Frank H. Lutz
We construct non-constructible simplicial $d$-spheres with $d+10$ vertices and non-constructible, non-realizable simplicial $d$-balls with $d+9$ vertices for $d\geq 3$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
We develop a tighter implementation of basic PL topology, which keeps track of some combinatorial structure beyond PL homeomorphism type. With this technique we clarify some aspects of PL transversality and give combinatorial proofs of a number of known results. New results include a combinatorial characterization of collapsible polyhedra in terms of constructible posets (generalizing face posets of constructible simplicial complexes in the sense of Hochster). The relevant constructible posets are also characterized in terms of Reading's zipping of posets, which is a variation of edge contract…
Durhuus and Jonsson (1995) introduced the class of "locally constructible" (LC) 3-spheres and showed that there are only exponentially-many combinatorial types of simplicial LC 3-spheres. Such upper bounds are crucial for the convergence of models for 3D quantum gravity. We characterize the LC property for d-spheres ("the sphere minus a facet collapses to a (d-2)-complex") and for d-balls. In particular, we link it to the classical notions of collapsibility, shellability and constructibility, and obtain hierarchies of such properties for simplicial balls and spheres. The main corollaries from …
In this paper, we consider constructibility of simplicial 3-balls. In many cases, examining 1-dimensional subcomplexes of a simplicial 3-ball is efficient to solve the decision problem whether the simplicial 3-ball is constructible or not. From the point of view, we consider the case where a simplicial 3-ball has spanning edges and present a sufficient condition for nonconstructibility.
Results of R. Stanley and M. Masuda completely characterize the h-vectors of simplicial posets whose order complexes are spheres. In this paper we examine the corresponding question in the case where the order complex is a ball. Using the face rings of these posets, we develop a series of new conditions on their h-vectors. We also present new methods for constructing poset balls with specific h-vectors. These results allow us to give a complete characterization of the h-vectors of simplicial poset balls up through dimension six.
A simplicial cell ball is a simplicial poset whose geometric realization is homeomorphic to a ball. Recently, Samuel Kolins gave a series of necessary conditions and sufficient conditions on $h$-vectors of simplicial cell balls, and characterized them up to dimension 6. In this paper, we extend Kolins' results. We characterize all possible $h$-vectors of simplicial cell balls in arbitrary dimension.
Top-heavy Lefschetz elements for regular unimodular embedded lattice simplicial balls with induced boundary
该研究证明带诱导边界的正则幺模嵌入格单纯球的Stanley-Reisner环的坐标阿廷约化存在显式重顶莱夫谢茨元。
In 1967, Chillingworth proved that all convex simplicial 3-balls are collapsible. Using the classical notion of tightness, we generalize this to arbitrary manifolds: We show that all tight simplicial 3-manifolds admit some perfect discrete Morse function. We also strengthen Chillingworth's theorem by proving that all convex simplicial 3-balls are non-evasive. In contrast, we show that many non-evasive 3-balls are not convex.
The Genocchi numbers are the coefficients of the generating function 2t/(e^t+1). In this note we will give an equation for simplicial balls which involves this numbers. It relates the number of faces in the interior of the ball with the number of faces in the boundary of the ball. This is a variation of similar equations given in [McMullen 2004] and [Herrmann and Joswig 2006].
We introduce the non-pure versions of simplicial balls and spheres with minimum number of vertices. These are a special type of non-homogeneous balls and spheres (NH-balls and NH-spheres) satisfying a minimality condition on the number of maximal simplices. The main result is that minimal NH-balls and NH-spheres are precisely the simplicial complexes whose iterated Alexander duals converge respectively to a simplex or the boundary of a simplex.
We construct the first explicit example of a simplicial 3-ball B_{15,66} that is not collapsible. It has only 15 vertices. We exhibit a second 3-ball B_{12,38} with 12 vertices that is collapsible and evasive, but not shellable. Finally, we present the first explicit triangulation of a 3-sphere S_{18, 125} (with only 18 vertices) that is not locally constructible. All these examples are based on knotted subcomplexes with only three edges; the knots are the trefoil, the double trefoil, and the triple trefoil, respectively. The more complicated the knot is, the more distant the triangulation is …
Mogami introduced in 1995 a large class of triangulated 3-dimensional pseudomanifolds, henceforth called "Mogami pseudomanifolds". He proved an exponential bound for the size of this class in terms of the number of tetrahedra. The question of whether all 3-balls are Mogami has remained open since, a positive answer would imply a much-desired exponential upper bound for the total number of 3-balls (and 3-spheres) with N tetrahedra. Here we provide a negative answer: many 3-balls are not Mogami. On the way to this result, we characterize the Mogami property in terms of nuclei, in the sense of Co…
We define and study the notion of a minimal Cohen-Macaulay simplicial complex. We prove that any Cohen-Macaulay complex is shelled over a minimal one in our sense, and we give sufficient conditions for a complex to be minimal Cohen-Macaulay. We show that many interesting examples of Cohen-Macaulay complexes in combinatorics are minimal, including Rudin's ball, Ziegler's ball, the dunce hat, and recently discovered non-partitionable Cohen-Macaulay complexes. We further provide various ways to construct such complexes.
Shellability of relative squeezed balls and spheres
研究相对挤压球及其边界复形的可剥壳性问题，通过证明其可剥壳，给出明确剥壳顺序并刻画限制面，为这类新复形建立强大组合结构。
We obtain a geometric realization of a minimal 8-vertex triangulation of the dunce hat in Euclidean 3-space. We show there is a simplicial 3-ball with 8 vertices that is collapsible, but also collapses onto the dunce hat, which is not collapsible. This 3-ball is as small as possible, because all triangulated 3-balls with fewer vertices are extendably collapsible. As we will see, the Alexander dual of the dunce hat is collapsible.
Strong collapsibility of the arc complexes of orientable and non-orientable crowns
本文证明了内部带标记点的多边形的弧复形是强可缩组合球，而边界带有限标记点的Möbius带的弧复形是单纯可缩但非强可缩的组合球。
The affine Tverberg theorem revisited
本文证明Bárány--Kalai--Tverberg猜想，将仿射Tverberg定理从单纯形推广至任意凸多胞体与单纯球，确保存在r个不相交面使仿射像有公共交点。
Spheres and balls as independence complexes
该研究探讨加须、嫁接构造的单纯复形的独立复形，证明其为广义Bier球（同胚于球或球面），并分类了极优覆盖图独立复形的同胚类型。
In 1988 Kalai construct a large class of simplicial spheres, called squeezed spheres, and in 1991 presented a conjectured about generic initial ideals of Stanley--Reisner ideals of squeezed spheres. In the present paper this conjecture will be proved. In order to prove Kalai's conjecture, based on the fact that every squeezed $(d-1)$-sphere is the boundary of a certain $d$-ball, called a squeezed $d$-ball, generic initial ideals of Stanley--Reisner ideals of squeezed balls will be determined. In addition, generic initial ideals of exterior face ideals of squeezed balls are determined. On the o…
We present extremal constructions connected with the property of simplicial collapsibility. (1) For each $d \ge 2$, there are collapsible (and shellable) simplicial $d$-complexes with only one free face. Also, there are non-evasive $d$-complexes with only two free faces. (Both results are optimal in all dimensions.) (2) Optimal discrete Morse vectors need not be unique. We explicitly construct a contractible, but non-collapsible $3$-dimensional simplicial complex with face vector $f=(106,596,1064,573)$ that admits two distinct optimal discrete Morse vectors, $(1,1,1,0)$ and $(1,0,1,1)$. Indeed…
Face Numbers of Shellable CW Balls and Spheres
本研究将多面体边界复形的面数不等式推广至可剥壳强正则CW球与球面，给出带边界项的面数下界及等号条件，还证明了兼具CL-可剥壳与对偶CL-可剥壳性质的CW球面的全维面数下界。
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。