Empty Simplices of Polytopes and Graded Betti Numbers

Nagel, Uwe

doi:10.1007/s00454-008-9057-y

Cited by 15 publications

(4 citation statements)

References 22 publications

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“…In Proposition 3.6 of [13], Kalai proved the special case of the following corollary for polytopal spheres. Also, in Corollary 4.8 of [19], Nagel proved the special case of this corollary for homology spheres with the Weak Lefschetz Property (WLP). Conjecturally, all homology spheres have WLP.…”

Section: Bistellar Moves and Shelling Movesmentioning

confidence: 97%

On k-stellated and k-stacked spheres

Bagchi

Datta

2013

Discrete Mathematics

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We introduce the class Σ k (d) of k-stellated (combinatorial) spheres of dimension d (0 ≤ k ≤ d+1) and compare and contrast it with the class S k (d) (0 ≤ k ≤ d) of k-stacked homology d-spheres. We have Σ 1 (d) = S 1 (d), and Σ k (d) ⊆ S k (d) for d ≥ 2k − 1. However, for each k ≥ 2 there are k-stacked spheres which are not k-stellated. For d ≤ 2k − 2, the existence of k-stellated spheres which are not k-stacked remains an open question.We also consider the class W k (d) (and K k (d)) of simplicial complexes all whose vertex-links belong to Σ k (d−1) (respectively, S k (d−1)). Thus,Let K k (d) denote the class of d-dimensional complexes all whose vertex-links are k-stacked balls. We show that for d ≥ 2k + 2, there is a natural bijection M → M from K k (d) onto K k (d + 1) which is the inverse to the boundary map ∂ :Finally, we complement the tightness results of our recent paper [5] by showing that, for any field F, an F-orientable (k + 1)-neighborly member of W k (2k + 1) is F-tight if and only if it is k-stacked.Mathematics Subject Classification (2010): 52B05, 52B22, 52B11, 57Q15.In parallel with these classes of homology spheres, we also consider the classes Σ k (d) and S k (d) of k-shelled d-balls and k-stacked homology d-balls, respectively. We have the filtrationof the class of all shellable d-balls, and the comparable filtrationof the class of all homology d-balls. The standard d-ball B d d+1 is the unique (d + 1)-vertex triangulation of the d-dimensional ball. It may be described as the face complex of the d-dimensional geometric simplex. The standard ball B d d+1 is the unique member of Σ 0 (d) = S 0 (d). We also have the equality Σ 1 (d) = S 1 (d) and for all d ≥ k we have the inclusion Σ k (d) ⊆ S k (d). However, for each k ≥ 2, there are k-stacked balls which are not k-shelled.While a k-stellated d-sphere is defined as a triangulated d-sphere which may be obtained from S d d+2 by a finite sequence of bistellar moves of index < k, a k-shelled d-ball is a triangulated d-ball obtained from B d d+1 by a finite sequence of shelling moves of index < k. A k-stacked homology d-ball is a homology d-ball all whose faces of codimension k + 1 (i.e., dimension d − k − 1) are in its boundary. A k-stacked homology d-sphere is a homology d-sphere which may be represented as the boundary of a k-stacked (d + 1)-ball. The boundary of any k-shelled (d + 1)-ball is a k-stellated d-sphere. Conversely, when d ≥ 2k − 1, any k-stellated d-sphere may be represented as the boundary of a k-shelled (d + 1)-ball. A homology ball is k-shelled if and only if it is k-stacked and shellable. Each k-stacked homology (respectively k-shelled) ball is the antistar of a vertex in a k-stacked homology (respectively k-stellated) sphere. Murai and Nevo [17] proved that, when d ≥ 2k, for any k-stacked homology d-sphere S, there is a unique k-stacked homology (d + 1)-ball S whose boundary is S. The ball S has a natural and intrinsic description in terms of the combinatorics of S. We point out the Murai-Nevo theorem is an immediate consequence o...

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Section: Bistellar Moves and Shelling Movesmentioning

confidence: 97%

On k-stellated and k-stacked spheres

Bagchi

Datta

2013

Discrete Mathematics

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“…It was shown by Kalai [Ka2,Proposition 3.6] and Nagel [Na,Corollary 4.6] that if ∆ is an (r − 1)stacked homology (d − 1)-sphere and r ≤ d 2 then ∆ has no missing k-faces for r ≤ k ≤ d − r (they write statements only for polytopes but this fact for homology spheres follows from Nagel's proof since an (r − 1)-stacked homology (d − 1)-sphere with r ≤ d 2 has the weak Lefschetz property and satisfies h r−1 = h r ). This fact can be generalized as follows.…”

Section: Vanishing Of Missing Facesmentioning

confidence: 94%

On r-stacked triangulated manifolds

Murai,

Nevo

2013

Discrete Mathematics &Amp; Theoretical Computer Science

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The notion of $r$-stackedness for simplicial polytopes was introduced by McMullen and Walkup in 1971 as a generalization of stacked polytopes. In this paper, we define the $r$-stackedness for triangulated homology manifolds and study their basic properties. In addition, we find a new necessary condition for face vectors of triangulated manifolds when all the vertex links are polytopal. Généralisant les polytopes simpliciaux empilés, McMullen et Walkup ont introduit en 1971 la notion de $r$-empilement pour les polytopes simpliciaux. Dans cet article, nous définissons la notion de $r$-empilement pour les variétés homologiques simpliciales et étudions ses propriétés élémentaires. En outre, nous donnons une nouvelle condition pour les $f$-vecteurs des variétés simpliciales lorsque tous les sommets ont un lien polytopal.

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“…In order to identify the facets in the unique (k + 1)-stacked triangulation S(D a ) of D a , we first identify the missing faces of D a , namely its minimal non-faces. Kalai [Kal94] and Nagel [Nag08] showed that if g i (R) 0 for a simplicial j-polytope R, then R has no missing ℓ-faces for i ≤ ℓ ≤ j − i . Combined with the simplicial GLBT, a subset A of d vertices of D a is a facet in S(D a ) iff all its subsets B of size ∂T * ast x (Lex a (C)).…”

Section: No Obvious Cubical Glbtmentioning

confidence: 99%

On the cone of $f$-vectors of cubical polytopes

Adin

Kalmanovich

Nevo

2019

Proc. Amer. Math. Soc.

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What is the minimal closed cone containing all f -vectors of cubical d-polytopes? We construct cubical polytopes showing that this cone, expressed in the cubical g-vector coordinates, contains the nonnegative g-orthant, thus verifying one direction of the Cubical Generalized Lower Bound Conjecture of Babson, Billera and Chan. Our polytopes also show that a natural cubical analogue of the simplicial Generalized Lower Bound Theorem does not hold. R. M. Adin, D. Kalmanovich and E. Nevo On the cubical g-vectorDenote by e i the i -th unit vector in R ⌊d /2⌋ . Stacked cubical polytopes show that the ray spanned by e 1 is in C d , and neighborly cubical polytopes show that the ray spanned by e ⌊d /2⌋ is in C d . Our main result is that all the rays spanned by the vectors e i are in C d . ThusThe conjecture of Adin and Babson-Billera-Chan isAn analogue of Theorem 1 was previously known for the much wider class of PL cubical spheres [BBC97, Theorem 5.7]. Also, Conjecture 2 holds for d ≤ 5, by combining the constructions above with Steve Klee's result [Kle11, Prop.3.7] asserting that g c k (Q) ≥ 0 for any cubical polytope Q of dimension 2k + 1. Sanyal and Ziegler [SZ10] showed how to construct, from any simplicial (d − 2)-polytope P on n − 1 vertices and a total order v 1 < v 2 < . . . < v n−1 on its vertices, a cubical d-polytope Q = Q(P, <) on 2 n vertices; it is the projection of a deformed n-cube in R n onto the last d coordinates. Further, they showed that if P is k-neighborly then the k-skeleton of Q is isomorphic to the k-skeleton of the n-cube. We apply their construction to the case where P n is the k-neighborly k-stacked (d − 2)-polytope on n − 1 vertices constructed by McMullen and Walkup [MW71], with 1 ≤ k ≤ ⌊ d −2 2 ⌋, and with a suitable total order <. Analyzing the cubical g-vectors of the resulting polytopes Q(k, d, n) = Q(P n , <), as n tends to infinity, gives Theorem 1. See Theorem 13 and Corollary 14 for the exact values and asymptotic behavior of the cubical g-vectors.The generalized lower bound theorem for simplicial polytopes (GLBT), conjectured by McMullen and Walkup [MW71] and proved by Murai and Nevo [MN13], asserts that for 1 ≤ k < ⌊ d 2 ⌋, a simplicial dpolytope P is k-stacked if and only if g k+1 (P ) = 0. The polytopes Q(k, d, n) demonstrate that the natural cubical analogue of the GLBT does not hold:Theorem 3. For any k ≥ 1 and n ≥ d ≥ 2k + 4, we have g c k+2 (Q(k, d, n)) = 0, and Q(k, d, n) is not cubical (k + 1)-stacked.The paper is organized as follows. Basic definitions and notation are given in section 2. In section 3 we define our variant of the McMullen-Walkup polytopes. A sketch of the Sanyal-Ziegler construction is given in section 4. In section 5 we construct the polytopes Q(k, d, n) mentioned above and analyze their cubical g-vector. In section 6 we prove Theorem 3, showing that Q(k, d, n) is not (k + 1)-stacked. The final section 7 concludes with remarks and open questions.

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Empty Simplices of Polytopes and Graded Betti Numbers

Cited by 15 publications

References 22 publications

On k-stellated and k-stacked spheres

On k-stellated and k-stacked spheres

On r-stacked triangulated manifolds

On the cone of $f$-vectors of cubical polytopes

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