulation of $\mathbb{S}^2\times \mathbb{S}^{d-3}$. We also construct the first balanced triangulation of $\mathbb{S}^2\times \mathbb{S}^{d-3}$ with $4d$ vertices, using a sphere decomposition inspired by handle theory.

math.CO↗
arXiv2018

Face enumeration on flag complexes and flag spheres

We give a survey on the recent results and problems on the face enumeration of flag complexes and flag simplicial spheres, with an emphasis on the characterization of face vectors of flag complexes, several lower-bound type of conjectures including the Charney-Davis conjecture and Gal's conjecture, and the upper bound conjecture for flag spheres and pseudomanifolds.

math.CO↗
arXiv2018

A lower bound theorem for centrally symmetric simplicial polytopes

Stanley proved that for any centrally symmetric simplicial $d$-polytope $P$ with $d\geq 3$, $g_2(P) \geq {d \choose 2}-d$. We provide a characterization of centrally symmetric $d$-polytopes with $d\geq 4$ that satisfy this inequality as equality. This gives a natural generalization of the classical Lower Bound Theorem for simplicial polytopes to the setting of centrally symmetric simplicial polytopes.

math.CO↗
arXiv2017

A characterization of homology manifolds with $g_2\leq 2$

We characterize homology manifolds with $g_2\leq 2$. Specifically, using retriangulations of simplicial complexes, we give a short proof of Nevo and Novinsky's result on the characterization of homology $(d-1)$-spheres with $g_2=1$ for $d\geq 5$ and extend it to the class of normal pseudomanifolds. We proceed to prove that every prime homology manifold with $g_2=2$ is obtained by centrally retriangulating a polytopal sphere with $g_2\leq 1$ along a certain subcomplex. This implies that all homology manifolds with $g_2=2$ are polytopal spheres.

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arXiv2016

Minimal Balanced Triangulations of Sphere Bundles over the Circle

We determine the minimum number of vertices needed to provide balanced triangulations of $\mathbb S^{d-2}$-bundles over $\mathbb S^1$. If $d$ is odd and the bundle is orientable, or $d$ is even and the bundle is non-orientable, the minimum number of vertices is $3d$; otherwise, it is $3d+2$. Similar results apply to all balanced simplicial complexes that triangulate homology manifolds with $β_1\neq 0$ and $β_2=0$, where $β_i$'s are the Betti numbers, computed with coefficients in $\mathbb Q$.

math.CO↗
arXiv2015

The flag upper bound theorem for 3- and 5-manifolds

We prove that among all flag 3-manifolds on $n$ vertices, the join of two circles with $\left\lceil{\frac{n}{2}}\right \rceil$ and $\left\lfloor{\frac{n}{2}}\right \rfloor$ vertices respectively is the unique maximizer of the face numbers. This solves the first case of a conjecture due to Lutz and Nevo. Further, we establish a sharp upper bound on the number of edges of flag 5-manifolds and characterize the cases of equality. We also show that the inequality part of the flag upper bound conjecture continues to hold for all flag 3-dimensional Eulerian complexes and find all maximizers of the face numbers in this class.

math.CO↗